Math

A collection of basic mathematics principles for practice and quick reference.

A
- associative property of addition
- associative property of multiplication
C
D
- distributive property of addition
I
N
- natural numbers
- number systems
R
- rational numbers
- real numbers

Number systems

Numbers can be classified into sets

See Number system on Wikipedia.

Natural Numbers

Can include \mathit{0} or not, both are used.

\mathbb{N}_0 = \left\lbrace 1, 2, 3, \dots \right\rbrace

\mathbb{N}_1 = \left\lbrace 0, 1, 2, 3, \dots \right\rbrace

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Integers

\mathbb{Z} = \left\lbrace -3, -2, -1, 0, 1, 2, 3 \right\rbrace

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Rational numbers

\mathbb{Q} = \frac{a}{b} where a and b are integers and b is not 0.

Rational numbers in decimal form either terminate or repeat. For example \frac{3}{4} = 0.75 (terminating) or \frac{1}{3} = 0.3333\dots (repeating).

Repeating sequences can be more than one digit. These are represented in standard notation with an overbar. For example, \frac{1}{7} = 0.142857142857\ldots = 0.\overline{142857}.

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Irrational numbers

Irrational numbers cannot be written as a simple fraction. Their decimal forms continue endlessly without a repeating pattern.

Examples:

\pi = 3.1415926535\ldots (It never ends and never repeats.)
\sqrt{2} = 1.41421\ldots (Square roots of non-perfect squares are often irrational.)
e (Euler’s constant) = 2.71828 \ldots

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Real numbers

\mathbb{R} = Any single point on the number line. The continuous number line contains all natural numbers, integers, rational numbers, and irrational numbers.

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}

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