Class 9th Mathematics AP Board Solution
Exercise 1.1- (a) Write any three rational numbers (b) Explain rational number is in your own…
- Give one example each to the following statements. i. A number which is rational…
- Find five rational numbers between 1 and 2.
- Insert three rational numbers between 2/3 3/5 .
- Represent 8/5 -8/5 on a numberline
- Express the following rational numbers as decimals numbers I. i. 242/1000 ii.…
- Express each of the following decimals in p/q form where q ≠ 0 and p, q are…
- Express each of the following decimal number in the p/q form i) 0.5 ii) 3. bar 8…
- Without actually dividing find which of the following are terminating decimals.…
Exercise 1.2- Classify the following numbers as rational or irrational. i. root 27 ii. root…
- Explain with an example how irrational numbers differ from rational numbers?…
- Find an irrational number between 5/7 7/9 . How many more there may be?…
- Find two irrational numbers between 0.7 and 0.77
- Find the value of √5 upto 3 decimal places.
- Find the value of √7 up to six decimal places by long division method.…
- Locate √10 on number line.
- Find at least two irrational numbers between 2 and 3.
- State whether the following statements are true or false. Justify your answers.…
Exercise 1.3- Visualise 2.874 on the number line, using successive magnification.…
- Visualilse 5. bar 28 on the number line, upto 3 decimal places.
Exercise 1.4- Simplify the following expressions. i. (5 + root 7) (2 + root 5) ii. (5 + root…
- Classify the following numbers as rational or irrational. i. 5 - root 3 ii. root…
- In the following equations, find whether variables x, y, z etc. represent…
- The ratio of circumference to the diameter of a circle c/d is represented by p.…
- Rationalise the denominators of the following: i. 1/3 + root 2 ii. 1/root 7 -…
- Simplify each of the following by rationalising the denominator: i. 6-4 root…
- Find the value of root 10 - root 5/2 root 2 upto three decimal places. (take √2…
- Find: i. 64^1/6 ii. 32^1/5 iii. 625^1/4 iv. 16^3/2 v. 243^2/5 vi. (46656)^-1/6…
- Simplify : root [4]81-8 cube root 343+15 root [5]32 + root 225
- If ‘a’ and ‘b’ are rational numbers, find the value of a and b in each of the…
- (a) Write any three rational numbers (b) Explain rational number is in your own…
- Give one example each to the following statements. i. A number which is rational…
- Find five rational numbers between 1 and 2.
- Insert three rational numbers between 2/3 3/5 .
- Represent 8/5 -8/5 on a numberline
- Express the following rational numbers as decimals numbers I. i. 242/1000 ii.…
- Express each of the following decimals in p/q form where q ≠ 0 and p, q are…
- Express each of the following decimal number in the p/q form i) 0.5 ii) 3. bar 8…
- Without actually dividing find which of the following are terminating decimals.…
- Classify the following numbers as rational or irrational. i. root 27 ii. root…
- Explain with an example how irrational numbers differ from rational numbers?…
- Find an irrational number between 5/7 7/9 . How many more there may be?…
- Find two irrational numbers between 0.7 and 0.77
- Find the value of √5 upto 3 decimal places.
- Find the value of √7 up to six decimal places by long division method.…
- Locate √10 on number line.
- Find at least two irrational numbers between 2 and 3.
- State whether the following statements are true or false. Justify your answers.…
- Visualise 2.874 on the number line, using successive magnification.…
- Visualilse 5. bar 28 on the number line, upto 3 decimal places.
- Simplify the following expressions. i. (5 + root 7) (2 + root 5) ii. (5 + root…
- Classify the following numbers as rational or irrational. i. 5 - root 3 ii. root…
- In the following equations, find whether variables x, y, z etc. represent…
- The ratio of circumference to the diameter of a circle c/d is represented by p.…
- Rationalise the denominators of the following: i. 1/3 + root 2 ii. 1/root 7 -…
- Simplify each of the following by rationalising the denominator: i. 6-4 root…
- Find the value of root 10 - root 5/2 root 2 upto three decimal places. (take √2…
- Find: i. 64^1/6 ii. 32^1/5 iii. 625^1/4 iv. 16^3/2 v. 243^2/5 vi. (46656)^-1/6…
- Simplify : root [4]81-8 cube root 343+15 root [5]32 + root 225
- If ‘a’ and ‘b’ are rational numbers, find the value of a and b in each of the…
Exercise 1.1
Question 1.(a) Write any three rational numbers
(b) Explain rational number is in your own words.
Answer:(a). A rational number is any number that can be expressed as a quotient or fraction p/q of two integers, where p is a numerator and q is a denominator, provided that p is non-zero i.e., p ≠ 0.
Any three rational numbers can be 1, -6.6, 4/5.
(this is because 1 can be written as 1/1, -6.6 can be written as -66/10 or -33/5 & 4/5 is already in p/q form)
(b). Rational number can be defined easily, as basically any sort of number that can possibly be written in a p/q form, numerator is p and denominator is q (q is non-zero).
Since, a rational number is merely an integer or fraction, we can say that we can find infinite rational numbers between two distinct numbers. It might be difficult to obtain rational numbers minutely from a number line graph and hence, formulas are used to find rational numbers.
Its range is from negative numbers in an axis to positive numbers, including zero. Infact all whole numbers, all positive and negative numbers are rational numbers.
Some examples:
1/2 is a rational number (p/q form, where q ≠ 0)
0.50 is a rational number (1/2)
1 is a rational number (1/1)
2.12 is a rational number (212/100)
-6.5 is a rational number (-13/2)
Question 2.Give one example each to the following statements.
i. A number which is rational but not an integer
ii. A whole number which is not a natural number
iii. An integer which is not a whole number
iv. A number which is natural number, whole number, integer and rational number.
v. A number which is an integer but not a natural number.
Answer:(i). Integer is any number that can be written without a fractional component. So, write a rational number in which numerator and denominator doesn’t have any common factor.
For example: 99/98, 2/3, 57/2 etc…
(ii). Whole numbers are all positive natural numbers including zero.
Natural numbers are the set of positive integers from 1 to infinity, excluding fractional and decimal parts. Basically, they are whole numbers without 0.
This clearly implies that 0 is the only whole number which is not a natural number.
(iii). Integers are set of positive as well as negative numbers, that can be written without a fractional component. While whole numbers are natural numbers, including 0.
Thus, all negative non-fractional numbers are integers but not whole numbers.
For example: -1, -2, -3, etc…
(iv). Natural numbers are counting numbers (positive, non-fractional and non-decimal). (1, 2, 3, 4, …)
Whole numbers are all positive natural numbers, including 0. (0, 1, 2, 3, 4, …)
Integer is any number that can be written without a fractional component. (…, -4, -3, -2, -1, 0, 1, 2, 3, …)
And rational numbers are numbers that can be written in the form p/q, where p is a numerator and q ≠ 0 is a denominator. (1, 2/1, 3/2, 55/3, …)
Drawing out a common number from these set, we can say
1, 2, 3, … are numbers which are natural, whole, integer and rational numbers.
(v). Natural numbers are positive counting numbers, excluding fractions and decimals.
Integers are basically any numbers that can be written without fractional component.
So, numbers which are integers but not natural numbers are:
-3, -2, -44, -1, …
Question 3.Find five rational numbers between 1 and 2.
Answer:We can represent a rational number between two numbers a and b as
(a + b)/2
Now, put a = 1 and b = 2.
Then,
Rational number between 1 and 2 ![](data:image/png;base64,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)
Rational number between 1 and 3/2 ![](data:image/png;base64,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)
Rational number between 1 and 5/4 = ![](data:image/png;base64,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)
Rational number between 5/4 and 3/2 ![](data:image/png;base64,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)
Rational number between 3/2 and 2 ![](data:image/png;base64,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)
Hence, five rational numbers between 1 and 2 are 3/2, 5/4, 9/8, 11/8 and 7/4.
Question 4.
Answer:We can represent a rational number between two rational numbers a and b as
(a + b)/2
Now, put a = 2/3 and b = 3/5.
Then,
A rational number between 2/3 and 3/5 = ![](data:image/png;base64,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)
A rational number between 2/3 and 19/30 ![](data:image/png;base64,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)
A rational number between 19/30 and 3/5 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL0AAAApCAMAAABX/thXAAAAAXNSR0IArs4c6QAAALFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmaQZpDbZrbbZrb/kDoAkDo6kDqQkGY6kLbbkLb/kNv/tmYAtmY6tmZmtpC2ttv/tv//25A625Bm27Zm27aQ29u229vb2/+22////7Zm/9uQ/9u2/9vb//+2///bXryROQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADaUlEQVRYR+1Y63baMAxOusGadVtbVtqwC03HtrBBSOlIMvv9H2y6OBATQ2yaFQ4H/eCQ5LP9SZZkyZ53JPKk6/F3sUkvBdwMeEGDJL5/g8uNh55ML6bwbxb4nalXfML/BhkP5cgHBANi379/QbbrSyXn9Ca58eTv+R0Qyi8mcgwv80uj9QEo7iYivFcAGe2fPVMVyD4Dw89QJeBZF6XTI+IJcAjsY2JK7MEvOiNkT0/rQkDR96+BPcP3b3tFNFd8xeAnso7rPkFA8XkhI9QCAQfAPuui40AE+mB0+evDhGOBQ6IqBBQhRq0CHAB7A08gZ4hbHYiAA2DPbr8uOSVQLTJ1IAL2z34DA3FL7l/x7TUgAo6fPZ0f4Imhf/a1vsXqawHRhJlsq8ze0RQyghDFGEVPZvvhi6XgHGx7SI8kCKkCFcDC9rOrkDJwf+jlgcpjq2HqK2axom/04JU+8uG7l57Bhn+DgHsAdiv2NaXb8xwK9uwVhFEZOFWl6avog15xPcvVo/Et+3PWgx9mb9TZxF4DWvs98cth2SJS/GrsZdRbFKhBgxSjIccimp6T+v/OOZxooSC8CtVZvXTGkoKHft9r8ns4k14z+4ynpJLGuGN8hmmyBOY/cAYAWPh9Zf76dparixCK3HJnttn/kUKHTc/EKbNz4QsPVCHjp7o/8RGAhzJOgQAb9jJ+w3YoyPQolWH8NQ8s2KPyHPhsemZPmqjCF8KH40Cpp9mB3405bujBgj14DDURMlK7rrEvv6aBTcYMOOmW3NgXlCpY+KJ5eUNMBT4BwfYdIE4AC/aa/u0+MHs2PhW+S/bKwvpySmmJuY0Ah8Ae7agKXzhUJOYgY3NCBgcXQ/YM2DN71dcWX/6owjcLcAtSw5lO2wBAOKcvpwzYc1/bqh9iMmKB4/x5YjWVVtzs+NBA02VWt6meZ58jG2213XY6tziV3YIn1MkCR2KBORbdbeuSDrB/hZLW1OU2LeYwWNxyH9mmyPFHKpeT8wV2W27iPNh9ie2EVIODfeyGjnTLeOfBG6o1N5ut0DK6HrwHz6Euy6YLry7kPDht2e3zoPc0hw5wJ/aug5OWyTPrpLvYyXMcB+OlSdZ8TeHgRnjzQd133HWPWrfBfI/WKvvlzQfco5/x5YaDlNcmOw12WOcEbbLAP8jKfcAig+Z3AAAAAElFTkSuQmCC)
Hence, three rational numbers between 2/3 and 3/5 are 19/30, 39/60 and 37/60.
