II Numbers

Natural Numbers

The numbers used to count objects, defined $\natnums = {1,2,3,4,… }$

Integers and Absolute Value

Integers are whole numbers ranging from negative infinite to positive infinite including the 0.

$\mathbb{Z} = { …,-3,-2,-1,0,1,2,3,… }$

$-(-x) = x$
$-(y) = -y$
$7 -(-9) = 7 + 9$

A number's absolute value is its distance from 0. It's written as a number between pipes: $|-5| = 5$, $|7| = 7$.

Rules for integer addition and subtraction:
Subtraction problems are first turned into addition problems by putting the subtracted amount into parantheses and prefixing those with a plus: $3 - 4 = 3 + (-4)$. If both operands share the same sign, they are to be added and the sign kept. If their signs differ, the smaller number is to be subtracted from the larger one and the larger number's sign is kept for the result.
$4 + (-18) = -14$
$-2 + (-5) = -7$

If an expression consists of more than two terms, individual terms are to be worked from one after another. e.g.
$(-2) + 4 + (-11)$
$= 2 + (-11)$
$= -9$

Multiplication and division have to be calculated first:
$-7 + 3(2) = -7 + 3 \times 2$
$-7 + 6 = -1$

Alternatively one can sum up all positive and negative integers individually before subtracting the latter from the former: $7 + (-12) + (-3) + 8 = 15 - 15$

Rules for integer multiplication and division: If both operands share the same sign, the result ist positive. Otherwise the result is negative. e.g.
$(-2) \times (-2) = 4$
$2 \times -5 = -10$ (the same applies for division problems)

Unary minus is basically short for $n \times -1$:
$(-6)^2 = (-6)(-6) = 36$
$-6^2 = -1 \times 6^2 = -36$
$-(-9) = -1 \times -9 = 9$

Linear equations have exactly one variable raised to the power of 1 (same as no exponent). They can always be written in the form $ax + b = 0$. Quadratic equations have the variable raised to a power of 2 or more. They possess as many solutions as the variables' power is high.

The order of simplification for these expressions is: 1. distribute, 2. combine like terms, 3. eliminiate fractions.

Whole Numbers

Ambiguous term referring to either one of the two above mentioned number sets.

Rational Numbers

All numbers that can be expressed as fractions with a non-zero denominator, including the range of integers.

$\mathbb{Q} = { x|x = \frac p q \land p \in \mathbb{Z} \land q \in \mathbb{Z} \land q \ne 0 }$

This set excludes numbers with non-breaking decimals and repeating decimals.

Real Numbers

As above but includes numbers with repeating and non-breaking decimals.