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.
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 ⇒ 2q2=p2 (1)
Since 2q2 is even, it follows that p2 is even. Then p is also even (in fact, if p is odd, then p2 Is odd). This means that there exists k ∈ Z such that
Substituting (2) into (1), we get
2q2 = (2k)2 ⇒ 2q2 = 4k2 ⇒ q2=2 = 2k2
Since 2k2 is even, it follows that q2 is even. Then q is also even. This is a contradiction.