Question 5.Represent
on a numberline
Answer:8/5 = 1.6 and -8/5 = -1.6
Thus, 1.6 lies between 1 and 2; -1.6 lies between -2 and -1.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAsMAAABgCAIAAAB3+eh8AAAAAXNSR0IArs4c6QAAFQNJREFUeF7tnXusVdWdxwtYBK68b5WpPFQwNMUWUqqIA8beEpVmSomxKRrJ0P5Bi5OJmrEyiTbi1DbxFXQeUmgympronZQ4Vu2ghGGo1yDaseHWzrQMIE8R7QXu9QLWSYT5dlaysrr3Puesvc9a556z7+f8Qc49d63f77c+v8f+7bU25w45e/bsJ3hBAAIQgAAEIACBQgSGFprFJAhAAAIQgAAEIPBHAnQSxAEEIAABCEAAAsUJ0EkUZ8dMCEAAAhCAAAToJIgBCEAAAhCAAASKE6CTKM6OmRCAAAQgAAEI0EkQAxCAAAQgAAEIFCdAJ1GcHTMhAAEIQAACEKCTIAYgAAEIQAACEChOgE6iODtmQgACEIAABCBQtk5izZo1OBUCEIAABMpKgCLfhJ4dUrJvyx4ypGwrasKgwSQIQAACA0WAIj9Q5KvoLdueRBMixiQIQAACEIBAiQnQSZTYuSwNAhCAAAQgEJ0AnUR0xCiAAAQgAAEIlJgAnUSJncvSIAABCEAAAtEJ0ElER4wCCEAAAhCAQIkJ0EmU2LksDQIQgAAEIBCdAJ1EdMQogAAEIAABCJSYAJ1EiZ3L0iAAAQhAAALRCdBJREeMAghAAAIQgECJCdBJlNi5LA0CEIAABCAQnQCdRHTEKIAABCAAAQiUmACdRImdy9IgAAEIQAAC0QnQSURHjAIIQAACEIBAiQnQSZTYuSwNAhCAAAQgEJ0AnUR0xCiAAAQgAAEIlJgAnUSJncvSIAABCEAAAtEJ0ElER4wCCEAAAhCAQIkJ0EmU2LksDQIQgAAEIBCdAJ1EdMQogAAEIAABCJSYAJ1EiZ3L0iAAAQhAAALRCdBJREeMAghAAAIQgECJCdBJlNi5LA0CEIAABCAQnQCdRHTEKIAABCAAAQiUmACdRImdy9IgAAEIQAAC0QnQSURHjAIIQAACEIBAiQnQSZTYuSwNAhCAAAQgEJ0AnUR0xCiAAAQgAAEIlJgAnUSJncvSIAABCEAAAtEJ0ElER4wCCEAAAhCAQIkJDDl79myrL2/nzp3PPfecWcV999137733mvdLly6dM2dOq68O+yEAAQgMcgIU+SYPgDJ0Er29vePHj0+AHjdu3L59+/RvkzsA8yAAAQhAoDoBinyTR0gZTjfULth9CIv7tttuo41o8uDDPAhAAAI+BCjyPpQGcEwZ9iSEL9GxsiExgCGFaghAAALBCVDkgyMNKLAMexLCkehY2ZAIGCKIggAEIDDgBCjyA+6CKgaUZE/C3ZZgQ6KZAw7bIAABCBQjYLclKPLFAMabVZI9CXdbgg2JeOGCZAgEJKD/aRVQWlpUVPlRhWstUeW3qHC7LRGvyEcl09JurZ6qA7An4f5HzcJ1pP/URzu6D77WfeBoT/+xvtM9J04d7zs9fsyIdw/vWzj/i5MnjZ8/e9qVs6eObju3sIrMiUGMr2RSVOEmiNOPpobiE1V4SxsflUxU4bGxDx069MyZM6EiMC0nqvyowrWWqPJbS7hb7d8/3v/b/9k/fOSYCWNHtY9vmzh21KT20QGrfVQyLe3Wpusk6nTVs1t+s6lrV/euI4qe+XOmTZk0tn1c28RxoyaMbTved2raJZ/ZvLXr0NG+13YeUJ8xe+anFy+cecOiy0JVqzqNr25GVOGtHsRR4SC8UmS2LhkCvkq1aRW3Zlb7z8+69MOTvar2x3pP9/SeClvto5Jp9ZisElE19iR0LrV///6w3+9U2FWKqqde+NWF549Ztni22ohhwzKOZlzhH398Rs1E56bud97/YPlXvxCknyhsvE8rE1V4qwdxVDgIp5PwyVB3TNSYaelsDUKmSrXPlB+q2gcxvgQ93LZt26655hr/pKjYSaiHePTRRx977DGdSK1Zs8ZfYs2RBVy1+0DP93+0Zcx5I5YvmTvvc1Py+un1tw499fybH5z8w/dWLbp0antNC/PKr0cgtcmTXoGw8ZRM1Sbg/UPFjowakIM5JmtW++rk66z2uNVEuC76Ojm9/fbb165d65MdGbf16iEk5eKLL5YgvfeREnXMi7/47S1/2/mVqz/zj3cvrd5GVDJDszRXEm5Z3fnCtv+Oai3CIQABCECgGAGqfTFukWZpN2HIkCF33HFHTfl/0kk0Ww8h69f/9PV/fvaX6++9YdniOTUXU32AJEjOE//6n5JZpyimQwACEIBAWAJU+7A8Q0nz6if0F7zM68SJE8329dLtlyyYuWj1OSNGhyIiOZImmZIcUCaiIAABCECgHgJU+3roNWau+RoP2zO4bz6R+DR9KKL/N5g5s/CH2i3xmfvSq7u+9M0f7T7we5/BdoyP8D0HeyRZ8nNJNoN95BcQ2wDhGF/FL63r1qiWEzMDFTMtTb5ATOaq9rnk5632uYQXKPVR5QcUnvi+APNNHtpuqLTk5HMSesJCQz0fsojXB/X2f/iD9f/+0N/8xYz6HpDMtHD6lImSLPnSEm8JSIYABCAAgZoEqPY1EQ3gANNDaCtCT09WObXI/o7LAe8n1v3LjiUdn50768JIBCV5ScesdZ07IslHLAQgAAEI+BCg2vtQavwYzx7CGFbt27JNP7F06dIGr6F717tbtu9e9Y35UfWu+saVW3bs1jdcRdWCcAhAAAIQqESAat+csaHrfs19CNfy2n93I+zXUvlQ27j51yu/Pq9t5HCfwYXHSP7KG+dt3PxWYQlMhAAEIACBeghQ7euhF2+urvu5/gdG7U4inq2Zks+cObv19b0d86Y3QK+0SJc0NkAXKiAAAQhAwCVAtS9NPDRdJ/Efv9z7+Zl/9qkJ5zUAsbRIlzQ2QBcqIAABCEDAJUC1L008NF0n0fXmvo55MxrGV7qksWHqUAQBCEAAAoYA1b40kdB0ncR7PSf1N7oaxle6pLFh6lAEAQhAAAKGANW+NJHQfJ3Esf4LJjbiaMO4ULreO9ZfGneyEAi0EIHEt98Etzyq/KjChSKq/CYRrtpboNoXNt6n2hcW7hm9seV7mhF82LCwf+fT074qf670H57e/u2vzzt3+DmeotLDcv0t1E+eM2zDT9/41g2X+6vLJd9frBkZVXhs+Rhfyd1RyUQVHjVmWtfyqFgaUAqikvcXXqza+8tP5KNPtS8s3LPUt7r8isss8H2fxaasWrWqq6ur5tw/v+Xx0x/+b81hNQdMnz695hgN0J8al0afkXbMddddt3fv3lxTPAc/8MADzzzzjOfgZh7mCb/AElodfjwyBWD6T/FMXn+BZqSiXTGfd1au8bGBN2AJudZbYHC8nDLGqFpKRaZhdVb7vM7NW+0bU5B1WVR+FXBc9SmRcraS0vCnG6+++qq+/Vuv66+/3u1f3n777QULFuhf81u9ZszIeLLy/AltR/McN3R2dqbVyYaVK1dKu/2txtx6663pfur94yelMbPPknnWVLsW2X/J/7/sMtMr9WlOrWGSY8dv3bp12bJl+tHq1RsfabnGiIOrNNfczMES6LL1h19F9YMPPpgOkiDw3ZBwDQgOXxwSISfV999/v5Tqc+tifVinC6w08alTlJmeTkzP5K2k3U0W18gnn3zyrrvuMhotkGKrUIYaCcFDMVHEpCLSEhL0tKJiKHzSykoOklNGo5tZomSLzMaNG++55x4NsD6yv/Ws9plF3maTrRUSq/dVll+l2tvwC1sTZKRrkjXVrcBPP/30nXfemcmn0lrSsZH+pM6czVtJwncSK1asMLfsutzaKqk3+twYZ/vTPXv2pM29YOLoXI9A3nTTTUadpFn3yDc33nijEW5veh5//PG0uveOnZTGTGoSaO+crPHKiptvvtmMt43kSy+9lIu77JTZRri6KzNXH3Z0dJj3bq+dS3L1waaNW7duXUCZ8uzmzZtdgf7wq5th+Fx77bU2ioLAP3jwoJEs99kkDw5fib1w4cLEAnXhNJ2iXnZ/zn5SzCmyXG419/eZvXIusaZkK6HcWf7JW0nX4cOHTZIqGR9++GEzzFzD7BR7r+N+6G+8csdIUDTay2SoULRmKBqVm/bHsEuwYk3j9fLLL/sv32ek/LhhwwZDyUIOklNGuyLZOlFF3lY29ej2vd3NNZ94VvvMIu9mk93KNY1ppVelaq/EMRLcJKq/Jqg/luVutBj+yn17QUkkQoJPeiHp2MiMlvpz1iei3DHhOwlbhq6++upXXnnFKNObK664wrxXhpgGMLPjvqD9vEPv9fkvQ4n9xhtvmPE2Xt0MX716daX9D0155/0PpLG6OjXUtty7WaFLslmIv7VmpAqcSmpi1vbt26+66irzoRgayWE3D1Q+FMdK8rwGVxov85TPerkD/OFXMcNWBEmbPHmyGRkEvpV80UUXTZ061UgODl/NpapSlQWqz0jcQBdziiw3DahCtP5rj7kYJCzxT95KS5BYc+kScGE3w5S2KhF2ign4wjs01q3KHXuZDBKK1kJ1hwp1d8Mm7BKsItUxecFtWYrFRmKWjDdbYu4rSE4lZKpBt1fKRLOlRbk7B57VPrPIu0p1wZbYxC54Glqlam+vRLpI2fui+muC7kXdOq9oUScqq8x1yih1+xX9mOCTXkI6NjKjpf6czR1ytous543Rau6k3QJq79rT52Qalnl4pr+F8Z2/e7aKMWrojDoj3KqzujQgffiq32aeyEqXNBp1tgdyDXOPyjQgfaClAT6nXLYuJK7l9t40TUO/ynsQmObm6jW/lSKfB1YyXWBpG5jygmmxi8F3VaThG0fbu4164LvxaX3tsq0fvi0Z1mC9cQNDP6YfgpENhX1hFiK9NrCNO+p/JeT4J29N1a7ktFgTAzWFVBpgXGDv6nLVgZpKLWf3qYLgS3DNUGyEfR7LbU2M5HpyqgoxN7PELR3h1k01q73Rki7ymdkkd1R/zsyt9pnFx17FTJ1MrLFAQXbT0y0I1rkqEWkvu2GcyTkdG4lPAuZszdQwA8LsSRhZ5rTC7nEJorn/cHdabKej7i/zdONLl0//9a53f3+84nc8mBZML51WqK3ThoH50SjSv+6WplWnu5/9+/cn+ixpkS5pNJ+bW3a93NMK7UfZDQl3J9CK0kaCz3GmPSsxE0122W2uRGdqxmiliX3mhP0+Pyb0+kypMsY6V7eAhrZuBXR7re0Zs7XuDz+hJQ3fOFrNtTmDqAe+G58SJX/p5sCGXxD4AmK0VDqtcDdj7dplhnb+63RK1Om5kre6JQoVk6fGBelTDH2iguiTTZmKjAuE1OzkFQ7FTOGqA2Z3Uxs/xsgYS4jqSgk3Fy27h19PTlUyVdnq7ny4ex52iq5zR4788U8n1qz2JlTSRT4zm7Q5p7PLSoYlqr07zBYfFWSz5xSkJvh4MzOKLB8fCekxAXPW34AwnURanwAp8czDCq7XFWemUiiIzVZP4jV06BDz5zA812CvtYlniMx0e3IsG9ytVPNb8wc+pLFAVljJKlj2+QZPmzVeW2dmsAlct+opDkwp1JuAJxGetvkPs12F2ZMwj6G4iVEdfnVF6SN/tyTVA19s3TaiMfAT9cLarw7Mnvr5k3dHqpEVGX2idcWIllzJW2kJ5gEd20aY9LfPG8ly05WasC/2nEQ6YEKFolmU7cjNnoSMDL6EYgHgP0thbw+CDeRQOeXa4N56uddjucPcEuiNujGzw+9Z7RNFvlI2SbU9IE5jqVntNUWHMuZcJkZBtkcn5lJlniN0j4HSfPyd644MkrO5VXvuXfgPsztR5rY7sYHmnn1Ukrnzd+90fGv9ydMf+Si1u8pmPyex66UrnCGS3u2R/I5vrt/5uyOVtMhyd5susVlq9RY7gLA7jeYuwTXPHt/I7LDbm+4Oh8+JjA9/s7NiT5rc3cUq8GtKTuANCN+93BrsMeC7WqQiscdrvR/k/0BazvVHi5ueel8geTM9ay1USJt8cZm7YVn4rMcCDx6KiRXZ043gS7CK3AoQJEKsZHt5MGnrCq+zoBkVgp+oAG5MWu3uGJ9qnyjyiWxKuD4zAqtXexuBFkiQmmDT3F4j7EJMnCeuSpl8EstJx0bik1A5W7NEJwYUP5X01JR5TlZz7g83bH3oiV/UHJYekPnsRaach57Y9sMfb/VXkXmg5T+9ysjMM78gkhssxB9+XsNaHX48MnlJ5hpfLHlrqsh8gqHmrFwDYgNvwBJyrbfA4Hg5ZY3x9ELeau8p1mWSq9o3piBnPqRSwI+JKZFytqZhf7LlmHtDI9qE3v4Pl/zVk2tXL5k768IYSt78r8N3PPDC8/+0YtzokTHkIxMCEIAABHwIUO19KDX5mFjPSdS5bF3g7/72l7/7yIt7DvbUKSo9fe+hY9995OeSTxsRnC0CIQABCOQiQLXPhas5Bw/M393wYTFj6sSR5w5f+5OuL8+fMWrEcJ8pPmOO9Z3+6x/8bMXSy7/W8Vmf8YyBAAQgAIGoBKj2UfE2QHjzdhJa/KwZF/Se/MOjP+maefGnJrVnfxNlLkZ6vvLOh168bsHMv/za3FwTGQwBCEAAAvEIUO3jsW2A5KbuJLT+L86aPGrkJ+965N9Gtw2/7NJJ9RDp3LTz7r9/+fblC27+ypx65DAXAhCAAASCE6DaB0faMIFN+sRlYv27D/Z8f92WMeeNWL5k7rzPTclL5/W3Dj31/Jv6Q3Df+86iS6e1553OeAhAAAIQaAwBqn1jOIfV0hqdhFnzs1t+89QLv7rw/DHLFs+eP3vasGE1Hhf9+OMzr3Uf6NzUra9bX/7VL9yw6LKw7JAGAQhAAAIxCFDtY1CNJ7OVOgnbT2zq2tW964iaiflzpk2ZNLZ9XNvEcaMmjG073nfqWO/pnt5Th472vbbzgNqI2TM/vXjhTHqIeAGEZAhAAAKRCKifoNpHYhtWbOt1Emb9/ac+2tF9UL3C0Z5+/XeMnhOnjvednjB2VPv4toljR+nxTPUZV86eOrrt3LC8kAYBCEAAAo0kQLVvJO1iulq1kyi2WmZBAAIQgAAEIBCWQJN+M1XYRSINAhCAAAQgAIFIBOgkIoFFLAQgAAEIQGBQEKCTGBRuZpEQgAAEIACBSAToJCKBRSwEIAABCEBgUBCgkxgUbmaREIAABCAAgUgE6CQigUUsBCAAAQhAYFAQoJMYFG5mkRCAAAQgAIFIBOgkIoFFLAQgAAEIQGBQEKCTGBRuZpEQgAAEIACBSAToJCKBRSwEIAABCEBgUBCgkxgUbmaREIAABCAAgUgE/g+9s5mZp4EVQgAAAABJRU5ErkJggg==)
Step 1: Divide 0 to -2 and 0 to 2 into equal parts. (…, -9/5, -8/5, …, 0, 1/5, 2/5, …, 1, 6/5, 7/5, …, 11/5, …)
Step 2: Mark -8/5 and 8/5 on the number lines.
Question 6.Express the following rational numbers as decimals numbers
I. i.
ii. ![](data:image/png;base64,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)
iii.
iv. ![](data:image/png;base64,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)
II. i.
ii. ![](data:image/png;base64,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)
iii.
iv. ![](data:image/png;base64,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)
Answer:I. (i). 242/1000 = 242 × 10-3
⇒ 242/1000 = 0.242
(∵ 242 ÷ 1000 = 0.242)
(ii). 354/500 = (354/5) × 10-2
⇒ 354/500 = 70.8 × 10-2
Or 354/500 = 70.8 ÷ 100
⇒ 354/500 = 0.708
(iii). 2/5 = 2 ÷ 5
⇒ 2/5 = 0.4
(iv). 115/4 = 115 ÷ 4
⇒ 115/4 = 28.75
II. (i). 2/3 = 2 ÷ 3
⇒ 2/3 = 0.6666666…
This is clearly a repeating or recurring decimal. Thus, round-off the decimal in 2-digit or so after the decimal point.
⇒ 2/3 = 0.67
(ii). -25/36 = -25 ÷ 36
⇒ -25/36 = -0.6944444…
This is clearly a repeating or recurring decimal. Thus, by rounding off in 4-digit or so after the decimal point.
⇒ -25/36 = -0.6944
(iii). 22/7 = 22 ÷ 7
⇒ 22/7 = 3.142857143…
This clearly is non-repeating, non-terminating decimal, which is a decimal number which continues endlessly with no group of digits repeating endlessly.
Thus, take only 2 or 3 digit after decimal point.
⇒ 22/7 = 3.14
This number is also called π (pi).
