# 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**

**N**_{0}** = 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*