2.1 Number Systems

Whole numbers (W) are the natural numbers plus the number 0.

W = { 0, 1, 2, 3, 4, 5, 6, … }

Integers (I) are all the whole numbers and their negatives.

I = { …, –4, –3, –2, –1, 0, 1, 2, 3, 4, … }

Rational numbers (Q) are numbers that can be expressed as a fraction of two integers , where . –4.5, 11, 0, 0.52,  , and are all examples of rational numbers. All rational numbers can be expressed as either a terminating decimal or a repeating decimal as seen in the examples below:

= 0.75, which is a terminating decimal
= , which is a repeating decimal 
= , which is a repeating decimal 

Irrational numbers () are numbers that cannot be represented by the fraction of two integers. π, , , and are all examples of irrational numbers. All irrational numbers can be expressed as non-terminating and non-repeating decimals as seen in the examples below:

π 3.14159265358979323…, which is a non-terminating and non-repeating decimal
1.73205…, which is a non-terminating and non-repeating decimal

Real numbers (R) are all the rational numbers and all the irrational numbers.

The relationship between the sets of numbers can be illustrated in a diagram:

Practice 1

Identify whether each number is rational or irrational.

a)

b) π

c)

d) 5.623

e)

Practice 2

Identify all the sets of numbers that the following numbers belong to. Write the answers from the largest set to the smallest set.

a) 8

b) 1.95856…

c)

d) 0

e)

f) –63

g) 0.40989898…

h)

i) 0.78

j)

Homework

1. Identify whether each number is rational or irrational.

   a)

   b)

   c) –2.356

   d) 3.370370…

   e)

   f)

   g) –

   h)

   i) 56.9645102…

   j)

2. Identify all the sets of numbers that the following numbers belong to. Write the answers from the largest set to the smallest set.

   a) 0

   b) π

   c) –11

   d) 47

   e) 8.23

   f) 7.8945…

   g)

   h) –

   i)

   j) 21.0

   k)

   l) –9.7

   m) 0.6666…

   n)

   o)

   p) –1

   q)

   r)

   s)

   t) 55

   u)

   v) 3.14

   w) 1

   x)

3. Which of the following statements is true?

   a) All integers are included in the set of whole numbers. b) All natural numbers are also integers. c) Irrational numbers are not real numbers. d) Rational numbers cannot have decimal representations.

4. Which of the following statements defines a rational number?

   a) Any number that is a non-repeating, non-terminating decimal. b) Any number that can be expressed as a finite decimal. c) Any number that can be expressed as , where a and b are integers and b ≠ 0. d) Any number that can be expressed as , where a and b are natural numbers and b ≠ 0.

5. Cross out the number(s) below that are not rational, then arrange the rational numbers from smallest to largest.

   ,  , 0, 6.3, , , , –1.5

    

Answers

2.

a) R, Q, I, W	b) R,	c) R, Q, I	d) R, Q, I, W, N
e) R, Q	f) R,	g) R, Q	h) R, Q, I
i) R,	j) R, Q, I, W, N	k) R, Q, I	l) R, Q
m) R, Q	n) R, Q, I, W	o) R, Q	p) R, Q, I
q) R, Q	r) R, Q	s) Not real	t) R, Q, I, W, N
u) R, Q, I, W, N	v) R, Q	w) R, Q, I, W, N	x) R, Q, I, W, N

3. b

4. c

5. –1.5, , 0, , , 6.3