(iv). 11/9 = 11 ÷ 9
⇒ 11/9 = 1.2222222…
This is a repeating or recurring decimal. Thus, round it off to 2 or so digits after the decimal point.
⇒ 11/9 = 1.22
Question 7.Express each of the following decimals in
form where q ≠ 0 and p, q are integers
i) 0.36 ii) 15.4
iii) 10.25 iv) 3.25
Answer:(i). 0.36 = 36/100
Simplify it in the same p/q form,
⇒ 0.36 = 18/50
On further simplifying it,
⇒ 0.36 = 9/25
This cannot be further simplified in p/q form.
9/25 is in p/q form, where p = 9 and q = 25 are integers and q = 25 ≠ 0.
Hence, the answer is 9/25.
(ii). 15.4 = 154/10
Simplifying it in the same p/q form,
⇒ 15.4 = 77/5
This cannot be further simplified in p/q form.
77/5 is in p/q form, where p = 77 and q = 5 are integers and q = 5 ≠ 0.
Hence, the answer is 77/5.
(iii). 10.25 = 1025/100
Simplifying it in the same p/q form,
⇒ 10.25 = 205/20
On further simplifying it,
⇒ 10.25 = 41/4
This cannot be further simplified in p/q form.
41/4 is in the form p/q, where p = 41 and q = 4 are integers and q = 4 ≠ 0
Hence, the answer is 41/4.
(iv) 3.25 = 325/100
Now, on simplifying,
3.25 = 13/4
13/4 is in the form p/q, where p = 13 and q = 4 are integers and q = 4 ≠ 0
Question 8.Express each of the following decimal number in the
form
i) 0.5 ii) ![](data:image/png;base64,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)
iii)
iv) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADQAAAAcCAYAAAAuh5ywAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAHjSURBVFjD7ZcxbgIxEEV9AC6AxKTjAIvJFRwr4gDZXdFFVMgNNTJ0K0VcIB0lReo0nIAbUHAD7pBkjBbYZey1gTTIhRvWzMyb+TM7y+bzOXuk81AwESgC3SNAxn6aTqxQBIpAEcgfSKu02wO2OU4R6G1SpbshRpVKEyH4FIBtQYwn1B3Kj8zV68U9PWxzxnb2Kcd3Q63bJJBWMuFcfoyKolV1Wv2T68xyPmAAW4RBhxQQ+gHG9lSAPJ8NLuy5xjbPlmSFimLUElxMS5i687ojLzACCP3IDluDzPV54vA3KuMZhwVVOWMHYCWVToJ66N5ARgVCvZMBGijYl0Gi3NK0GnAlLpCregEaA8s4W9bLegtQoy9PeY8FTCjbzsY2/dOXX/Us/AfQsUIeybPJjQQ6lf7UeMCzRShUKNBB2r2NT3VscnNWCMFU/vxWTqMQ6VwDhM3v26c2uQUNhZDRHQrkCjBEbt6bQkizhgLhPWrqXSM3byDMIOvIdUgf+QAdXuS5DpFyUzW9p8+9ewif47ChVqLzbSVEbhUgYwh3JrO7qaQ0gM1ad2wq1pD948pCjOHzYUMd+/73JzciCdbV54U/fVYMQ/+bWjtcQLY9rbyLMO5P6tOmQPVy0ySM30MRKALddn4BIAFMf7R+/yoAAAAASUVORK5CYII=)
Answer:(i). 0.5 = 5/10
Simplifying it in the same p/q form,
⇒ 0.5 = 1/2
It cannot be further simplified in p/q form.
1/2 is in the form p/q, where p = 1 and q = 2 are integers and q = 2 ≠ 0.
Hence, the answer is 1/2.
(ii). ![](data:image/png;base64,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)
This has to be converted in p/q form.
Let
,
That is, x = 3.88888… …(i)
Multiply 10 on both sides,
10 × x = 10 × 3.88888…
⇒ 10x = 38.8888… …(ii)
Subtracting equations (i) from (ii), we get
10x – x = 38.8888… - 3.88888…
⇒ 9x = 35
⇒ x = 35/9
35/9 is in the form of p/q, where p = 35 and q = 9 are integers and q = 9 ≠ 0.
Thus,
.
(iii). ![](data:image/png;base64,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)
This has to be converted in p/q form.
Let
,
That is, x = 0.363636… …(i)
Multiply 100 on both sides,
100 × x = 100 × 0.363636…
⇒ 100x = 36.363636… …(ii)
Subtracting equations (i) from (ii), we get
100x – x = 36.363636… - 0.363636…
⇒ 99x = 36
⇒ x = 99/36
Simplifying it further,
⇒ x = 11/4
11/4 is in the form of p/q, where p = 11 and q = 4 are integers and q = 4 ≠ 0.
Thus,
.
(iv). ![](data:image/png;base64,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)
This has to be converted in p/q form.
Let
,
That is, x = 3.127777… …(i)
Multiply 100 by equation (i),
100 × x = 100 × 3.127777…
⇒ 100x = 312.7777… …(ii)
Multiply 1000 by equation (ii),
1000 × x = 1000 × 3.127777…
⇒ 1000x = 3127.7777… …(iii)
Subtract equation (ii) from (iii), we get
1000x – 100x = 3127.7777… - 312.7777…
⇒ 900x = 2815
⇒ x = 2815/900
⇒ x = 563/180
563/180 is in the form of p/q, where p = 563 and q = 180 are integers and q = 180 ≠ 0.
Thus,
.
Question 9.Without actually dividing find which of the following are terminating decimals.
i.
ii. ![](data:image/png;base64,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)
iii.
iv. ![](data:image/png;base64,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)
Answer:(i). We have
![](data:image/png;base64,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)
Therefore, it clearly is non-terminating decimal. (since, it is recurring or repeating).
Justification:
First, converting it in the form p/q, where p and q are integers and q ≠ 0.
So, let
.
That is, x = 0.55555… …(i)
Multiply 10 by equation (i),
10 × x = 10 × 0.55555…
⇒ 10x = 5.55555… …(ii)
Subtracting equations (i) from (ii), we get
10x – x = 5.5555… - 0.5555…
⇒ 9x = 5
⇒ x = 5/9
Now, by factorizing the denominator, we get
9 = 3 × 3 = 32
∵ Denominator is not in the form of 2n × 5n.
∴
is non-terminating.
(ii). We have 11/18.
Now, without actual division, we got to factorize the denominator of the rational number.
By factorizing the denominator, we get
18 = 2 × 3 × 3
⇒ 18 = 2 × 32
∵ Denominator is not in the form of 2n × 5n.
∴ 11/18 is non-terminating.
(iii). We have 13/20.
Now, without actual division, we got to factorize the denominator of the rational number.
By factorizing the denominator, we get
20 = 2 × 2 × 5
⇒ 20 = 22 × 51
∵ Denominator is in the form of 2n × 5n (i.e., 22 × 51)
∴ 13/20 is terminating decimal.
(iv). We have 41/42.
Now, without actual division, we got to factorize the denominator of the rational number.
By factorizing the denominator, we get
42 = 2 × 3 × 7
∵ Denominator is not in the form of 2n × 5n.
∴ 41/42 is non-terminating.
(a) Write any three rational numbers
(b) Explain rational number is in your own words.
Answer:
(a). A rational number is any number that can be expressed as a quotient or fraction p/q of two integers, where p is a numerator and q is a denominator, provided that p is non-zero i.e., p ≠ 0.
Any three rational numbers can be 1, -6.6, 4/5.
(this is because 1 can be written as 1/1, -6.6 can be written as -66/10 or -33/5 & 4/5 is already in p/q form)
(b). Rational number can be defined easily, as basically any sort of number that can possibly be written in a p/q form, numerator is p and denominator is q (q is non-zero).
Since, a rational number is merely an integer or fraction, we can say that we can find infinite rational numbers between two distinct numbers. It might be difficult to obtain rational numbers minutely from a number line graph and hence, formulas are used to find rational numbers.
Its range is from negative numbers in an axis to positive numbers, including zero. Infact all whole numbers, all positive and negative numbers are rational numbers.
Some examples:
1/2 is a rational number (p/q form, where q ≠ 0)
0.50 is a rational number (1/2)
1 is a rational number (1/1)
2.12 is a rational number (212/100)
-6.5 is a rational number (-13/2)
Question 2.
Give one example each to the following statements.
i. A number which is rational but not an integer
ii. A whole number which is not a natural number
iii. An integer which is not a whole number
iv. A number which is natural number, whole number, integer and rational number.
v. A number which is an integer but not a natural number.
Answer:
(i). Integer is any number that can be written without a fractional component. So, write a rational number in which numerator and denominator doesn’t have any common factor.
For example: 99/98, 2/3, 57/2 etc…
(ii). Whole numbers are all positive natural numbers including zero.
Natural numbers are the set of positive integers from 1 to infinity, excluding fractional and decimal parts. Basically, they are whole numbers without 0.
This clearly implies that 0 is the only whole number which is not a natural number.
(iii). Integers are set of positive as well as negative numbers, that can be written without a fractional component. While whole numbers are natural numbers, including 0.
Thus, all negative non-fractional numbers are integers but not whole numbers.
For example: -1, -2, -3, etc…
(iv). Natural numbers are counting numbers (positive, non-fractional and non-decimal). (1, 2, 3, 4, …)
Whole numbers are all positive natural numbers, including 0. (0, 1, 2, 3, 4, …)
Integer is any number that can be written without a fractional component. (…, -4, -3, -2, -1, 0, 1, 2, 3, …)
And rational numbers are numbers that can be written in the form p/q, where p is a numerator and q ≠ 0 is a denominator. (1, 2/1, 3/2, 55/3, …)
Drawing out a common number from these set, we can say
1, 2, 3, … are numbers which are natural, whole, integer and rational numbers.
(v). Natural numbers are positive counting numbers, excluding fractions and decimals.
Integers are basically any numbers that can be written without fractional component.
So, numbers which are integers but not natural numbers are:
-3, -2, -44, -1, …
Question 3.
Find five rational numbers between 1 and 2.
Answer:
We can represent a rational number between two numbers a and b as
(a + b)/2
Now, put a = 1 and b = 2.
Then,
Rational number between 1 and 2
Rational number between 1 and 3/2
Rational number between 1 and 5/4 =
Rational number between 5/4 and 3/2
Rational number between 3/2 and 2
Hence, five rational numbers between 1 and 2 are 3/2, 5/4, 9/8, 11/8 and 7/4.
Question 4.
Answer:
We can represent a rational number between two rational numbers a and b as
(a + b)/2
Now, put a = 2/3 and b = 3/5.
Then,
A rational number between 2/3 and 3/5 =
A rational number between 2/3 and 19/30
A rational number between 19/30 and 3/5 =
Hence, three rational numbers between 2/3 and 3/5 are 19/30, 39/60 and 37/60.
Question 5.
Represent on a numberline
Answer:
8/5 = 1.6 and -8/5 = -1.6
Thus, 1.6 lies between 1 and 2; -1.6 lies between -2 and -1.
Step 1: Divide 0 to -2 and 0 to 2 into equal parts. (…, -9/5, -8/5, …, 0, 1/5, 2/5, …, 1, 6/5, 7/5, …, 11/5, …)
Step 2: Mark -8/5 and 8/5 on the number lines.
Question 6.
Express the following rational numbers as decimals numbers
I. i. ii.
iii. iv.
II. i. ii.
iii. iv.
Answer:
I. (i). 242/1000 = 242 × 10-3
⇒ 242/1000 = 0.242
(∵ 242 ÷ 1000 = 0.242)
(ii). 354/500 = (354/5) × 10-2
⇒ 354/500 = 70.8 × 10-2
Or 354/500 = 70.8 ÷ 100
⇒ 354/500 = 0.708
(iii). 2/5 = 2 ÷ 5
⇒ 2/5 = 0.4
(iv). 115/4 = 115 ÷ 4
⇒ 115/4 = 28.75
II. (i). 2/3 = 2 ÷ 3
⇒ 2/3 = 0.6666666…
This is clearly a repeating or recurring decimal. Thus, round-off the decimal in 2-digit or so after the decimal point.
⇒ 2/3 = 0.67
(ii). -25/36 = -25 ÷ 36
⇒ -25/36 = -0.6944444…
This is clearly a repeating or recurring decimal. Thus, by rounding off in 4-digit or so after the decimal point.
⇒ -25/36 = -0.6944
(iii). 22/7 = 22 ÷ 7
⇒ 22/7 = 3.142857143…
This clearly is non-repeating, non-terminating decimal, which is a decimal number which continues endlessly with no group of digits repeating endlessly.
Thus, take only 2 or 3 digit after decimal point.
⇒ 22/7 = 3.14
This number is also called π (pi).
(iv). 11/9 = 11 ÷ 9
⇒ 11/9 = 1.2222222…
This is a repeating or recurring decimal. Thus, round it off to 2 or so digits after the decimal point.
⇒ 11/9 = 1.22
Question 7.
Express each of the following decimals in form where q ≠ 0 and p, q are integers
i) 0.36 ii) 15.4
iii) 10.25 iv) 3.25
Answer:
(i). 0.36 = 36/100
Simplify it in the same p/q form,
⇒ 0.36 = 18/50
On further simplifying it,
⇒ 0.36 = 9/25
This cannot be further simplified in p/q form.
9/25 is in p/q form, where p = 9 and q = 25 are integers and q = 25 ≠ 0.
Hence, the answer is 9/25.
(ii). 15.4 = 154/10
Simplifying it in the same p/q form,
⇒ 15.4 = 77/5
This cannot be further simplified in p/q form.
77/5 is in p/q form, where p = 77 and q = 5 are integers and q = 5 ≠ 0.
Hence, the answer is 77/5.
(iii). 10.25 = 1025/100
Simplifying it in the same p/q form,
⇒ 10.25 = 205/20
On further simplifying it,
⇒ 10.25 = 41/4
This cannot be further simplified in p/q form.
41/4 is in the form p/q, where p = 41 and q = 4 are integers and q = 4 ≠ 0
Hence, the answer is 41/4.
(iv) 3.25 = 325/100
Now, on simplifying,
3.25 = 13/4
13/4 is in the form p/q, where p = 13 and q = 4 are integers and q = 4 ≠ 0
Question 8.
Express each of the following decimal number in the form
i) 0.5 ii)
iii) iv)
Answer:
(i). 0.5 = 5/10
Simplifying it in the same p/q form,
⇒ 0.5 = 1/2
It cannot be further simplified in p/q form.
1/2 is in the form p/q, where p = 1 and q = 2 are integers and q = 2 ≠ 0.
Hence, the answer is 1/2.
(ii).
This has to be converted in p/q form.
Let ,
That is, x = 3.88888… …(i)
Multiply 10 on both sides,
10 × x = 10 × 3.88888…
⇒ 10x = 38.8888… …(ii)
Subtracting equations (i) from (ii), we get
10x – x = 38.8888… - 3.88888…
⇒ 9x = 35
⇒ x = 35/9
35/9 is in the form of p/q, where p = 35 and q = 9 are integers and q = 9 ≠ 0.
Thus, .
(iii).
This has to be converted in p/q form.
