Real number: A real number is a value that represents a quantity along a continuous line. The real numbers include number we normally use, such as 1, 15.82, -0.1, 3/4, etc…Positive or negative, large or small, whole numbers or decimal numbers are called Real Numbers.

Let a, b, and c represents real numbers.

- Closure properties of Addition; a=b is a real number.
- Closure propertied of multiplication; ‘ab’ is a real number.
- Commutative property of Addition; a + b = b + a.
- Commutative Property of Multiplication; a . b=b.a
- Associative Property of Addition; (a + b) +c = a = ( b + c)
- Associative Property of Multiplication; (ab)c = a(bc)
- Distributive Properties; a(b +c )= ab+ ac

(a + b)c =ac + bc - Multiplicative Identity Property a . 1 = 1. a =a

Irrational number is any number which can’t be expressed as the ratio of integers. This type of real number cannot represented as repeating decimals or terminating. Example for irrational number: Euler's number e, the golden ratio φ, he ratio of a circle's circumference to its diameter π.

Example: We prove that √2 is irrational. Assume to the contrary that √(2 ) is rational, that is √2 =P/q

Where p and q are integers and q ≠ 0. Moreover, let p and q have no common divisor > 1. Then

2=p2/q2 ⇒ 2q^{2}=p^{2} (1)

Since 2q^{2} is even, it follows that p^{2} is even. Then p is also even (in fact, if p is odd, then p^{2} Is odd). This means that there exists k ∈ Z such that

p=2k (2)

Substituting (2) into (1), we get

2q^{2} = (2k)^{2} ⇒ 2q^{2} = 4k^{2} ⇒ q^{2}=2 = 2k^{2}

Since 2k^{2} is even, it follows that q^{2} is even. Then q is also even. This is a contradiction.

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