1.1 Real Numbers

Real Numbers

What number system have you been using most of your life?

*The real number system.

A real number is any number that has a decimal representation.

(i)     Real Numbers

Counting numbers (also called positive integers) N = { 1, 2, 3, …… }

Whole Numbers:

W={0}∪N={0,1,2,3,⋯}

(ii)    Integers

Natural numbers, their negatives, and 0.

Z = {……, –2, –1, 0, 1, 2, ……}

(iii)   Rational Numbers, Q

Numbers that can be represented as a/b, where 

a and b are integers and b ≠  0.

All rational number can be represented by:

(a)     terminating decimal numbers

such as 5/2 = 2.5, 1/2 = 0.5, −3/4 =−0.75

 (b)     non-terminating repeating decimal numbers 

          such as -2/3 = -0.666..., 2/15 = 0.1333...

(iv)   Irrational Numbers

Numbers which cannot be expressed as a ratio 
of two integers. They are non-terminating & non-
repeating decimal numbers.

(v)    Real Numbers, R

Rational and irrational numbers.

 Operation of Real Numbers

(i)    Commutative Law

•                  Addition : a + b = b + a

•                    Multiplication : ab = ba

(ii)   Associative Law

• Addition : a+(b+c)=(a+b)+c

• Multiplication : a(bc) =(ab)c

(iii)   Distributive Law

• a(b+c)=ab+ac

• 
  
• a(b−c)=ab−ac

(iv)   Identity Law

* Addition : a + 0 = 0 + a = a

a + identity = a

* Multiplication : a×1 = 1×a = a

a × identity = a

(v)  Inverse Law

       *Addition : a + (-a) = (-a) + a = 0

   a + inverse = identity

       *Multiplication : a×1/a= 1/a× a = 1

   a × inverse = identity

Real Number Intervals

For any two different real numbers, a and b, with a<b:

i. The open interval is defined as the set

(a, b) ={x: a<x<b}

ii. The closed interval is defined as the set

[a, b] ={x: a≤x≤b}