Let ,
That is, x = 0.363636… …(i)
Multiply 100 on both sides,
100 × x = 100 × 0.363636…
⇒ 100x = 36.363636… …(ii)
Subtracting equations (i) from (ii), we get
100x – x = 36.363636… - 0.363636…
⇒ 99x = 36
⇒ x = 99/36
Simplifying it further,
⇒ x = 11/4
11/4 is in the form of p/q, where p = 11 and q = 4 are integers and q = 4 ≠ 0.
Thus, .
(iv).
This has to be converted in p/q form.
Let ,
That is, x = 3.127777… …(i)
Multiply 100 by equation (i),
100 × x = 100 × 3.127777…
⇒ 100x = 312.7777… …(ii)
Multiply 1000 by equation (ii),
1000 × x = 1000 × 3.127777…
⇒ 1000x = 3127.7777… …(iii)
Subtract equation (ii) from (iii), we get
1000x – 100x = 3127.7777… - 312.7777…
⇒ 900x = 2815
⇒ x = 2815/900
⇒ x = 563/180
563/180 is in the form of p/q, where p = 563 and q = 180 are integers and q = 180 ≠ 0.
Thus, .
Question 9.
Without actually dividing find which of the following are terminating decimals.
i. ii.
iii. iv.
Answer:
(i). We have
Therefore, it clearly is non-terminating decimal. (since, it is recurring or repeating).
Justification:
First, converting it in the form p/q, where p and q are integers and q ≠ 0.
So, let .
That is, x = 0.55555… …(i)
Multiply 10 by equation (i),
10 × x = 10 × 0.55555…
⇒ 10x = 5.55555… …(ii)
Subtracting equations (i) from (ii), we get
10x – x = 5.5555… - 0.5555…
⇒ 9x = 5
⇒ x = 5/9
Now, by factorizing the denominator, we get
9 = 3 × 3 = 32
∵ Denominator is not in the form of 2n × 5n.
∴ is non-terminating.
(ii). We have 11/18.
Now, without actual division, we got to factorize the denominator of the rational number.
By factorizing the denominator, we get
18 = 2 × 3 × 3
⇒ 18 = 2 × 32
∵ Denominator is not in the form of 2n × 5n.
∴ 11/18 is non-terminating.
(iii). We have 13/20.
Now, without actual division, we got to factorize the denominator of the rational number.
By factorizing the denominator, we get
20 = 2 × 2 × 5
⇒ 20 = 22 × 51
∵ Denominator is in the form of 2n × 5n (i.e., 22 × 51)
∴ 13/20 is terminating decimal.
(iv). We have 41/42.
Now, without actual division, we got to factorize the denominator of the rational number.
By factorizing the denominator, we get
42 = 2 × 3 × 7
∵ Denominator is not in the form of 2n × 5n.
∴ 41/42 is non-terminating.
Exercise 1.2
Question 1.Classify the following numbers as rational or irrational.
i. ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,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)
iii. 30.232342345…
iv. 7.484848…
v. 11.2132435465
vi. 0.3030030003.....
Answer:(i) On prime factorization 27 = 33
√(33) = 3√3
√3 is an irrational number and so √27 is an irrational number
(ii) On prime factorization 441 = 72 × 32
√(72 × 32) = 21
Since 441 is a perfect square so square root of it is a rational number
(iii) The decimal is not terminating in case of this number.
So it can never be expressed in the form of fraction i.e. numerator by denominator.
So the number is an irrational number.
(iv) The decimal is not terminating in case of this number.
So it can never be expressed in the form of fraction i.e. numerator by denominator.
So the number is an irrational number.
(v) The decimal is a terminating decimal number.
So it can be expressed in the form of fraction i.e. numerator by denominator.
So the number is an rational number.
So it can never be expressed in the form of fraction i.e. numerator by denominator.
So the number is an irrational number.
(vi) The decimal is not terminating in case of this number.
So it can never be expressed in the form of fraction i.e. numerator by denominator.
So the number is an irrational number.
Question 2.Explain with an example how irrational numbers differ from rational numbers?
Answer:The rational numbers can be expressed in the form of fraction i.e. numerator by denominator.
The irrational numbers doesn’t have a terminating decimal number, so they can never be expressed in the form of fraction i.e. numerator by denominator.
For example:
i) ![](data:image/png;base64,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)
The number 4.43 can be expressed in the form of fraction i.e. numerator by denominator. So it is a rational number.
ii) √5 = 2.236067977…
This number doesn’t have a terminating decimal number, so they can never be expressed in the form of fraction i.e. numerator by denominator.
So it is an irrational number.
Question 3.Find an irrational number between
. How many more there may be?
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
When two numbers are irrational then their mean is also irrational
![](data:image/png;base64,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)
Mean ![](data:image/png;base64,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)
⇒ Mean = 0.746031746…
So
is an irrational number between the two given numbers.
Question 4.Find two irrational numbers between 0.7 and 0.77
Answer:First we square the two given numbers
0.72 = 0.49
0.772 = 0.5929
Square root of any two non perfect square numbers between 0.49 and 0.5929 will give an irrational number since they are not perfect squares.
√0.5 = 0.7071067811…
√0.53 = 0.728010988…
So √0.5 and √0.53 are the two irrational numbers between 0.7 and 0.77
Question 5.Find the value of √5 upto 3 decimal places.
Answer:Step 1: Since 5 is not a perfect square so we have to take the next greatest perfect square just less than 5 which is 4.
Step 2: Since the remainder is less than the quotient so a decimal is added and two zeroes are added. This becomes the dividend for the next division.
Step 3: The quotient in the previous step is now added with the divisor. A single digit is added at the end and this resulting number is multiplied with the same single digit such that it is less than the dividend for the next division.
Step 4: The remainder in this step is again added with two zeroes and this process continues until we get the number of digits required after decimal point.
![](data:image/png;base64,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)
Answer is 2.236
Question 6.Find the value of √7 up to six decimal places by long division method.
Answer:Step 1: Since 7 is not a perfect square so we have to take the next greatest perfect square just less than 7 which is 4.
Step 2: Since the remainder is less than the quotient so a decimal is added and two zeroes are added. This becomes the dividend for the next division.
Step 3: The quotient in the previous step is now added with the divisor. A single digit is added at the end and this resulting number is multiplied with the same single digit such that it is less than the dividend for the next division.
Step 4: The remainder in this step is again added with two zeroes and this process continues until we get the number of digits required after decimal point.
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)
Square root is 2.645751
Question 7.Locate √10 on number line.
Answer:According to Pythagoras theorem:
Hypotenuse = √ (Perpendicular2 + Base2)
√10 = √ (32 + 12)
To plot √ 10 we can use the above concept
Step 1: Draw a number line and mark a distance of 3 units from 0 to 3 and mark the point as C
Step 2: Draw a perpendicular of 1 unit from C and mark it as B. Mark the point 0 as A.
Step 3: Join AB
Step 4: Δ ABC becomes a right angled triangle with AB = √10 units from Pythagoras theorem.
Step 5: Taking AB as radius draw an arc taking A as centre until it meets the number line. Mark the point as P.
Step 6: The point P is the required point √10 to be plot on the number line
![](data:image/png;base64,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)
Question 8.Find at least two irrational numbers between 2 and 3.
Answer:First we square the two given numbers
22 = 4
32 = 9
Square root of any two numbers between 4 and 9 will give an irrational number since they are not perfect squares.
√5 = 2.236067977…
√7 = 2.645751311…
So √5 and √7 are the two irrational numbers between 2 and 3
Question 9.State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every rational number is a real number.
(iii) Every real number need not be a rational number
(iv) √n is not irrational if n is a perfect square.
(v) √n is irrational if n is not a perfect square.
(vi) All real numbers are irrational.
Answer:(i) The statement is true.
Reason: Each and every irrational number can be plotted in the real number line and hence they are all real.
(ii) The statement is true.
Reason: Each and every rational number can be plotted in the real number line and hence they are all real.
(iii) The statement is true.
Reason: The set of rational number is a subset of real number. The set of real number is composed of both rational and irrational number.
(iv) The statement is true
Reason: Square root of a positive real number is either a rational number if it’s a perfect square or else it is an irrational number.
(v) The statement is true
Reason: Square root of a positive real number is either a rational number if it’s a perfect square or else it is an irrational number.
(vi) The statement is false.
Reason: The set of irrational number is a subset of real number. The set of real number is composed of both rational and irrational number.
Classify the following numbers as rational or irrational.
i.
ii.
iii. 30.232342345…
iv. 7.484848…
v. 11.2132435465
vi. 0.3030030003.....
Answer:
(i) On prime factorization 27 = 33
√(33) = 3√3
√3 is an irrational number and so √27 is an irrational number
(ii) On prime factorization 441 = 72 × 32
√(72 × 32) = 21
Since 441 is a perfect square so square root of it is a rational number
(iii) The decimal is not terminating in case of this number.
So it can never be expressed in the form of fraction i.e. numerator by denominator.
So the number is an irrational number.
(iv) The decimal is not terminating in case of this number.
So it can never be expressed in the form of fraction i.e. numerator by denominator.
So the number is an irrational number.
(v) The decimal is a terminating decimal number.
So it can be expressed in the form of fraction i.e. numerator by denominator.
So the number is an rational number.
So it can never be expressed in the form of fraction i.e. numerator by denominator.
So the number is an irrational number.
(vi) The decimal is not terminating in case of this number.
So it can never be expressed in the form of fraction i.e. numerator by denominator.
So the number is an irrational number.
Question 2.
Explain with an example how irrational numbers differ from rational numbers?
Answer:
The rational numbers can be expressed in the form of fraction i.e. numerator by denominator.
The irrational numbers doesn’t have a terminating decimal number, so they can never be expressed in the form of fraction i.e. numerator by denominator.
For example:
i)
The number 4.43 can be expressed in the form of fraction i.e. numerator by denominator. So it is a rational number.
ii) √5 = 2.236067977…
This number doesn’t have a terminating decimal number, so they can never be expressed in the form of fraction i.e. numerator by denominator.
So it is an irrational number.
Question 3.
Find an irrational number between . How many more there may be?
Answer:
When two numbers are irrational then their mean is also irrational
Mean
⇒ Mean = 0.746031746…
So is an irrational number between the two given numbers.
Question 4.
Find two irrational numbers between 0.7 and 0.77
Answer:
First we square the two given numbers
0.72 = 0.49
0.772 = 0.5929
Square root of any two non perfect square numbers between 0.49 and 0.5929 will give an irrational number since they are not perfect squares.
√0.5 = 0.7071067811…
√0.53 = 0.728010988…
So √0.5 and √0.53 are the two irrational numbers between 0.7 and 0.77
Question 5.
Find the value of √5 upto 3 decimal places.
Answer:
Step 1: Since 5 is not a perfect square so we have to take the next greatest perfect square just less than 5 which is 4.
Step 2: Since the remainder is less than the quotient so a decimal is added and two zeroes are added. This becomes the dividend for the next division.
Step 3: The quotient in the previous step is now added with the divisor. A single digit is added at the end and this resulting number is multiplied with the same single digit such that it is less than the dividend for the next division.
Step 4: The remainder in this step is again added with two zeroes and this process continues until we get the number of digits required after decimal point.
Answer is 2.236
Question 6.
Find the value of √7 up to six decimal places by long division method.
Answer:
Step 1: Since 7 is not a perfect square so we have to take the next greatest perfect square just less than 7 which is 4.
Step 2: Since the remainder is less than the quotient so a decimal is added and two zeroes are added. This becomes the dividend for the next division.
Step 3: The quotient in the previous step is now added with the divisor. A single digit is added at the end and this resulting number is multiplied with the same single digit such that it is less than the dividend for the next division.
Step 4: The remainder in this step is again added with two zeroes and this process continues until we get the number of digits required after decimal point.
Square root is 2.645751
Question 7.
Locate √10 on number line.
Answer:
According to Pythagoras theorem:
Hypotenuse = √ (Perpendicular2 + Base2)
√10 = √ (32 + 12)
To plot √ 10 we can use the above concept
Step 1: Draw a number line and mark a distance of 3 units from 0 to 3 and mark the point as C
Step 2: Draw a perpendicular of 1 unit from C and mark it as B. Mark the point 0 as A.
Step 3: Join AB
Step 4: Δ ABC becomes a right angled triangle with AB = √10 units from Pythagoras theorem.
Step 5: Taking AB as radius draw an arc taking A as centre until it meets the number line. Mark the point as P.
Step 6: The point P is the required point √10 to be plot on the number line
Question 8.
Find at least two irrational numbers between 2 and 3.
Answer:
First we square the two given numbers
22 = 4
32 = 9
Square root of any two numbers between 4 and 9 will give an irrational number since they are not perfect squares.
√5 = 2.236067977…
√7 = 2.645751311…
So √5 and √7 are the two irrational numbers between 2 and 3
Question 9.
State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every rational number is a real number.
(iii) Every real number need not be a rational number
(iv) √n is not irrational if n is a perfect square.
(v) √n is irrational if n is not a perfect square.
(vi) All real numbers are irrational.
Answer:
(i) The statement is true.
Reason: Each and every irrational number can be plotted in the real number line and hence they are all real.
(ii) The statement is true.
Reason: Each and every rational number can be plotted in the real number line and hence they are all real.
(iii) The statement is true.
Reason: The set of rational number is a subset of real number. The set of real number is composed of both rational and irrational number.
(iv) The statement is true
Reason: Square root of a positive real number is either a rational number if it’s a perfect square or else it is an irrational number.
(v) The statement is true
Reason: Square root of a positive real number is either a rational number if it’s a perfect square or else it is an irrational number.
(vi) The statement is false.
Reason: The set of irrational number is a subset of real number. The set of real number is composed of both rational and irrational number.
Exercise 1.3
Question 1.Visualise 2.874 on the number line, using successive magnification.
Answer:We know that 2.874 lies between 2 and 3. So, let us decide the part of the number line between 2 and 3 into 10 equal parts and look at the portion between 2.8 and 2.9 through a magnifying glass.
