### 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. 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... 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)