Real numbers and Irrational Numbers Assignment Help
Introduction to the Real number
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, etcPositive or negative, large or small, whole numbers or decimal numbers are called Real Numbers.
Properties of 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 cant 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 , he ratio of a circle's circumference to its diameter .
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.