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
Associative Property For Addition
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
Additive Identity In Algebraic Properties
Any number added to additive identity gives the same number.
a+0=a
Example: 5+0=5,
Here ‘0’ is the additive identity in addition to any number added.
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.
Additive Inverse Property
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
Addition Property For Equality
This property states the addition of the same value on both sides of equality does not change resultant value.
i,e
If a=b
adding c on both sides
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.
| Algebraic Expressions | Algebra Multiplication Rules |
| Algebraic Equations List | AP and GP |
