Write the numbers whose multiplicative inverses are the numbers themselves
A Number System is a way of representing numbers. Representation of numbers is done by using digits or symbols. The numbers that are represented by digits or symbols have the value and the value depends on the place, base, and value of the digits used. There are many types of numbers in the decimal number system, for instance, real numbers, complex numbers, natural numbers, whole numbers, and so on. Let’s take a look at their definitions,
Types of numbers
In the number system, the decimal number system is majorly used. In the decimal number system, there are many types of numbers based on their different characteristics. Lets take a brief look at their definitions,
The Real Numbers
The Real Numbers are the Numbers which consist of All the numbers ie., all of the Rational Numbers and Irrational Numbers. Some example of real numbers are 3, 1.444…, 55, 9.73409…, etc.
The Complex Numbers
The Complex Numbers are the numbers that are represented in (a+ib) form where b≠0 , where a and b are real numbers and i is an imaginary unit with value √-1, when the value is b. It is a real number since in (a+ib) b=0 then a+i*0 = a which is a real Number.
The Natural Numbers
The Natural Numbers are the numbers that start from 1 and counts to Infinity. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11…..up to Infinity are Natural numbers. It is a Subset of Whole numbers as whole number contains all natural numbers and 0 as well.
The Whole Numbers
The Whole numbers are the Natural numbers with Including an extra number zero ie., the numbers that start from 0 and counts to infinity are called Whole Numbers. It is a superset of natural numbers. Some examples are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,….upto Infinity.
Integers set are the number sets that consist of negative and positive numbers including zero. All the basic operations of number system i.e., Addition, Multiplication, Subtraction except Division will result in an Integer. The division may or may not result in Integer because when a smaller numerator and bigger denominator are used in division. It results in a fraction, that may be rational Or irrational
A Number is called irrational if it cannot be expressed in the form of p/q, where p and q are Integers. In other words, if it cannot be expressed as a ratio of two numbers. When a number is expressed in decimal form, if it never terminates it is an irrational number. For example,
- The value π(PI) is 3.1415926…. the value never terminates. It is rounded off to 3.14 and used in calculations like this.
- √2 is 1.41421356237309… which is an irrational number. So, square root of those numbers which are not perfect squares come under this category.
- While, square Roots of Perfect Squares are always Rational
A Number is called Rational if it can be expressed in the form of p/q, where p and q are Integers. In other words, if it can be expressed as a ratio of two Numbers. When a number is expressed in decimal form, if it terminates it is a Rational Number. Some numbers that are never terminating But are recurring are Rational Numbers. For example, 0.0833333….. it is the decimal representation of 1/12 is a rational Number. √4 is rational it can be expressed in p/q form i.e., 2/1.
There are many operations done on these numbers and one of them is the multiplicative inverse, lets see what are multiplicative Inverses and if it is possible to obtain the same term after performing multiplicative inverse on a number,
What are the numbers whose multiplicative inverses are the numbers themselves?
An Multiplicative Inverse of a number is defined as a value we multiply with the number to get the product as 1. In other words, Multiplicative Inverse is defined as the reciprocal of a number. For example, Multiplicative Inverse of number x : As per definition the multiplicative Inverse is Reciprocal of the Number i.e, multiplicative Inverse of x is 1/x or x-1. Similarly,
Multiplicative Inverse of numbers,
- 11 is 1/11 or 11-1
- -20 is -(1/20) or (-20)-1
- 3 is 1/3 or 3-1
The Numbers that are equal to its Multiplicative Inverse are -1 and 1.
Because Reciprocal of 1 is 1/1 which is 1.
i.e, Multiplicative Inverse of 1 = 1/1 = 1.
Reciprocal of -1 is -1/1 is -1
i.e, Multiplicative Inverse of -1 = -1/1 = -1
-1 and 1 are the Only Numbers which are equal to its Multiplicative Inverse.
Note There is No Multiplicative Inverse of 0 because Reciprocal of 0 is not defined. hence, Multiplicative Inverse of 0 does not exist.