Which rational number does not have a multiplicative inverse?
In today’s life, we can’t imagine the representation of any quantity without numbers. We can’t count things without numbers. So to remove all these difficulties we use the number system. The number system was discovered a long time ago by our ancestors. They used to represent different things by number and symbol. Now we can apply the different operations on the number system for calculation purposes. We can express them in words and numbers, for example, 7 can be written in numerals itself and words also i.e., ‘Seven’.
There are different types of numbers:
- Natural Numbers
- Whole Numbers
- Real Numbers
- Rational Numbers
- Complex Numbers
What are Rational Numbers?
A rational number is a subset of the number system. A rational number is represented in p/q form, where q can not be equal to zero. Here ‘p’ is called the numerator and ‘q’ is known as the denominator. All the integers are also called rational numbers because they can be represented in p/q form.
For example: 3/4, -2/3, 0/6, etc.
There are two types of rational numbers:
(i) Positive Rational Number: Both the numerator and denominator of a rational number are either both positive or both negative then the rational number is known as a positive rational number.
Example: 2/5, 3/7, (-2)/(-9) etc.
(ii) Negative Rational Number: When the numerator of a rational number is positive and the denominator is negative or numerator is negative and the denominator is positive then such type numbers are called negative rational numbers.
Example: (-5)/(3), (3)/(-7) etc.
What is a Multiplicative Inverse?
When a number is multiplied by another number and if their product is 1, then these two numbers are called as multiplicative inverse of each other. Multiplicative Inverse is also termed as inverse of the number. In multiplicative inverse, basically we change numerator to denominator and denominator to numerator along with their sign.
How to Find Multiplicative Inverse?
Step 1: Suppose the rational number is given in p/q form.
Step 2: To find the multiplicative inverse, change the numerator by denominator and the denominator by the numerator.
Step 3: If the given number is a negative rational number then exchange their numerator and denominator along with their sign.
Step 4: We can verify our answer in the last. If the product of a rational number with their multiplicative inverse is 1 then our answer is correct.
Which rational number does not have any multiplicative inverse?
A rational number is represented in p/q form, where q can not be equal to zero.
‘All the integers are rational numbers.’
It means that 0 is also a rational number. It can be represented as 0/1, 0/-2, 0/6, etc.
In general form we can write 0 as rational number is 0/q and here q is not equal to zero.
Now find out the multiplicative inverse of 0/q i.e. q/0.
Here q/0 is not defined and denominator of a rational number can not be equal to zero.
q/0 is something like we are dividing q things to 0 objects, which is senseless.
So we can not find the multiplicative inverse of 0.
Question 1: Find out the multiplicative inverse of -6/7.
Compare -6/7 with p/q.
We got p = -6 and q = 7.
To find out the multiplicative inverse, exchange numerator and denominator.
Multiplicative inverse of -6/7 = 7/-6
Question 2: By which number we should multiply 5/9, to get the answer 1?
If the product of two numbers is 1 it means that they are multiplicative inverse of each other.
So we have to find out the multiplicative inverse of 5/9.
Compare 5/9 with p/q.
We got p = 5 and q = 9.
Now change the numerator with denominator and denominator with numerator, that will be the multiplicative inverse of given rational number.
Multiplicative inverse of 5/9 is 9/5.