The set of natural numbers is a subset of the set of integers, which is a subset of the set of rationals, which is a subset of the set of real numbers, which is a subset of the set of complex numbers.
This is represented mathematically as N ⊂ Z ⊂ Q ⊂ R ⊂ C. The various number systems can be represented by the Venn diagram below:
C R Q Z N
_ −3
2 –11 2
− 31 __
4 0 4 6 − 38
1
3 + 2i Complex numbers are so important that Section 6 of this book is devoted to them.
ACTION ACTIVITY
ACTION Working with absolute values
OBJECTIVE To position various numbers on the number line
| − 1 _
| √ | = √
__ 2
2
1.6 Absolute value Y 7
–5 –4
The absolute value or modulus | x | of a real number x is simply its distance to the origin.
–3 –2 –1
| −3 | = 3 [The distance of –3 to 0.] | 3 | = 3 [The distance of 3 to 0.]
| = 1 _
2 [The distance of − 1 _
__ 2
| −7⋅35 | = 7⋅35 EXERCISE 5
1. Evaluate: (a) | −72 |
(b)
| − 69 ___
4 | (c) | 1·72 | (d) (e)
| | |
___ √
1
__ 2
(f ) | 0 | (g) | − π | − 3 √
____ 4
__ 2
| (h) | |
__ 6
5
2. Evaluate: (a) | 3 − 5 | (b) | 5 − 3 | (c) | √
__ 2 − 1 |
(d) | π − 3 |
(e) | 3 − π | (f) | √
__ 2 − √
__ 3 |
(g) | a − b | if a > b (h) | a − b | if b > a
27 2 to 0.] 0 1 2 3 4 5
The modulus is never negative as it is a distance.
| 7 − 3 | = | 4 | = 4 | 3 − 7 | = | −4 | = 4 This means | a − b | = | b − a |
