What are Real Numbers?
Real numbers are all the numbers that can be represented on the number line. Real numbers are the set of all rational and irrational numbers. The symbol used for Real Numbers is R
Real numbers are used in mathematics, science, engineering, and everyday calculations.
Classification of Real Numbers
Given below is the classification of real numbers
Properties of Real Numbers
Properties of Real Numbers include Closure Property, Commutative Property, Associative Property, Distributive Property, Identity, Inverse property
Closure Property
If a and b are real numbers:
a + b is real
a − b is real
a × b is real
a ÷ b is real (if b ≠ 0)
Commutative Property
a + b = b + a
a × b = b × a
Example:
3 + 5 = 5 + 3
4 × 2 = 2 × 4
Associative Property
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
Distributive Property
a × (b + c) = ab + ac
Example:
3(4 + 5) = 3×4 + 3×5
= 12 + 15
= 27
Identity Elements
Additive identity:
a + 0 = a
Multiplicative identity:
a × 1 = a
Inverse Property
Additive inverse:
a + (-a) = 0
Multiplicative inverse:
a × 1/a = 1
Decimal Representation of Real Numbers
Types of decimals:
- Terminating decimal
1/2 = 0.5
3/4 = 0.75
- Non-terminating repeating decimal
1/3 = 0.333...
2/11 = 0.181818...
- Non-terminating non-repeating decimal (irrational)
√2 = 1.414213...
π = 3.141592...