Number Terminology

1. Prime Numbers - A number that can't be divided/grouped by any other number except 1 and itself. Examples are 2, 3, 5, 7, 11, 13, etc.
2. Composite Numbers - A number that can be divided by numbers other than 1 and itself. Examples are 4, 6, 8, 9, 10, etc. This is nothing but a product of prime numbers.
3. Factors - Numbers that can be multiplied together to get another number. It means, the factors that result in the number. For example, the factors of 6 are 1, 2, 3, and 6 because: 1 * 6 = 6 and 2 * 3 = 6
4. Multiples - Numbers that can be divided by a given number without leaving a remainder. For example, the multiples of 3 are 3, 6, 9, 12, etc.
5. Rational numbers - Numbers that are ratio of whole numbers. For example, 1/2, 3/4, 5, –2, etc.
6. Irrational numbers - Numbers that can't be expressed as a ratio of two integers. Examples include $\sqrt{2}$, $\pi$, $e$.
7. Real numbers - All rational and irrational numbers combined.
8. Imaginary numbers - Numbers that involve the square root of a negative number, denoted by 'i' where $i^2$ = –1.
9. Whole numbers - All natural numbers including zero (0, 1, 2, 3, ...).
10. Natural numbers - Positive integers starting from 1 (1, 2, 3, ...). The trick is to remember that natural numbers are counting things naturally, meaning things that exist. Hence 0 isn't part of it.
11. Integers - All whole numbers and their negative counterparts (..., –3, –2, –1, 0, 1, 2, 3, ...).
12. Factorial - The product of all positive integers less than or equal to a given number. For example, $n!$ = 5 × 4 × 3 × 2 × 1 = 120.

Factors vs Multiples

• Factors divides number into groups of different sizes.
• Multiples are increasing groups by the same size at each step.

prime factors vs regular factors

Prime Factorization is the process of expressing a composite number as the product of its prime factors. For example, the prime factorization of 28 is 2 × 2 × 7.

Even if the factors are only big numbers, we will keep breaking them down until we reach prime numbers.

Rational vs Irrational Numbers

It's best to understand this with examples and the method used to convert decimals to fractions. It's clear that the terminating decimals are proper rational numbers. But in the below examples, we see how any repeating decimals are also rational numbers.

irrational numbers can't be fractions

Irrational numbers are can never be represented as fractions of two integers. They have non-terminating and non-repeating decimal expansions.

$$\begin{aligned} x &= 0.\overline{6} \\ 10x &= 6.\overline{6} \\ 10x - x &= 6.\overline{6} - 0.\overline{6} & \text{(subtract x from both sides. Value of x taken from first line.)}\\ 9x &= 6 \\ x &= \frac{6}{9} = \frac{2}{3} \end{aligned}$$ $$\begin{aligned} x &= 0.\overline{27} \\ 100x &= 27.\overline{27} \\ 100x - x &= 27.\overline{27} - 0.\overline{27} & \text{(subtract x from both sides. Value of x taken from first line.)} \\ 99x &= 27 \\ x &= \frac{27}{99} = \frac{3}{11} \end{aligned}$$

Square roots as middle point

When we build factor pairs for a number, both sides of the factor pair will have one number greater than the square root and other number less than the square root.

square root tricks

• Any factor bigger than $\sqrt{N}$ will have its pair factor smaller than $\sqrt{N}$.
• If no number $\leq \sqrt{N}$ divides N, then nothing will and N is a prime number.

$$\begin{aligned} N = 36 \\ \sqrt{N} = 6 \\ \text{all factor pairs of 36.} \\ 1 × 36 \\ 2 × 18 \\ 3 × 12 \\ 4 × 9 \\ 6 × 6 \\ \text{LHS increases and stops at } \sqrt{N} \\ \text{RHS decreases and stops at } \sqrt{N} \\ \end{aligned}$$