Operations on Rational Numbers

You know how to add, subtract, multiply and divide integers as well as fractions. Let us now study these basic operations on rational numbers.

Addition

Let us add two rational numbers with same denominators, say 73 and −53. To visualize the addition let us do this on a number line. The number line for rational numbers is similar to the number line for integers. Draw a line and divide into equal parts. Since we need to show the addition between 73 and −53 we have to look at the denominators and find the distance between two points. Here the denominator is 3. So The distance between two consecutive points is .

We find 73 + −53

So adding −53 to 73 will mean, moving to the left of 73 , making jumps.

Where do we reach? We reach at .

73 + −53= 23

Similarly, lets calculate for −78+ 58

We should divide the number line in to equal parts. The distance between points should be

We find −78 + 58

We start from −78 and make jumps to reach

So, we find that while adding rational numbers with same denominators, we add the numerators keeping the denominators same.

How do we add rational numbers with different denominators? As in the case of fractions, we first find the LCM of the two denominators. Then, we find the equivalent rational numbers of the given rational numbers with this LCM as the denominator. Then, add the two rational numbers

For example, let us add −75 and −23

LCM of 5 and 3 is . That means we need to make the denominator as 15

Lets convert −75. To make the denominator 15, we need to multiply both the numerator and denominator with to get

Now lets do −23. To make the denominator 15 we need to multiply both the numerator and denominator with to get

Thus, −75 + −23 =

Additive Inverse

What will be −47+47 = ?

−47+47=−4+47 = . Also, 47+−47 = .

Similarly −23+23 = = 23+−23

In the case of integers, we call – 2 as the additive inverse of 2 and 2 as the additive inverse of .

For rational numbers also, we call −47 as the additive inverse of and 47 as the additive inverse of .

Similarly,−23 is the additive inverse of 23 and 23 is the additive inverse of .

Subtraction

Lets find the difference of two rational numbers 57 and 38 in this way:

57−38=40−2156=

We know that for two integers a and b we could write a – b = a + (– b)

So 57−38=57+−38=1956

So what did we do? We converted a subtraction problem into an addition problem. How? We took the rational number to be subtracted(38) and found it's additive invers(−38) and added the additive inverse.

So, we say while subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number.

Try These

Answer the following:

What will be 27−−56?

27−−56=27 + additive inverse of −56=27+56 = /

Multiplication

Let us multiply the rational number −35 by 2, i.e., we find −35×2

On the number line, since the denominator is 5, we will divide the number line into equal parts of . So −35×2 will mean start from 0 and jumps of to the left.

Where do we reach? We reach at −65. Let us find it as we did in fractions.

−35×2=−3×25 =

So, we find that while multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping the denominator unchanged.

Let us now multiply a rational number by a negative integer

29×−5

Remember, –5 can be written as = −51
This gives us:
Multiplying the number we get
Hence the result is -10/9

So, as we did in the case of fractions, we multiply two rational numbers in the following way:

−58×−97

Multiply the numerators and denominators of the two rational numbers.
We get:
Hence the result is

Division

We have studied reciprocals of a fraction earlier. What is the reciprocal of 27? It will be .

We extend this idea of reciprocals to non-zero rational numbers also. The reciprocal of −27 will be 7−2i.e.,(−7)/2; that of −35 would be −53.

Product of reciprocals

The product of a rational number with its reciprocal is always 1.

For example, −49×reciprocalof−49

=−49×−94=

Similarly,−613×−136=

This shows, to divide one rational number by the other non-zero rational number we multiply the rational number by the reciprocal of the other.

Thus, −65÷−23=6−5 × reciprocal of −23=6−5×3−2=

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Rational Numbers

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