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.

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.

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.