Identity Property Definition
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 , Euler's number e, the golden ratio , 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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