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The Natural Numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12…
The Whole Numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12… The Integers are …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5… 5x = 1 x = 1/5 In this equation, we need the number 3/2 to solve this equation. These numbers are called Rational Numbers. The Rational Numbers are in the format of p/q where as q is not equal to zero. The Natural Numbers between 3 and 9 are 4, 5, 6, 7 and 8. The Whole Numbers between 0 and 6 are 1, 2, 3, 4 and 5. The Integers between -3 and 4 are -2, -1, 0, 1, 2 and 3. In these cases the numbers between those two numbers are countable. But the number of Rational Numbers between any two rational Numbers is not countable. That means that there are innumerable Rational Numbers between any two rational Numbers. The Rational Numbers between 2/11 and 6/11 are innumerable. You may feel that there are only 3/11, 4/11 and 5/11. But 2/11 and 6/11 can be written in their respective equivalent forms as 20/110 and 60/110. Now between 20/110 and 60/110, there are many more rational numbers such as 21/110, 22/110, ………58/110, 59/110. In this way, there are innumerable Rational Numbers between any two rational Numbers. Properties of Rational Numbers : Closure Property : Let us add two rational numbers. Example : 1 (1/2) + (4/5) = ([5 + 8]/10) = 13/10 13/10 is also a rational number. Example : 2 (3/8) + (1/7) = ([21 + 8]/56) = 29/56 29/56 is also a rational number. When two rational numbers are added, what we get is only another rational number. So, rational numbers are closed under addition. Now, Let us subtract a rational number from another rational number. Example : 1 (4/5) - (1/2) = ([8 - 5]/10) = 3/10 3/10 is also a rational number. Example : 2 (3/8) - (1/7) = ([21 - 8]/56) = 13/56 13/56 is also a rational number. When two rational numbers are subtracted, what we get is only another rational number. So, rational numbers are closed under subtraction. Now, let us multiply two rational numbers. Example : 1 (2/3) x (4/9) = (2x4)/(3x9) = 8/27 8/27 is also a rational number. Example : 2 (3/11) x (4/7) = (3x4)/(11x7) = 12/77 12/77 is also a rational number. So, rational numbers are closed under multiplication. Now let us divide rational numbers. Example : 1 (2/3) ÷ (4/9) = (2/3) x (9/4) = (2x9) / (3x4) = 18/12 = 3/2 3/2 is a rational number. Example : 2 (3/11) ÷ (4/7) = (3/11) x (7/4) = (3x7) / (11x4) = 21/44 21/44 is also a rational number. So, rational numbers are closed under division. Rational Numbers : Exercise - 1
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