![](data:image/png;base64,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)
Now, 2.874 lies between 2.8 and 2.9. Now, we imagine to divide this again into 10 equal parts. The first mark will represent 2.81, then next 2.82 and so on.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAlsAAABNCAIAAAAetzaJAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAHs9JREFUeF7t3Xu8VFXZB3ABpSLJTLmUopKSl0wTtOSihiBampEpqZQpZZroS6YVlWkW9ap08xUtKyJTunmJzDIMyUyIMk1LMSAVCRXRTAVvoPB+Zb3vdj4zc4Bzzt5zZuY8+4/5zJmz91rP+j1rPfe1dpe1a9duElcgEAgEAoFAINDpEegSGrHTz4EAIBDoLAisWrVq9uzZy5Yte+SRR5YvX1722adPn969e5d99u3b98ADD+zevXtnwahzjzM0Yufmf4w+EOgECPz1r3+98cYbZ82a5XPYsGE77LBDqeZLKtBFRyY1WaopFy9efMstt4wYMWLkyJE+99prr04AWOcdYmjEzsv7GHkg0MQIiH5Nnz49KcLNN9886TNXz549WzvqFStWaCc1tXLlytTU2LFju3Tp0tqm4v46RyA0Yp0zKMgLBAKB1iHwzDPP/M+6a+jQoUkLDhgwoHVNtHz3okWLknacM2fOf627evTokVfj0U6HIxAascNZEAQEAoFAPgg89thjF110EV04evRouqrQCKdIrI5mzJiho9NOO23rrbfOZwzRSoci0LVDe4/OA4FAIBDIAYF//etfn/zkJ7fddttHH330j3/847Rp0wpVhyjWvl70pUf96h0NOYwkmuhQBEIjdij80XkgEAi0G4Hvf//7b3rTm9asWbNgwYJLLrlkl112aXeTG9uAvvSoX72jASUb+2TcV5cIRNS0LtkSRAUCgcBGIPDiiy+ecsopt956K7W07777bsQTBd4yb948xOyzzz6I6datW4E9RdOFIRA+YmHQRsOBQCBQJAJqWwYNGkT33HbbbR2uDg0UDSjp2rUrqtBW5NCj7aIQCI1YFLLRbiAQCBSHwNe//vVRo0adfvrpHLL62QWBkm9961uoQhsKixt+tFwQAhE1LQjYaDYQCASKQuDDH/6wjfN04c4771xUH+1r9x//+Mf48eMdBTB16tT2tRRP1xSB0Ig1hTs6CwQCgXYiMGHChPvvv//aa69tZzs1ePzwww/v37//hRdeWIO+ootcEIioaS4wRiOBQCBQCwTOOuusv//971dffXUtOmt3H+hELZrb3VI0UCMEQiPWCOjoJhAIBNqJwHnnnTdz5kxqZrPNNmtnU7V5HJ2oRTPKa9Nj9NJOBEIjthPAeDwQCARqgcDFF1982WWXUTBbbrllLfrLqQ/Uohnl6M+pyWimQAQij1gguNF0IBAI5IIApSJ9eMMNN+y22265NFjjRubPn6/6VELxfe97X427ju5ahUBoxFbBFTcHAoFAByDgaJivfvWrhx12WAf0nVOX11133ZlnnqkGNaf2oplCEOj2hS98oZCGSxr93ve+9+c//3nvvfcuuqO2tf+d73zn9ttvt6m2bY8X/ZTtTX/7298GDhxYdEdta18siP371re+tW2PF/2Us5gXLly45557Ft1R29r/5je/ed99973lLW9p2+NFP/W1r31tyZIlu+++e9Edrb99lSmvec1rPvWpT5XdNnny5IceeujNb35zx5LXUu/nn3++9yxmTq0z3uzf/8tf/uL9w/VA8Fe+8pV///vftTzxrlWj/vKXv/zEE08ArcabTWuRR/zVr371i1/8olVw1PJmZdy//OUva9ljq/oC3a9//etWPVLLm3/+859ff/313kVXy043vq9rrrnmN7/5Td2Sd+WVV4oE1i15P/vZz7z2qGPJ84oJwUbysZLpP/nJT373u991LHnrmYo/+tGPbr755lLyJk2aZCxGtPETuLg7r7jiij/84Q91i94Pf/jDuXPnFjf8llquohG9Xcw7o2tPSvQYCAQCgUAZAp/73Ocokm222aYJkDEKqt2ImmAsDTcEbya55557Nkh2uUb8/e9/77Umd9xxxwafjBsCgUAgECgUAa+S8M56NTWF9lLLxr1M0YjiFRm1xDz19eyzz8qhTJkyRSh7Pb2/rBFF5OUUve7L2y9PPfXU2lMcPQYCgUAgUIoAd6pqvLShUQo3sUPYJyWp6KF3797wl4pavXp1VTJeqjUVJv32umvYsGGHHnroK17xiqeffvrJJ5/MhW6tKR/QPdfz+eefz6XNHBtBnvIBydtPfOIT9UneBRdcgEhmcn2SZ+vx5ptvzoSqT/LM/te97nUf+9jH6pC87t27f+lLX+rbt+9HP/rR+iTv3HPP9S7cj3zkIx1CnoKy3/72t2eccUbV9W5RfP7zn3dG2rhx4zqEvPVLIcxVEEQKf+hDH6okj8w56KCD9thjjxxFWauaQt5nP/tZRUkf+MAH6hO9iRMnqtc7+uijV61a1aqhVb2ZmjNhtthiC2cmLF26lGp8/etfb/W9/e1vL7v/JY1I/5144omzZ8/2ci81jbSXK69TIbT/qle9KjmtNa4a2hgcG4I8RD733HN1iB6EX/nKVwZ5GzPTqt5jaXjTbN0yt2PJU4RCo7zzne9sCV7keT8igV6fS2M95KmV++c//ymC2uaZ0/4HGxe9to3dQjNbmAImzPTp0+0vcFL8u9/97rLJ8/J+RPWWs2bNskfimGOO2XTTTdvWazwVCAQCgUD7EeAZOO3F1pQ+ffq0v7V6a0Eq641vfON//vMfArreaGtuehYsWPDTn/6UduQEVi3Xenk/oveqUIc2kMr6mov9+vVrAmhY38uWLTPzLLBXv/rVLY3Ixpfly5cLFPN4yqyBlIaNiQsEu5cee+wxKIkftBRCeOGFFyD2+OOPlwHuKb8/9dRTLDLhi7ymFqMvdadaAX+9rLVqy/ICpgEuu6Gyd78bWs+ePevT1cgLq8Zqh9jCU1vyMVcQS2S+Je74b2KulZvWqVX/4IMPmmxmncu//MIlaml61B4Zw5k3b553HdftVtTaY1J0j+aDYKnS0Xe9613HHXecHa7VexTyKrvYZdOmTav8va2/rHlh9fPPr1q9pq3Pt/k560SwPi0DyaRLL72UmK5szVkSgjMJnSOOOIJNkN1DATAUZFjbTEPTPGhzlYxXQmnkyJFO9K8cGtnknXBJ5Wy99dYXXXSRAIXb7PjMEH7b295mR38usFCHXs2auiMufcevypYx1AFaiXJb9dP2xOxasWKFV5/LllkwuVBV2ci9994LChlNYUCGV0u9/OlPf5KUdVvaxJYuk9NO6v9ed33xi190mFleRGIWraNxF+ncUrMWkTMiZFzIBImP0tvUstsjjyoJv7yoytoZPXr08OHDk4azhCW92LWVvej6gAMOSMzdZ5990vbEm266qUx9UjyyR/kSKeyZ2KqC8dFHH20VW01dBYw273ur8AMPPJAvYaWtcYnSbk7KoCqA6WYbE00wM0FlZfa43JkZYuuLQ1ktk+KIrE3Ld911l+2/VVVAKQGbFEvNqoevmXbOsWOOPuaIo06dfMWSleW9ERCKMnbaaSc5XtPLRMmRHlJbUYDlYUtsCtkDpax96n/HHXccO3ase0giK8fLSNM9tGM6U5jTnCNVZU2peho6dOiAAQOkkcm7qggwlq0cECGVymH/aoQ3Jho+ZMgQP1Iz5m5xRGrZosIgh25IgWy//fZe/CbyUNajc7bkrlEF52984xt0lW087rHZ1oIUuGegkQIjRoxAfPuptUpPOumkGTNmaJnVQm6S3WXNWgBSBRShLbaIP/bYY8GY9LTLfxXdKD/bbrvtCtKIiMQjaXxKlxcrDFPVIHCu02tf+1phnDe84Q29evVS9Z0oPOqoo8xAARuPb7XVViZnJextQ/Kcc87RkWbZLlr+7ne/W9mO1cEiTLf5dNgKj9xtZqnHkep3tubll1/eNhpaeorLLghxwgknMAgsTKV5QKjshYY2l0x+ctxZMO9973vBCx+YMy9uXXdBm0/A18wLt0Sz2cKQytiKhlJLOhsXK9yKSGyFM8PCvzisJ598Mh+F2oaqU2Mc6ZUvgKk1BiKpktjUo0cPhZPeK1nZEdmCmGzuqfpxj7DK8ccfL2TtWUOAM2+7CCLrrc1iNeKtF47b46AJc+9b/vjiO849ZuCYs68sHb9ZlZTBZz7zGRaTGaPYuiCAGHQEdDqjofTyi9mQTl1xMXgVX6XvTD/1ThZ8VWGRC53sdBauajQmMKOYRfzjH/+4suVPf/rTBMTHP/5xQDlSa/Dgwe5xOJm1ROWruPMskeGlM7lQtcFGmBdWV6WIURj2jne8I5mTbB16yKkTZa3ZD6Q+MNNJG+xr42+ga7Gv7H5zjP4+++yz0+8y5TwJFkb6kzr84Ac/yAQm2grSiPpi6KTGmWj4KJ9fRiQF4GXrDr1MkBJJzhRMEOEscy1faZ56V8bJyfNFRwcffLCiu8pelNqZb9xEt919992EJpvGd4rKfCtuXahpOOSQQzKUKGBMZGOV4cYzI7IVY6ffGZeMj7KpRQco6cx9aZjk2Jqm+p133ilg6zDISvJYWhZvApY/TWH74hgg6MHfGJ2pZCkp3SiCxcLFiCRkdMpo0KlTfsqItG9d7CepahdDZ7/99vNFSIDCTi4Ea4PVqPC47Nmm/LPQU9zWPL7pTqdNmji4f68tt9/z5NMPW7LwmodKYrcPP/zwnDlzWMe8deqHcE9zpYhr0aJFcoTsnbLGqWRe15gxYxikAoN4zwxP91DSIgmMOAuyCJK0iSQHZZl8PDAevWQ7oVnWF2PNPKabSQRAcbbYmOw1ekWMiBFKDbiB/c4VK4jO0mYtA+ZFFgUt/Zc1QxGS5nw1DiXDOasv9xQjmiRFqjhM7nnZlC2upEp2kOcKNEYua+MHP/gBFZgyynAjDqZOnUqckUcFQacvwQZkaH/XXXclwUFR1hc0SByuv0NE/Ys5z2lLmVpSidw3K4hdoi1HIoW+05mWeuc3V82xMRRMP24Wn14ejq+QBkJuOmVULJoTJiZcOaJ20kn7lh6mik3GznQua5YNTc0INdPo5hX+WsVlU0uAxygMtp0klT0uEYitPhNbsa8SBCa430Vr0e82AGKrL4IBkMRWI2LXIpjizJe81JpeEGk6+Q5Pxn0lkUQQGE08dCYi/ekL19xOPEPz3YrmEAvA5s7oIkbd3jZrpudnTjrqkOPOf6akP+YzK8m6EvETTDdr+e9F0EOgYG1LwU8qmTfDNQQl+ygxPrtqmUcUPaaDKxHgQbKRHRxsaYmnZV5sdicXhK9JCRWBXlmb/C2mbtXECZsUNwX64Emg099SEelxX9KJzCxillC+dLIP9CuCWtXQJrUpZoYOicDLoZX1zp+QpuKc+S4cR1EV5COWjlSAF4srQ/fuMUXRgEIxXmMhj9KD73//+7MV/p73vEeZSb7QaY0Rs9dee6VYWdmFa0ceeSQC2BMmWApKM8jA6BECN8l3ailfqlgtWSkDHu2///62C1dlrhhPynSI9FgjHK9SSjiIlM16sqS5kM3Pw9aqwXBpTiY1RZjYyqxJPXLd2DoIxnRavAgHsWxozGWiTBi8csgizFiJEkSKMKfwLyOSMZeyHpILYj8iQAXRyS8SILErl6W1nkQ758Ea5xuUiehcmJg1UmzUNOvm3hmT99v3gKvuKU/P8snkSKgrtonUVBHBNGuGs9KStpA7JIbSJKBvOIsWf6lwrJlG5GDRFlVrQxDGxGNRMt/Y9fJhZZOAG+T3omU6UUg3wydF2yov0jPFfyR4FLCIxliH2W3cOM4ldU4/bTC/vfGznG0rgWQ5VU3+w4T45hpqkDphcHDEkcGr4BqmFFqy9H2xJXfj+23tnYQ1Hl111VWVD0JMvoDCcw+jXiQjC/OyPBYvXsx9pMhNVGHMfKUSfcMVEJ7JbJdS8uTg+QcMVpF5C4FYxESQ4qzICsKAbx86ZyhfIaXTlFrThf2IojVlRT2JSNKTvkmQGkgiEnOzIVA2BcWcsy7EY7GVUqzKVmkOk9Nh7qLf2JrSw8jGa042gmkd0k8wv7XTqVX3O45fL2U1ZakFfJeOoe3810xAZLJvIC/sxINUSyH04ovkTr5zLxHAFyLZyAR2tvUo8VmptoUoHFFihUp5WL+mX2U6plWArOfmGmjENfOvvGDU8CGTby4vqbLOVYWIv5siHDiOBRMgr4GldpSBSC9lacL0I76SyKmGxVxhrGVJYwqSuVRaSGm6i0zmS1Vla2YhozszIUtvMBt40oQCl1p+xayV8knOTbpk5kx3gaZCiWSsOBzESi5LsKfzHFLXZrPMXLZsLDPvR/VnaR2N6Bb3USwuF2otHgtJzLmsNfxNndIotF16v4SLwZGmHBkkn8cQ9t1SFwwUn8i9HDGjihsKHDW3VUedgqVZeYVJyLNJZZOlF2uDeshRKkFPNIzRnSVWy3qUU6AsU48UIV9NQS92i0NmPpyoKT1UNf/dZhancCLDC2iV6cOMuYQp3LLySOdQMq8VdqV+xQb4PUXUwWbjYmmh0KqsOlIeKvIy+xVDsZVDZhrwaAX5jdFIyXrhiqoqv80Alj5onjt3pcx7zm4QsUBkZtB4TxEiQecGIR8DtEZYijRW7tZYokEvJnxarcAhjeFZNslTEpSs9jv5w8PhcOe4EErhKlojrrlxytkj9j/kqrurBMqU/zE9TItEkKkssMYfymUeaISFyw/gjxN/ZqGYFcT9Dl/9pjXMSTU7bdJIKS4GCDmeRLwiArzhYSgMw7OC6o/1Za8oBlvPVQduxpgNZmr6L5lOSZumSUhx2hjUVUvI8oJRO4i0g4cclOxkZAg5QiaVHfKqSdX0XdmwLVbyOjSQggIJkmT8Gh0fHcIytRSS0HQutab8FbIY75i3GkeVfpFKeTN1Wb7WjPwTS0vdCna7h23hkVQqkl18oEKjpnLkeFS2X4VYR08KtSXJLnqfSDLZ0M/5MBD577TyTQMDydFOF5hlTFSW+ZCGZLd4gE4VqggJJGGNAD4iX9x3IkkhYiLMyuJG5Kt4pN+QRzSL4+FpYq4igyQQ+QqpBI9SwU0ejHgaMpAHoizKImJhZRUkN/WuAhaAZZYollkaKYjC4MDWzHQgjggTzCXZGYUkj1Vj4hEvTI2WjJL2rGJjZywqxCv1mzXIHgWXqeU7xiEyC11Y3WpThTFL+xUtx5EkPAu9uEaynmgo45oJIIiYZVsIaoZIjpqidFDFasRFV56/9+5DL7/9wRdWPbdyxVPPPrfq8SeeHDPmqFTypGgeM4jOtIGa/0Fw5GgrMXDEc8hldkfan6QL/eK3714PloBIFpBf3CkWlAGdXtIrrOFZEiqvXXSl6BuslIn1wDuxlijdtDAoDAGfTO1xW4kG68f9XBmryHIiDpiW4v6MdM/Sjnk5XpWTPiUImRcUXkLS9+R5g9TsTDqGGBULSuklogqXk+ajI9PxEASBxFhLur+1i02Sg9DUpqqQRBU2pdMYSgOMzAUBeZS7QbC0MqtEWbKECoo5k+xMLoEKAjrlaSgVFIIUzeabxZ8MDtOPC2ifgEigWB80POtByIu5+WJQVXOQrcXN/SYSSkgZTjxwdOozIYOh8ltpkxxNAzdyX+gPlzEx2a8kFNjNWyNiJ2V+ZBsoqfoIg5Xm0HIpc+XqkI1NyDbNktBEDysnTQBDIFJSg6BjOBYnxHl+gu1yGRlbxbeEnQRvTDPF2CkQhY8ZW4UELXa0kTDAxGXooZw9wdDMC7rSdrgcehd8yohEWCIAkcmqIDQwF6SCTy7EZGGeVLIkdkpLVYZhiiBY5QFjq7JGwczEX/SgloR01C3xIq5bBA0vn+LW3hKdKs8vPe3IUd+e/ez+g3frZnY8/9ygg0ePP+nEHXttYUhSX8Zj96hgukGaGRxEOizHl8UTx2R0KV3kuMVm4kLWJMiOp0GdC+ilB1u4JytBLPtXXlgx0Mg7lhG5k6K4pIDYqcVjapKPnCpd84EIHUFm/bLgoKfQgM9BkcPNKJJTy/dihxZxMEfSdmU1t7BCdrJgfE86CWLpTrIM1AkoN9DZ2OEePybl1P4rJSw1njWlfavXpxmFgOx4mmwm+KXytB3o4X56sP1UlbVAZCuJksJM6KGZXuExI0k9CLVESvodDapek5FB//kRne6RoOJSpNJi4bWqJb5toDmZVsS6LhKD0i54