Algebra Properties of Real Numbers

Introduction to algebra properties of real numbers:

In algebra, numbers that are used to measure the real-world quantities like area, speed, length, probability, temperature, volume, rate and so on. Those numbers should be called as real numbers. Few examples for that number should be 9, -12, 0, 2.523, `sqrt(3)` , `(sqrt(5))/(3)`, 3×106, `pi` so on. Algebra properties of real numbers should express in terms of addition as well as multiplication. Here we will see about the algebra properties of real numbers.

List of Properties:

The lists of properties in algebra for the real numbers are shown below:

Commutative properties
Associative properties
Distributive properties
Identity properties
Inverse properties

Explanation of Properties:

Let us consider the numbers a, b and c are the real numbers.

Commutative properties:

i) a + b = b + a

ii) a × b = b × a

This property says that the order in which we add or multiplication with real number that should not affect the solution.

Example:

2 + 5 = 7 = 5 + 2
3 × 4 = 12 = 4 × 3


Associative properties:

i) (a+b)+c = a+(b+c)

ii) (a×b)×c = a×(b×c)

This property tells that when the real numbers are grouping in terms of adding or multiplied could not matter. Here, the parentheses should not make a sense.

Example:

(5+6)+2 = 13 = 5+(6+2)
(2×3)×4 = 24 = 2×(3×4)


Distributive properties:

i) a×(b+c) = (a×b) + (a×c)

ii) (b+c)×a = (b×a) + (c×a)

Example:

3×(2+5) = (3×2) + (3×5) = 6 + 15 = 21
(2+5)×3 = (2×3) + (5×3) = 6 + 15 = 21


Identity properties:

i) a+0 = 0+a = a

ii) a×1 = 1×a = a

Here, 0 -> Additive identity and 1 -> Multiplicative identity.

Example:

3+0 = 0+3 = 3
4×1 = 1×4 = 4

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Inverse properties:

i) For every real number a. -a is called additive inverse of a, when, a+(-a) = (-a)+a = 0.

ii) For every real number a`!=` 0. `(1)/(a)` is called multiplicative inverse of a, when a × `(1)/(a)` = `(1)/(a)` × a.

Example:

5+(-5) = (-5)+5 = 0
2 × `(1)/(2)` = `(1)/(2)` × 2 = 1
These are the properties of real numbers in algebra.