# Algebraic Properties Of Real Numbers

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Algebraic properties of real numbers help in the simplification part thereby make solving the problem of algebraic equations easily. Properties of algebraic expression involve two types addition and multiplication.

## Algebraic Properties Of Real Numbers

### Commutative Property For Addition In Algebraic Properties Of Real Numbers

This property states that the order of adding numbers does not change its resultant sum.
i,e a+b=b+a
Example:
9+10=10+9
19=19

### Commutative Property For Multiplication In Algebraic Properties Of Real Numbers

This property states that the order of multiplying numbers does not change its resultant product.
i,e a × b=b × a
Example:
9 × 10=10 × 9
90=90

This property states that whichever way the numbers are grouped and added the resultant sum is the same.
(a+b)+c=a+(b+c)
Example:
(9+10)+11=9+(10+11)
30=30

### Associative Property For Multiplication

This property states that whichever way the numbers are grouped and multiplied the resultant product is the same.
(a . b) . c = a . (b . c)
Example:
(9 × 10) × 11=9 × (10 × 11)
990=990

### Distributive Property In Algebraic Properties Of Real Numbers

The property says when a sum is multiplied by a number, then all the numbers added should be multiplied by the number outside the parenthesis.
a × (b+c)=a × b+a × c (or) (b × c) × a=b × a+c × a

a+0=a
Example: 5+0=5,

### Multiplicative Identity Property

The value which multiplied with any number gives the same number is the multiplicative identity of that number.
a × 1=a
Example: 4 × 1=4
Here in multiplication ‘1’ is the multiplicative identity for any number multiplied.

The value which when added to the given number results to zero then that value is additive inverse of given number.
a+(-a)=0
Example: 5+(-5)=0
‘-5’ is the additive inverse of ‘5’

### Multiplicative inverse property

The value which when multiplied with a given number results to form ‘1’. Then, that value is called the multiplicative inverse of the given number.
a × 1⁄a = 1
Example:
9 × 1⁄9 = 1
Here ‘1⁄9’ is the multiplicative inverse of ‘9’.

### Zero Property Of Multiplication

Any value multiplied with ‘0’ results to zero only.
a×0=0
Example:
19×0=0

This property states the addition of the same value on both sides of equality does not change resultant value.

i,e
If a=b
a+c=b+c

### Subtraction Property For Equality

This property states the subtraction of the same value on both sides of equality does not change resultant value.

i,e
If a=b
Subtracting c on both sides
a-c=b-c

### Multiplication Property For Equality

The property says multiplying the same value on both sides of equality does not change the resultant value.

i,e
If a=b
Multiplying c on both sides
a×c=b×c

### Division Property For Equality

The property says Dividing the same value on both sides of equality does not change the resultant value.

i,e
If a=b
Dividing c on both sides
a ⁄ c =b ⁄ c

### Reflexive Property

Every real number is equal to itself. i,e a=a.

### Symmetric Property

These property states quantities on both sides of equity can be read or written in any order.
i,e a=b ⇒ b=a

### Transitive Property For Equality

The Property states when two real numbers are equal to the same number then, the two real numbers are also equal to each other.

If a=b and b=c
then a=b

### Law Of Trichotomy In Algebraic Properties Of Real Numbers

This law states when two real numbers are given, then they satisfy any one of these that is, they may be either equal (or) any one of them is greater than the other (or) less than the other number.

I,e if a and b are given
then a=b (or)
a>b (or)
a<b

Knowledge of these algebraic properties helps in how to multiply algebraic expressions and simplify them correctly.