Number Theory

Fractions

A fraction has been simplified when the numerator and the denominator have no common factor greater than one.

\frac{28}{42} = \frac{2 \times 14}{3 \times 14} = \frac{2}{3}

\frac{4.5}{10.5} = \frac{45}{105} = \frac{3 \times 15}{7 \times 15} = \frac{3}{7}

Remember

to write a fraction as a decimal, divide the numerator by the demoninator;
to change a fraction, or a decimal, into a percentage, multiply by 100;
to change a percentage into a fraction, rewrite % as divided by 100.

\frac{3}{25} = \frac{12}{100} = 0.12

\frac{7}{20} = \frac{7 \times 100 %}{20} = 7 \times 5 % = 35 %

Directed Number

3 + 4 = 7

3 - 4 = -1

3 + -4 = 3 - 4 = -1

3 - -4 = 3 + 4 = 7

-3 + 4 = 1

-3 + -4 = -3 - 4 = -7

-3 \times -4 = 12

Order of Operation

The order in which operations are performed can be remember by the mnemonic

BIDMAS

Brackets - Indices - Division and/or Multiplication - Addition and/or Subtraction

This will help in remembering the correct priority of operations when completing calculations. Those on the lelf have priority over those on the right.

Common
Percentages, Decimals
&
Fractions

Sequences
& Series

In index notation, the number being multiplied by itself is called the base. The number written above the base is called the index or power or order.
For example: if the base is 3 and the index is 4 then

3 \times 3 \times 3 \times 3 = 3^{4}

Indices is the plural of index

The laws of indices

Some powers of 10 have a name called a prefix.

Prefix	Letter	Power	Number
tera	T	10¹²	1,000,000,000,000
giga	G	10⁹	1,000,000,000
mega	M	10⁶	1,000,000
kilo	k	10³	1,000
deci	d	10^-1	0.1
centi	c	10^-2	0.01
milli	m	10^-3	0.001
micro	μ	10^-6	0.000 001
nano	n	10^-9	0.000 000 001
pico	p	10^-12	0.000 000 000 001

See Units & Prefixes

Prime Factorization

Units & Prefixes: SI (or "Systeme Internationale d'Unites"); SI Prefixes; SI is Not the Only Measurement System

Set
Theory

Divisibility
Rules!

Multiplication
Practice

Number Systems
- Cuneiform Numbers