AXklNiSmRCYF7F8SNvZ6ElIi8w7rNz9Td1xJ8Uz2B23Ky8mLp6lx0ynLC2SjQyEeIZXfX5W5pQdOmW+gTicqF3GpPLDrw/ATW30K2Mpf6BdbqTq60O/mVWKr7zLEkuhJ5nBkPW4JC2ZaswoIiiCSeCFm2dkZkawKItdsZJYJwrOH9ItmXiDHwPwXIWOzJhkilcNBF18RL6XCi1gdpaO2KOTOXFR4GRqYbhGpu4EngxuGTGGuZHZmSI7oFakR1z57z113PvjI05usXUNoOWZ1q2367brLLjfNvH6nAQMo+TQM1pw8iilOFxYxwhzByr2ptJWYeEotY7xVLfjDshM0g4YoUPqXCAwn1SKHW4LOs+lsjqQSTHpWpzBX0RM3dxCiwUCgDAEzn7Ywn5sbGaawQtn1v66vuRHIRidSKI9OKZbuuikbuxsETrkEtKNQAQlZhKwrUiN2EmbGMAOBQCBXBLjCPAAOTa6t1l1jMnkClaIIdUdZDQkSUhLLlUlRfZbOCEuXuIjKEt5z2Tuz5Lz43zSoyEQRGrHQHfo1xDW6CgQCgWZBIJViNstoWhyHXHKL5003/eD/f4DKuNSUMQ4kiWSsRcXSOZpKE1TqpYpC90qoiaaKI4rrSnwWpA519PK7LzoNC2KggUAgUNcIKAlRtyJ3VddUtps4SUQJb/K93S01cAN8RNWLirmEyqXYXUqi1MemKgGlPRKZhsdCYkAIG8h2K2IqwjtMIEbUtIEnU5AeCDQlAhJFak9UdjTl6LJBqVRSQycA2NzDbKzRRdS0sfgV1AYCzY+AsvO0c7y5L2NMRxvGVT8IhI9YP7wISgKBQOAlBGwPEEMTKFvPW74bHSl7SBSSyJalXS5x1QkC4SPWCSOCjEAgEPg/BCgJ2xwdpNLEiBidMYY6rDcWh0asN44EPYFAILCJY1aaXiOm94rEVVcIRNS0rtgRxAQCgcBLCDidRFDRwRTOLGw+ROzKd3CgsHDuLwptPqxqPKLwEWsMeHQXCAQCG0aAquBCOfBzw7c24B3cXyeohTqsQ9aFRqxDpgRJgUAg0MyBU5rei7WDx3WIQERN65ApQVIgEAi8hICDxb1/wxH2zQSHd5h7h5ojOptpUE0zlvARm4aVMZBAoNkQ8BIiLy1qslEZkXE12aCaZjihEZuGlTGQQKDZEBg3bpzj3LxyuWkG5mVMRmRcTTOiJhtIRE2bjKExnECgqRDwtjivJvY+wvTS6Ya+vOLUoZ1ehlz5CsCGHlczER8asZm4GWMJBJoQgbPOOosumTZtWqOP7fjjj992220nTZrU6ANpYvpDIzYxc2NogUCTIMC18jp4h7w07niuu+66M888k7PbuEPoDJRHHrEzcDnGGAg0NgJKUbwSb/78+Q06DJSjPwpq6p994SPWP4+CwkAgENjk4osvnjJlysyZM7fbbrvGgmPJkiXe8+f9t+PHj28syjshtaEROyHTY8iBQEMicN5551199dU33HCDA94aZQCOahs1apQTaiZOnNgoNHdmOkMjdmbux9gDgQZDQJXN3LlzeYqbbbZZ/ZO+evVq3uGQIUOimqb+mZUoDI3YKJwKOgOBQOAlBCZMmHD//fdfe+219Q/H4Ycf3r9/f3sQ65/UoDAhEJU1MRMCgUCgkRCgYHr16jVixIh6rttcsGABCtEZ6rCR5lZoxMbiVlAbCAQCEJg6deqhhx46aNCgyy67rA4BQdXAgQNRiM46JC9IWg8CETWN6REIBAINicCcOXNUbw4ePPiSSy7p0qVLPYxh7dq1SJLpVBk7dOjQeiApaGgVAhE1bRVccXMgEAjUCwJUzm233bZmzRrO4rx58zqcLDSg5MUXX0RVqMMOZ0fbCAiN2Dbc4qlAIBDoeAS6det26aWX2uo3fPjwM84444EHHugQmvSrdzSgBD2o6hAyotP2IxAasf0YRguBQCDQkQh4lcTChQu7du268847n3LKKbWsuNGXHvWrdzTESy06ch7k0XdoxDxQjDYCgUCgQxHo16/f5MmTly5dqrxTZvGEE07w0oxCKdK+XvSlR/3qHQ2F9hiN1wCBqKypAcjRRSAQCNQOgWeeecYrFV2SebZAuAYMGJBX94sWLbpx3aWu57/WXT169Mir8WinwxEIjdjhLAgCAoFAIH8ElH1Onz6d6po1a5aX9I4cOTJpx549e7a2sxUrViQtqKmVK1empsaOHVsnBa6tHU7cvx4EQiPG9AgEAoEmR0CEM+kzn8OGDdthhx169+7dp0+f9Jm+uJavux5Zd6UvPhcvXnzLLbdQgUkRxst+m3uuhEZsbv7G6AKBQOBlBFatWjV79uxly5aV6rzse6YjS/Vl3759DzzwwO7duweOnQGB0IidgcsxxkAgEAgEAoENIxC1phvGKO4IBAKBQCAQ6AwIhEbsDFyOMQYCgUAgEAhsGIHQiBvGKO4IBAKBQCAQ6AwIhEbsDFyOMQYCgUAgEAhsGIHQiBvGKO4IBAKBQCAQ6AwI/C/h4JHzYiROxgAAAABJRU5ErkJggg==)
Again, 2.875 lies 2.87 and 2.88. Let us imagine and divide into 10 equal parts.
![](data:image/png;base64,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)
This is representation of numbers on the number line through a magnification glass.
Thus, we can visualize that 2.871 is the first mark 2.874 is the 4th mark in these subdivisions.
Question 2.Visualilse
on the number line, upto 3 decimal places.
Answer:Step 1:
Here, the number
lies between 5 and 6. First, draw the number line.
![](data:image/png;base64,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)
Step 2:
Divide the above part into 10 equal parts and make first point to the right of 5 as 5.1, the second 5.2 and so on.
![](data:image/png;base64,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)
Step 3:
Now, 5.28 lies between 5.2 and 5.3. So, by focusing on this portion, divide it into 10 equal parts and mark first point to the right of 5.2 as 5.21, the second 5.22 and so on.
![](data:image/png;base64,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)
Step 4:
Now, 5.282 lies between 5.28 and 5.29. So, focus on this portion and again divide it into 10 equal portions and mark first point to the right of 5.28 as 5.281 and the second one as 5.282.
![](data:image/png;base64,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)
The number 5.282 is the 2nd mark in these subdivisions.
Visualise 2.874 on the number line, using successive magnification.
Answer:
We know that 2.874 lies between 2 and 3. So, let us decide the part of the number line between 2 and 3 into 10 equal parts and look at the portion between 2.8 and 2.9 through a magnifying glass.
Now, 2.874 lies between 2.8 and 2.9. Now, we imagine to divide this again into 10 equal parts. The first mark will represent 2.81, then next 2.82 and so on.
Again, 2.875 lies 2.87 and 2.88. Let us imagine and divide into 10 equal parts.
This is representation of numbers on the number line through a magnification glass.
Thus, we can visualize that 2.871 is the first mark 2.874 is the 4th mark in these subdivisions.
Question 2.
Visualilse on the number line, upto 3 decimal places.
Answer:
Step 1:
Here, the number lies between 5 and 6. First, draw the number line.
Step 2:
Divide the above part into 10 equal parts and make first point to the right of 5 as 5.1, the second 5.2 and so on.
Step 3:
Now, 5.28 lies between 5.2 and 5.3. So, by focusing on this portion, divide it into 10 equal parts and mark first point to the right of 5.2 as 5.21, the second 5.22 and so on.
Step 4:
Now, 5.282 lies between 5.28 and 5.29. So, focus on this portion and again divide it into 10 equal portions and mark first point to the right of 5.28 as 5.281 and the second one as 5.282.
The number 5.282 is the 2nd mark in these subdivisions.
Exercise 1.4
Question 1.Simplify the following expressions.
i. ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,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)
iii. ![](data:image/png;base64,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)
iv. ![](data:image/png;base64,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)
Answer:i. (5 + √7) (2 + √5)
We know that (a + √b) (c + √d) = ac + a√d + √b c + √b (√d).
Comparing with the given expression,
⇒ a = 5; b = 7; c = 2; d = 5
∴ (5 + √7) (2 + √5) = 5(2) + 5(√5) + √7(2) + √7(√5)
= 10 + 5√5 + 2√7 + √35
ii. (5 + √5) (5 - √5)
We know that (a + √b) (a - √b) = a2 – b.
Comparing with the given expression,
⇒ a = 5; b = 5
∴ (5 + √5) (5 - √5) = 52 – (√5)2
= 25 – 5 = 20
∴ (5 + √5) (5 - √5) = 20
iii. (√3 + √7)2
We know that (√a + √b)2 = a + 2√(ab) + b.
Comparing with the given expression,
⇒ a = 3; b = 7
∴ (√3 + √7)2 = 3 + 2(√3) (√7) + 7
= 3 + 2(√21) + 7
= 10 + 2√21
∴ (√3 + √7)2 = 10 + 2√21
iv. (√11 - √7) (√11 + √7)
We know that (√a - √b) (√a + √b) = a – b.
Comparing with the given expression,
⇒ a = 11; b = 7
∴ (√11 - √7) (√11 + √7) = 11 – 7
= 4
Question 2.Classify the following numbers as rational or irrational.
i. ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,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)
iii. ![](data:image/png;base64,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)
iv. ![](data:image/png;base64,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)
v. 2π
vi. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAA5CAYAAAC4YUKZAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAHSSURBVFjD7ZhPboJAFMafez1AE4dduy/PXoFMGrs0qZBuWRDDxgWLxqA7ktYDdFcP0G039QL1BHbhCdozQB0rjdDhj+CANUPyEkKA+WXme/O9NzCZTKDuqB1AQpwexPpq1AZhWT1V03BECHwQbTCsHGJsYBcUZakosFzPRFALRARGQkgICSEhJEQO43L7eMMgAPVZGSMr9JFr00sC8Lke2N+NojMiixoJISH+DwTbcKqKNAi/qjhNTRRdhkNDBEWW4eAQtfeixwARHAuEXztEVrYIheDpwbX750jg/TcjiLroWfcXIiEiS8HKPkT6YHpeMwqEqzvXPRMGEd57ntnUUBuFALF69AuNcVc4REZRLAYiz/7A3tERZoDGs6i+IzE12eDsTEslsIAOfTEdpyUMgvecaYMSeNvxi4CgPo1rJRdEVo5nLQWDsY2r222TlNoqptlysI1GGdMKhZmWpryf71ovF2QLmXur3ohzHwgO0B+D2tewBhoZQpvOk3SRR4B+fDaSIHg6chyzRdsw30sTaXYdDsCD2GzRAKsfr7BU9i4D0JFMCRqPpVM0rg+eHjap2VGeIkcFpPNKDfv6YC6apI/Kq+0QpPaSv2wVJRvio4f4BgOibyzn/qeFAAAAAElFTkSuQmCC)
vii. ![](data:image/png;base64,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)
Answer:i. 5 - √3
Here, 5 is a rational number and √3 is an irrational number.
We know that subtraction of a rational number and an irrational number always gives an irrational number.
∴ 5 - √3 is an irrational number.
ii. √3 + √2
Here, both √3 and √2 are irrational numbers.
We know that sum of two irrational number is always an irrational number.
∴ √3 + √2 is an irrational number.
iii. (√2 – 2)2
We know that (√a - b)2 = a - 2√a(b) + b2.
Comparing with the given expression,
⇒ a = 2; b = 2
On simplification, we get
⇒ (√2 – 2)2 = 2 – 2 (√2) (2) + 22
= 2 – 4√2 + 4
= 6 – 4√2
Here, 6 is a rational number and 4√2 is an irrational number.
We know that subtraction of a rational number and an irrational number always gives an irrational number.
