Number properties quick reference
NUMBER SETS
Irrational numbers H
(H+ and H-)
Real numbers R
(R+ and R-)
Complex numbers C
Set / Symbol
Natural numbers N
N0 = N+{0}
Integers Z
(Z+ and Z-)
Rational numbers Q
(Q+ and Q-)
Description
Positive integers
Positive, negative integers and 0
Numbers of the form a/b, where a and b are integers, b≠0. These are fractions, finite decimals and infinite periodical (repeating) decimals
Infinite non-periodical decimals
Rational and Irrational numbers
Ordered pairs of real numbers
Examples
1,2,3,4, ...
The natural numbers, their opposite numbers and 0, e.g., 0, 1,2,3, ... -1, -2, -3, . . . (The Z comes from the German word “Zahl” – “Number”)
5/19, -11/7, 6.13, ..., 1/3=0.33333...
All real numbers that are not rational numbers, e.g.
=1.4142...
1,2,3,4, ...0, -1, -2, -3, . .
5/19, -11/7, 6.13, ..., 1/3=0.33333...
=1.4142...
5+6i
example: The number -11 belongs to the sets Z, Q,R.
The number 12.45 belongs to the sets Q,R.
The number 3.π belongs to the sets H,R.
NUMBER RULES REFERENCE:
Basic properties
a + (b + c) = (a+ b)+ c = a + b + c
a(bc) = (ab)c = abc
0 + a = a + 0 = a
1.a = a .1 = a
a + (-a) = (-a) + a = 0
a.a-1 = a-1.a = 1
a + b = b + a
ab = ba
a(b + c) = ab + ac
(a+b)(c+d) = ac+ad+bc+bd
a.0 = 0
If a.b = 0, then either a = 0 or b = 0
-(-a) = a
(-a)(-b) = ab
-ab = (-a)b = a(-b) = -(-a)(-b)
(-1)a = -a
Ordering properties:
a < b, if and only if b-a is positive
Absolute value properties:
|a| = ( a if a >= 0; -a if a < 0)
|a| is called absolute value of a
|a| ≥ 0
|-a|=|a|
|ab|=|a||b|
|a+b|≤|a|+|b|
Fractions properties:
Order of operations
"PEMDAS" - Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right)
If a < b, then a+c < b+c