In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number.

Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor’s proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.[1]

When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common.

Perhaps the best-known irrational numbers are: the ratio of a circle’s circumference to its diameter π, Euler’s number e, the golden ratio φ, and the square root of two sqrt2.[2][3][4]

Introduction on Multiplicative Identity:

In mathematics, the multiplicative Identity defines so as to the multiplication of whichever number and one (=1) is the number itself. Therefore, the multiplicative identity satisfying the following definition, such as

a x 1 = 1 x a = 1.

or

a x `(1)/(a)` = `(1)/(a)` x a

A number e designed for which definition is (a).(e)=(e).(a)=a for each factor a of a set. Here, the sets are N (natural numbers), Z (integers), Q (Rational numbers), R (real numbers), C (complex numbers) the multiplicative identity is 1.

Multiplicative identity is as well labeled the identity property of one (=1) or the multiplications of identity property.

Multiplicative Identity Examples:

Example 1:

Check the following expression satisfy multiplicative identity property?

49 × 1 = 49

Solution:

Given : 49 x 1 = 49.

Here, when the number 49 is multiplied by multiplication identity 1, then the product is the number itself.

i.e., a × 1 = a

Hence, the above expression satisfied the multiplicative identity property.

Some more Examples on Multiplicative Identity:

Example 2

Check whether the following expressions are satisfying the multiplicative identity?

A. 44 + 1 = 44

B. 33 × 1 = 33

C. 66 × 1 = 54

D. 45 × 1 = 0

Solution:

(A) 44 + 1 = 44

Here, 44 + 1 = 45.

Therefore, the addition of 44 and 1 is produced the number is 45.

Hence, the above expression is not satisfied the multiplicative identity property.

(B) 33 x 1 = 33

Here, 33 x 1 = 33.

Therefore, the multiplication of 33 and the multiplicative identity 1is produce that number itself.

Hence the above expression satisfied the multiplicative identity property.

(C) 66 x 1 = 54

Here, 66 x 1 = 66, but the given product of result is not same.

Hence, the above expression not satisfies the multiplicative identity definition.

(D) 45 x 1 = 0

Here, 45 x 1 = 0.

That is, the multiplication of 45 as well as multiplicative identity 1 is produce the result is 0.

Hence, the above expression not satisfied the multiplicative identity definition.

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Example 3:

Verify the following expression satisfy the multiplicative identity definition?

78 x `(1)/(78)` = 1

Solution:

Given: 78 x `(1)/(78)` = 1

Here, 78 x `(1)/(78)` = 78 x 1 = 78.

Therefore, the multiplication of 78 and it’s reciprocal number produce the multiplicative identity 1.

Hence, the above expression is satisfied the multiplicative identity definition.

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