Number properties quick reference

NUMBER SETS

 

Set / Symbol

Description

Examples

Natural numbers N

N0 = N+{0}

Positive integers

1,2,3,4, ...

Integers Z

(Z+ and Z-)

Positive, negative integers and 0

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”)

Rational numbers Q

(Q+ and Q-)

Numbers of the form a/b, where a and b are integers, b≠0. These are fractions, finite decimals and infinite periodical (repeating) decimals

5/19, -11/7, 6.13, ..., 1/3=0.33333...

Irrational numbers H

(H+ and H-)

Infinite non-periodical decimals

All real numbers that are not rational numbers, e.g.

=1.4142...

Real numbers R

(R+ and R-)

Rational and Irrational numbers

1,2,3,4, ...0, -1, -2, -3, . .

5/19, -11/7, 6.13, ..., 1/3=0.33333...

=1.4142...

Complex numbers C

Ordered pairs of real numbers

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

If a < b, then a+c < b+c

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)

 

 


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