Page 26 - Math_Genius_V1.0_C8_Flipbook

E:\Working\Focus_Learning\Math_Genius-8\Open_Files\01_Chapter_1\Chapter_1
\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Existence of Multiplicative Inverse

Every non-zero rational number has its multiplicative
−3 Think and Answer
inverse or reciprocal. For example, a rational number
4
−4 −3 −4 −4 Find two rational numbers
has a multiplicative inverse , as × = 1 . So, is the which are also the multiplicative
3 4 3 3 inverse of themselves.
−3 5 7
multiplicative inverse or reciprocal of . Similarly, × = 1 .
4 7 5
7 5
So, is the multiplicative inverse or reciprocal of .
5 7

In general,
p q p q
if is a non-zero rational number, then there exists a rational number such that × = 1 .
q p q p
p
This q is called the multiplicative inverse or reciprocal of and vice versa.
p q

There is no reciprocal or multiplicative inverse for the rational number zero, as 0 cannot be a
divisor.
Distributive Property

1
2
Let us consider any three rational numbers, −3 , and .
5 3 3
−3  2  1 −3   3 −3 3 −3
We have, ×  +  = ×   = × =
5  3  3 5   3 5 3 5
−3 2  −  3 1 −2  −  1 −3
Also, × +   × = +   =
5 3  5  3 5  5  5

Clearly, −3 ×  2 +  1  = −3 × 2 +   −  3  × 1

5  3  3 5 3  5  3
According to this property, multiplying a rational number by the sum of two other rational numbers
is the same as multiplying the rational number by each of the two numbers individually and then
adding the results.
It is called the distributive property of multiplication over addition for rational numbers.

5
2
Again, let us consider other three rational numbers, −3 , , and .
4 3 6
−3  2  5 −3  4  5 −3  −  1 3 1
We have, ×  −  = ×  −  = ×   = =
4  3  6 4  6  6 4  6  24 8
−3 2  −  3 5 −6  −15  −12  −15  3 1
Also, × −   × = −   = −   = =
4 3  4  6 12  24  24  24  24 8
−3  2  5 −3 2  −  3 5
Clearly, ×  −  = × −   ×
4  3  6 4 3  4  6
It shows the distributive property of multiplication over subtraction for rational numbers.
Thus, we can say that, for rational numbers, multiplication distributes over both addition and
subtraction.

Mathematics-8 24