∴ 6 - 4√2 is an irrational number.
iv. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAAlCAMAAACeRoI0AAAAAXNSR0IArs4c6QAAAGxQTFRFAAAA///b//+22///2/+2tv///9uQ29u2kNv//7Zm27a227ZmkLbbZrb/25Bm25A6tpBmkJCQZpDbOpDbtmZmkGaQZma2Oma2kDqQkDo6AGa2kDoAOjqQZjoAADqQZgA6ZgAAOgA6AAA6AAAAdZV9KgAAAAF0Uk5TAEDm2GYAAACzSURBVHjazZLRFoIgDIY3KJLMMCtLLTV5/3fssGEacpnntCu2/fxn7ANgnTCWI6xftlG5KD/H3dPa25joArAhH9EdIB0SXz8mgHcJWFGmXr6OV8nX2MCcATAvABTbY0W61HX3eStBZyyvSc2XsMn8lCx3XqQz7aac5KJ3Y7m66E7FzH22gyGJPlY95O8XbuMB/8aXcdYLvowzzleP/+GbL8kjfInPkq/vhnx9N+Q7DbMK3zdroREnrmJZ+gAAAABJRU5ErkJggg==)
On simplification, we get
⇒ ![](data:image/png;base64,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)
which is a rational number.
v. 2π
Here, 2 is a rational number and π is an irrational number.
We know that the product of a rational number and an irrational number is always an irrational number.
∴ 2π is an irrational number.
vi. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAhCAMAAADnC9tbAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAA///b//+22///2/+2tv///9u2/9uQkNv//7Zm27a227aQ27ZmkLbbZrb/Zrbb25Bm25A6tpBmkJCQZpDbOpDbtmZmtmY6kGaQtmYAZma2kGY6OmZmkDo6AGa2kDoAAGaQOjqQZjoAOjoAADqQADo6ZgA6OgA6OgAAAAA6AAAARyw32AAAAAF0Uk5TAEDm2GYAAACUSURBVHjatZDJDoIwFEVPQW1xKBTrACo4IEL//wNdICR0YUyIZ/eme/Mu/Ig4FKM6Od8Lb8VMb4jsGo7mzjm34384jwlSxtfYz8cLQe5dRBaSh2vtEKpCXDasW/WJ49ilIV8KkVqQncTiFMMyrUIiDZC5ZwyIUg+mq0YDpprlQFArZKOBoN5agKTpbU1v1iNv4Zcv3+kaDBm4uDEbAAAAAElFTkSuQmCC)
Here, 1 is a rational number and √3 is an irrational number.
We know that division of a rational number and an irrational number gives an irrational number.
∴
is an irrational number.
vii. (2 + √2) (2 - √2)
We know that (a + √b) (a - √b) = a2 – b.
Comparing with the given expression,
⇒ a = 2; b = 2
∴ (2 + √2) (2 - √2) = 22 – (√2)2
= 4 – 2 = 2
Here, 2 is a rational number.
∴ (2 + √2) (2 - √2) is rational.
Question 3.In the following equations, find whether variables x, y, z etc. represent rational or irrational numbers.
i. x2 = 7 ii. y2 = 16
iii. z2 = 0.02 iv. ![](data:image/png;base64,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)
v. w2 = 27 vi. t4 = 256
Answer:i. x2 = 7
⇒ x = √7
Here, √7 is an irrational number.
∴ x is an irrational number.
ii. y2 = 16
⇒ y = √16 = ±4
Here, 4 is a rational number.
∴ y is a rational number.
iii. z2 = 0.02
⇒ z = √0.02
Here, √0.02 is an irrational number.
∴ z is an irrational number.
iv. u2 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAgCAMAAAD68tKbAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAABmADqQAGaQAGa2OgAAOjoAOmZmOma2OpDbZjoAZjpmZrbbZrb/kDoAkDo6kDqQkGaQkNv/tmYAtmY6tv//25A625Bm29uQ2////7Zm/9uQ//+2///b6UDxpgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAfUlEQVQoU62RyQ6AIAxEcd/FFVS0//+Z0goED0YPzgVepp2GwtirYE50DfBAS9+WvCLuNwbdiN0SGaUKOiwb27Eydca3NkzxRt2XLXFOqQdc4X8L0z39Hf+Ut6etb0GX3Vg23GdVgM9HLYA3A23JrSiIHNP33PLYmobiy9tOMqIFp1VB4DQAAAAASUVORK5CYII=)
⇒ u =
= ![](data:image/png;base64,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)
Here, √17 is an irrational number and 2 is a rational number.
We know that division of a rational number and an irrational number gives an irrational number.
∴ u is an irrational number.
v. w2 = 27
⇒ w = √27 = 3√3
Here, √3 is an irrational number.
∴ w is an irrational number.
vi. t4 = 256
⇒ t =
=
= 4
Here, 4 is a rational number.
∴ t is a rational number.
Question 4.The ratio of circumference to the diameter of a circle
is represented by p. But we say that π is an irrational number. Why?
Answer:When we measure the length with a scale, we only get an approximate rational value, so we may not realize that in c and d which one is irrational.
So, either c or d is irrational and hence c/d is irrational i.e. π is irrational.
∴ π is an irrational number.
Question 5.Rationalise the denominators of the following:
i. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAAA5CAYAAACGRC3XAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAM9SURBVGje7ZrPTxNREMdn2yty1/j2pnikO+U/IOszIImHDWybnkw2ZEOaSCF7QGh7a0Q9e1M8evDmRfwDVO4aE/gDEP6E4vpmW+qyLvRtf1DafU3mQLO0M583833z5hXq9Tqk2VIdvAKgACgAowEgXlrbsgC+lioAFLjrWmiauMMY/GLmWiVVAGolXABd/6Hr8FPA8FMHoAOiiIsKgAKgACgACoACoAAoAGkHsJG6swAdgrbt3BIBAGPl/SgPRde/8hU+qwMci6CbYWPcXVfzAAVAAVAAFAAFQAEYHYBWQzLZ1g1Ac9JNlcCkABhE+o87AL9tPaf/JABoprkE/NRqQLumUw3AHzoA8cqEFDRz0+s/6i9NnnoCQP+45Vr3kcHXjoIy47vlejM3sf7JX8+1ZgwGB52dgRkH5rK7KAPhvze23Hkjn+cvnEZjiv6ulu17LRjGUcnz7ty0+hf+Cve0k/B2pwH8oWdy9vZSIgCNhjPFkT8/D/7cqmWeYwCnWKwtyjrqB8PPwQ86w/Uf+Mu0fcbtnfCCPWTwuQUBj0rV6u2+RZCygihjqbYg6+iaySrDmPmH658GrHN87Wn0mQDMXfgCwE55uZrrGUB7hJ2xMbMHWHibxNFhApB5roCwJzLgsOcM+CACp9/yBOKC5kfL8aZHDUB2/+9kgMSiXfoBVEdtUTkLbnCMldeO17g1YgBS7W+rZPFbt9XvWgIEolycWxYC+Puqayy/85O3f7Zqsk1mrm7GnMq0bie8ftKffCkgeyWrV1LkSWwEhJPLVNXlbD168tKCzNHOou9f5th5erct2wsAghtkXoJbJun0KyK8k9lWei2BEKRYCN3qn4Kv2vg4blcYCAAR0AYwvh/tEQatAWEIsvXf6l7nDUR7J66lTwTAiumlPceabjUc8qnVjwiGIGS7pT/5STfNJNJRvak4Tx4g8t2rFg0udnyi7QU4pF7acl2kD3FE8EVkLxkWdq9zG4yWQhyAoOZtLFCXGnr+gjFeXk/UCpt5/U1YtHQ9/4kXnz1KGgAJYz9X3lE9iANAO9TVYzD9mFdqs2M7D4hAaA7lO8Zg+NEc1PBjbCdCw1r9sR2JKQAKwODsL3HxSF26ObblAAAAAElFTkSuQmCC)
ii. ![](data:image/png;base64,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)
iii. ![](data:image/png;base64,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)
iv. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFAAAAA9CAYAAAA6e+4pAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAARnSURBVHja7Zu/T9tAFMfPYaV0BnHeWjoSX/IfoPQQFKmDRZ0oE5JVWVFUEioPKThsUX/N3SgdGbp1KXSH8gd0gT8AyJ8QSH1JbIzj2OfEic/kkJ6QEid++uTd99679wz29/cBt+GNQ3jsAM2/mTDGAfYDbJvWojUOsD/6WnwJjxB9XAM5QA4wkQCToH+sA2xTXCPcpzBHKQ6QEiABp6ryUwWjiiiCf50UBuUPOEAKgGZSKJQUlIdAuAEQneLC9qraaMzyJUyhfyTy3ubge/PfLcRahWtgyOjbVdIbJrw7VuAlCqCuFxckAC4BxMeyqs456t8UB0gBsJSDO+T1rCl6OCN+69W+twBKf2VNXyLLeyoABp2a9N5/ALDRUGfxIjgBAN6YAN9Zm8Z2Aa9CAG7M15srWk169AAtOD2bGRR97g3EMIrziCzfRXzi3nHLGFYE8hkp/2MaADqPnjwhei1fo4zTZqQ1zVzvsO89Cy5Al0XDmJ8aDbQgUgG0IHlE2f3ynjKAjuU646d/NiQIjgGQLou6vuAB8A/ZneNIqOMG+CAKvfTPsnoRrZlad5dWdjecO25NW0GkKkHF+tpUpjHOKAw6QMgjcCAAeJ3b1NbJZ3RNXkJQOIurDmYFoL2h0JzAlHB2SwTgqvs58SqLS1tTn0g7UhvmD1BZBdjy0z8OkBJi0uAxWwtzgBwgNw4wDoBhB3mm0YIAtriFG17iy5AFDWR5fi8Kn/18j7oUY25+L8DnkZZv1AATVUlEVXcz5cyEly8bAKN0ZtKnP6wATNwxVJSSMxZnyLQAK5MDtJLj9pm2UR+pM+SmNU1+jiA4tXev3uQAq/pHfCatAQmCczubgNJ5r20gjBWg25matiJlMvij1R0zysqzLsz+bhory9dqSjnTFdK8IteRBta4Adr612kvIvzB3Vq0muKoUF9nbfl226XCMcTKnvNHfwnB7y7E4F7z2MWYRGWcbUc/gPUqXvZqSjlmcZq4bKTHCtDnPTK/nFJQ6jDOtuOwKVcegUMzAi/GFoF+zhyZ4DRNRh1hRrmfsqrPJSl9sSOQ4oeP3Blyc6Ih9vyeeR2U3nxV9caTpJRvXdlBZzSzNoGnFcM6Q0CWC9lNcwO57kDMlXYGXdt+8LhCkLWFoNOVUQASX/IIfqHVbD9HAuf4aG5AhLo7BDl4R9MwrNCeiHjNR1tD6aP6THS7lIPVMDPYXl/inuHrcyhs/VtA4Ps4x88sKaHwueUHz1DQq7CjIoGOedW6YWvJznzzhMbP/Hz2g0d0DyFlz6vEG2kT8ZomHeSM7FFH6qo8101WJ/doQhifia/k8Qmy0bn1tqq+foEQ/uT3w4eqOPzG0DplGwAXpI6UNQ2Ra1UTXgHBzxDlP8V1ZOXnM7CfegJN54CT0yAuV0ZOY9zaMmiKNHf/+EHHRDHzizyKFVPK4va5T3JIluC/aYlXuFpfjiQPdDmUiOP7QXoY6T2GcSgJ8Cbl89ApQ9IgMgOQGwfIAbJk/wEho94vaqqEsgAAAABJRU5ErkJggg==)
Answer:i. ![](data:image/png;base64,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)
Here, the conjugate of the denominator (3 + √2) is (3 - √2).
By rationalizing,
⇒ ![](data:image/png;base64,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)
We know that (a + √b) (a - √b) = a2 – b.
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,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)
Here, the conjugate of the denominator (√7 - √6) is (√7 + √6).
By rationalizing,
⇒ ![](data:image/png;base64,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)
We know that (√a + √b) (√a - √b) = a – b.
![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
iii. ![](data:image/png;base64,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)
By rationalizing the denominator,
⇒ ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
iv.![](data:image/png;base64,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)
Here, the conjugate of the denominator (√3 - √2) is (√3 + √2).
By rationalizing,
⇒ ![](data:image/png;base64,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)
We know that (√a + √b) (√a - √b) = a – b.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAH0AAAAcCAMAAAB79kDoAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgBmZjoAZjo6ZjqQZrbbZrb/kDoAkDo6kDpmkGY6kLaQkLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///b0SG2ZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACI0lEQVRIS+1W7U7DMAxMGFAGA8ZnYUBg3djasrz/6xEnaWJ7aVqBhIS0/hnoLj77HLsV4vD8Fwfau4ffp1qff4wKUsn4wIH67A2d42guJOE2xZga9BMWE6Ih4hzNiTNuUzwOV7+732KSLk/xvwzNhuPc6oRETp6tiZqoj4gVDM2qc+6Y4pXzR78XNlVFEwZUr6ZSHtP+pNKw3PVt4DIbU0f089JeteJyA7+7ObHCouroVbQl9kSXiZY6rnwQ7XxiY4rK//Y71lxAxXV3QZviGnMtqiChSiLBpPo+l3UxkUQFarHiGosYzZCLwnUk1SO3C5Jr/Je1+h36GTVDugg1hHYx+0CpM3XKNWb5TK2RKmwT1Lt2IcHQ3Q20SE1ezGW5NC5XjkJQY42UMzw9RJ1yzeFQMmsjSv9zYzO0U6JLebUVa8jUq2PUHloXnfN7tXCuLrt29atDiqAGd0mXYdiC8xF1KdcSXUfmPOGiOet3HjQnS7ccvToMSLgBEXXqpA6mTrjqNPQou26Mkps3f01g0USRgDonyTDwOx8jCbtea+dTduJMyStHawqT8AqGLc5eQDUsmtY1xz9cHUWCd5RWLmx+21Ry6ldIA/vx1doQVALaLgo3EL3qZhf5SP5OWvWBTbvrViIKHN1KoYhI/kxyh94yFX2/sYQTaJ+6SHHHvGF5QPp10Ss3DAyVno5Av6yGVXoY476s9g//6Vflj6s7HMw78A2WMDY2r5KDMAAAAABJRU5ErkJggg==)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG8AAAAZCAMAAAAIYICOAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpDbZgAAZgBmZjoAZjqQZrbbZrb/kDoAkDo6kDpmkGY6kLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bV2iAPAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABeUlEQVRIS+1V21bCMBDcBaEiirdoNUpAbGPz/z9o7s2t9pQ+Sh7TnZndmSQFuCzpAMPV9/lGTEaLl/fz1WA6unucMR1MRzfrGePBdDR9nqM3hhaHDeJVkJh43QOI40O8O9BCVmfQKWeApos34GTRC7Y3Mj6KT8B3S4kNlyDp6FmdQaecoZ5Ki2FPxO7kBk13NaKgNwEdiupJfk5q68OP2gRdDOjpbVOXomnqjtfj9fYTgNeomu3uvYkZIp9Pc6i6FG04ZS5u+cS6HeJWmv510vz9eW6rNK2ynq6L0Y5z4JQdKz17W8nomBMRRAXpV6FT883VxWiwnGXFBhW3IMu9f44Eya99aT5fF6F1qpJzoEvdmw5en2cdyjp/1Up6fV2Idm5l4xk37EkUZHWwJjL1i2giR0v3AYI6j444E0Whrjon9gfEcGPia6/lrRB0VC+qc+iYMxHkdYV4a73r3JNifR/Vi+o8OuIcOKNmm836N8xD/9nY5eM/cOAXK34irTbz2CoAAAAASUVORK5CYII=)
∴ ![](data:image/png;base64,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)
Question 6.Simplify each of the following by rationalising the denominator:
i. ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,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)
iii. ![](data:image/png;base64,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)
iv. ![](data:image/png;base64,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)
Answer:i.![](data:image/png;base64,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)
Rationalizing the denominator by its conjugate,
⇒ ![](data:image/png;base64,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)
We know that (a - √b) (a + √b) = a2 – b.
We know that (a - √b)2 = a2 – 2a√b + b.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 17 – 12√2
ii.![](data:image/png;base64,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)
Rationalizing the denominator, we get
⇒ ![](data:image/png;base64,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)
We know that (√a - √b) (√a + √b) = a – b.
We know that (√a - √b)2 = a - 2√(ab) + b.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 6 - √35
iii. ![](data:image/png;base64,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)
Rationalizing the denominator,
⇒ ![](data:image/png;base64,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)
We know that (√a - √b) (√a + √b) = a – b.
⇒
![](data:image/png;base64,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)
iv.![](data:image/png;base64,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)
Rationalizing the denominator,
⇒ ![](data:image/png;base64,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)
We know that (√a - √b) (√a + √b) = a – b.
⇒ ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Question 7.Find the value of
upto three decimal places. (take √2 = 1.414 and √5 = 2.236)
Answer:Given √2 = 1.414 and √5 = 2.236
Given ![](data:image/png;base64,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)
Rationalizing the denominator,
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒
= 0.327
Question 8.Find:
i.
ii. ![](data:image/png;base64,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)
iii.
iv. ![](data:image/png;base64,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)
v.
vi. ![](data:image/png;base64,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)
Answer:i. ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
We know that (am)n = amn.
⇒ ![](data:image/png;base64,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)
∴
= 2
ii. ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
We know that (am)n = amn.
⇒ ![](data:image/png;base64,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)
∴
= 2
iii. ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
We know that (am)n = amn.
⇒ ![](data:image/png;base64,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)
∴
= 5
iv.![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
We know that (am)n = amn.
⇒ ![](data:image/png;base64,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)
∴
= 64
v. ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
We know that (am)n = amn.
⇒ ![](data:image/png;base64,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)
∴
= 9
vi.![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
We know that (am)n = amn.
⇒ ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
Question 9.Simplify :
![](data:image/png;base64,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)
Answer:Given ![](data:image/png;base64,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)
It can also be written as
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
We know that (am)n = amn.
⇒ ![](data:image/png;base64,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)
⇒ 3 – 8(7) + 15(2) + 15
⇒ 3 – 56 + 30 + 15 = -8
∴
= -8
Question 10.If ‘a’ and ‘b’ are rational numbers, find the value of a and b in each of the following equations.
i. ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,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)
Answer:i. Given a + b√6 = ![](data:image/png;base64,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)
Rationalizing the denominator,
⇒ a + b√6 = ![](data:image/png;base64,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)
We know that (√a - √b) (√a + √b) = a – b.
We know that (√a + √b)2 = a + 2√(ab) + b.
⇒ a + b√6 = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= 5 + 2√6
Comparing it with a + b√6, we get
⇒ a = 5 and b = 2
ii. Given a – b√15 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADcAAAAlCAMAAADRG4HkAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6ADo6ADqQOgAAOgA6OjoAOjqQOma2OpDbZgAAZgA6ZjoAZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkJCQkLbbkNv/tmYAtmY6tmZmtpBmtv//25A625Bm27Zm27aQ27a229u22/+22////7Zm/9uQ/9u2//+2///bVy9+8wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABcElEQVRIS9WU2VqDMBCFh1pc0UrrglpDq7aYvP8DOpNJQlb4eldzw3LmJCcD+QH+5RCVGUn6skKlH5+l7ZYVdMiHkq2skOO4AqBAiy72k9JfVQu8ZEaP9aK1glq7W0BFbV5hl86I1er5UPBpBcdwGSVRW0ww0PYo540u4vVGBeD36TFK+bVdHuBocg11C/KOv0jrKW/VRezTk9teq/XtuF6g7HHGaIjlD8VUmw5oPZcTk7MiG6f4Vtm8UJPVO34GTmP7aRTo6+x3ENke0wRlRff4mnudjrJSMJzRa3vgTrye0Q5OijKNEfoHuBHxpJMYoeJ8wTRG6Od2BPq+t+cUX+cAE2AFC/hZNj41QsDwqQiw0nf8rCVHjQgwbvNhARm0JPB45wBjfRorHmcMZnaEoRgwji8AjJWxwGBGjPSKAONyaqx4BNrX2EK0aZClgNG+EStcwM+Gd+RLAcPrOayYggQz0xihRHkCzWJktmD+KPwBhUUhdoDgMIAAAAAASUVORK5CYII=)
Rationalizing the denominator,
⇒ a – b√15 = ![](data:image/png;base64,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)
We know that (√a - √b) (√a + √b) = a – b.
⇒ a – b√15 = ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG4AAAAkCAMAAABfRv/DAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOjqQOmZmOma2OpDbZgAAZgA6ZjoAZma2ZpDbZrbbZrb/kDoAkDo6kDqQkGY6kGaQkJCQkLbbkNv/tmYAtmY6tmZmtpBmtv//25A625Bm27Zm27aQ27a229u22/+22////7Zm/9uQ/9u2//+2///b07bQ2AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABsklEQVRYR+2W61KDMBCFQ7UWtVpQoNYLtUK9FJq8/9u5lwRSVMg408zosD8okK9nd0+SpkKM4d2BIujEDxW4cgMNPL+4dejK9avJG7dsDpxaz1BLrYLJrS1aXrQvKhxhEO2aPLXgexwEV+YRue8xUcckV85jSlfMdvU5yagsxevyUWyNbgkDGixwsMVkZGECuC6m5eQiEXXIieAiFykMXLc6eKfzC3W/a8BOOhvTHOk1mE5XncDs5226CBrI2VbqDmK/SoTagAM1Tx3rgJlkXoOJAqq0uENMc1j5PvuSTi54xUPGPDhNxNtmuhMV5ycdbDpMLUxsMbvFWRiYZuSgzHncb+ZriA6nQi9vo8Omm+4KXigtd4hZLsgIufwMZyafHi4VCe5CG9DT9IO9JFAt9WsjU0E26t5wHcyamxiqpN8B+JSwUJOmTrgpQ94IMnqgDcKgWsM+II67Y7fojrkuZtKpDKbGJQp7m/V8wZUbyFlf0jYYDFduUGgE3BzonmPHfXaraaRGB/6MAyrDLaMPwONXre74MPIYeNT5Cw/N2f9RKm8zR6e555n7z835W5K/z/QJViotDex4KdIAAAAASUVORK5CYII=)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Comparing with a – b√15, we get
⇒ a =
and b = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAADlQTFRFAAAAAAAAAAA6AGa2OgAAOma2OpDbZrbbZrb/kDoAkDo6kDqQkNv/tmY625A62////7Zm/9uQ///bHGcVowAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAUklEQVQYV2NgQAd8jIyMTDwMDHwcEBk4DRRnhQgJskDkhDjZGIS4eMB8IV6gNnYM03AKAE0DAeI1QFUKcYK0MTMIcfOD7QYBAYi74FxmfFxUGwEWtwI/+vNk+wAAAABJRU5ErkJggg==)
Simplify the following expressions.
i.
ii.
iii.
iv.
Answer:
i. (5 + √7) (2 + √5)
We know that (a + √b) (c + √d) = ac + a√d + √b c + √b (√d).
Comparing with the given expression,
⇒ a = 5; b = 7; c = 2; d = 5
∴ (5 + √7) (2 + √5) = 5(2) + 5(√5) + √7(2) + √7(√5)
= 10 + 5√5 + 2√7 + √35
ii. (5 + √5) (5 - √5)
We know that (a + √b) (a - √b) = a2 – b.
Comparing with the given expression,
⇒ a = 5; b = 5
∴ (5 + √5) (5 - √5) = 52 – (√5)2
= 25 – 5 = 20
∴ (5 + √5) (5 - √5) = 20
iii. (√3 + √7)2
We know that (√a + √b)2 = a + 2√(ab) + b.
Comparing with the given expression,
⇒ a = 3; b = 7
∴ (√3 + √7)2 = 3 + 2(√3) (√7) + 7
= 3 + 2(√21) + 7
= 10 + 2√21
∴ (√3 + √7)2 = 10 + 2√21
iv. (√11 - √7) (√11 + √7)
We know that (√a - √b) (√a + √b) = a – b.
Comparing with the given expression,
⇒ a = 11; b = 7
∴ (√11 - √7) (√11 + √7) = 11 – 7
= 4
Question 2.
Classify the following numbers as rational or irrational.
i.
ii.
iii.
iv.
v. 2π
vi.
vii.
Answer:
i. 5 - √3
Here, 5 is a rational number and √3 is an irrational number.
We know that subtraction of a rational number and an irrational number always gives an irrational number.
∴ 5 - √3 is an irrational number.
ii. √3 + √2
Here, both √3 and √2 are irrational numbers.
We know that sum of two irrational number is always an irrational number.
∴ √3 + √2 is an irrational number.
iii. (√2 – 2)2
We know that (√a - b)2 = a - 2√a(b) + b2.
Comparing with the given expression,
⇒ a = 2; b = 2
On simplification, we get
⇒ (√2 – 2)2 = 2 – 2 (√2) (2) + 22
= 2 – 4√2 + 4
= 6 – 4√2
Here, 6 is a rational number and 4√2 is an irrational number.
We know that subtraction of a rational number and an irrational number always gives an irrational number.
∴ 6 - 4√2 is an irrational number.
iv.
On simplification, we get
⇒
which is a rational number.
v. 2π
Here, 2 is a rational number and π is an irrational number.
We know that the product of a rational number and an irrational number is always an irrational number.
∴ 2π is an irrational number.
vi.
Here, 1 is a rational number and √3 is an irrational number.
We know that division of a rational number and an irrational number gives an irrational number.
∴ is an irrational number.
vii. (2 + √2) (2 - √2)
We know that (a + √b) (a - √b) = a2 – b.
Comparing with the given expression,
⇒ a = 2; b = 2
∴ (2 + √2) (2 - √2) = 22 – (√2)2
= 4 – 2 = 2
Here, 2 is a rational number.
∴ (2 + √2) (2 - √2) is rational.
Question 3.
In the following equations, find whether variables x, y, z etc. represent rational or irrational numbers.
i. x2 = 7 ii. y2 = 16
iii. z2 = 0.02 iv.
v. w2 = 27 vi. t4 = 256
Answer:
i. x2 = 7
⇒ x = √7
Here, √7 is an irrational number.
∴ x is an irrational number.
ii. y2 = 16
⇒ y = √16 = ±4
Here, 4 is a rational number.
∴ y is a rational number.
iii. z2 = 0.02
⇒ z = √0.02
Here, √0.02 is an irrational number.
∴ z is an irrational number.
iv. u2 =
⇒ u = =
Here, √17 is an irrational number and 2 is a rational number.
We know that division of a rational number and an irrational number gives an irrational number.
∴ u is an irrational number.
v. w2 = 27
⇒ w = √27 = 3√3
Here, √3 is an irrational number.
∴ w is an irrational number.
vi. t4 = 256
⇒ t = =
= 4
Here, 4 is a rational number.
∴ t is a rational number.
Question 4.
The ratio of circumference to the diameter of a circle is represented by p. But we say that π is an irrational number. Why?
Answer:
When we measure the length with a scale, we only get an approximate rational value, so we may not realize that in c and d which one is irrational.
So, either c or d is irrational and hence c/d is irrational i.e. π is irrational.
∴ π is an irrational number.
Question 5.
Rationalise the denominators of the following:
i.
ii.
iii.
iv.
Answer:
i.
Here, the conjugate of the denominator (3 + √2) is (3 - √2).
By rationalizing,
⇒
We know that (a + √b) (a - √b) = a2 – b.
∴
ii.
Here, the conjugate of the denominator (√7 - √6) is (√7 + √6).
By rationalizing,
⇒
We know that (√a + √b) (√a - √b) = a – b.
∴
iii.
By rationalizing the denominator,
⇒
∴
iv.
Here, the conjugate of the denominator (√3 - √2) is (√3 + √2).
By rationalizing,
⇒
We know that (√a + √b) (√a - √b) = a – b.
∴
Question 6.
Simplify each of the following by rationalising the denominator:
i.
ii.
iii.
iv.
Answer:
i.
Rationalizing the denominator by its conjugate,
⇒
We know that (a - √b) (a + √b) = a2 – b.
We know that (a - √b)2 = a2 – 2a√b + b.
⇒
⇒
⇒
⇒ 17 – 12√2
ii.
Rationalizing the denominator, we get
⇒
We know that (√a - √b) (√a + √b) = a – b.
We know that (√a - √b)2 = a - 2√(ab) + b.
⇒
⇒
⇒
⇒ 6 - √35
iii.
Rationalizing the denominator,
⇒
We know that (√a - √b) (√a + √b) = a – b.
⇒
iv.
Rationalizing the denominator,
⇒
We know that (√a - √b) (√a + √b) = a – b.
⇒
=
Question 7.
Find the value of upto three decimal places. (take √2 = 1.414 and √5 = 2.236)
Answer:
Given √2 = 1.414 and √5 = 2.236
Given
Rationalizing the denominator,
⇒
⇒
⇒
⇒
⇒
⇒ = 0.327
Question 8.
Find:
i. ii.
iii. iv.
v. vi.
Answer:
i.
⇒
We know that (am)n = amn.
⇒
∴ = 2
ii.
⇒
We know that (am)n = amn.
⇒
∴ = 2
iii.
⇒
We know that (am)n = amn.
⇒
∴ = 5
iv.
⇒
We know that (am)n = amn.
⇒
∴ = 64
v.
⇒
We know that (am)n = amn.
⇒
∴ = 9
vi.
⇒
We know that (am)n = amn.
⇒
∴
Question 9.
Simplify :
Answer:
Given
It can also be written as
⇒
⇒
We know that (am)n = amn.
⇒
⇒ 3 – 8(7) + 15(2) + 15
⇒ 3 – 56 + 30 + 15 = -8
∴ = -8
Question 10.
If ‘a’ and ‘b’ are rational numbers, find the value of a and b in each of the following equations.
i.
ii.
Answer:
i. Given a + b√6 =
Rationalizing the denominator,
⇒ a + b√6 =
We know that (√a - √b) (√a + √b) = a – b.
We know that (√a + √b)2 = a + 2√(ab) + b.
⇒ a + b√6 =
=
= 5 + 2√6
Comparing it with a + b√6, we get
⇒ a = 5 and b = 2
ii. Given a – b√15 =
Rationalizing the denominator,
⇒ a – b√15 =
We know that (√a - √b) (√a + √b) = a – b.
⇒ a – b√15 =
=
=
=
Comparing with a – b√15, we get
⇒ a = and b =