## Number Sets

• Integers, denoted by $\Z$, are any whole number.
• Natural numbers, denoted by $\N$, are all non-negative integers and $0$.
• Rational numbers, denoted by $\mathbb{Q}$, are all numbers that can be written as a ratio $\frac ab, \space a,b\in\Z,\space b\not=0$..
• Irrational numbers, denoted by $\mathbb{P}$, are numbers that cannot be written as a ratio.
• Real numbers, denoted by $\R$, are any number.

### Prime Numbers

Prime numbers are positive integers with only two distinct positive factors.

• The only positive factors of a prime number $x$ are $1$ and $x$.
• $2$ is the only even prime number.
• $2$ is the smallest prime number.

Prime numbers can be used to factorize other numbers.

The Prime Factorization of a number $n$ is a way to write $n$ as a product of prime numbers.

$$\begin{array}{c} \text{Let n=64. What is the prime factorization of n?}\\ \hline\\ 64=2\cdot32=2\cdot2\cdot16\\ 2\cdot2\cdot16=2\cdot2\cdot2\cdot8=2\cdot2\cdot2\cdot2\cdot4\\ 2\cdot2\cdot2\cdot2\cdot4=2\cdot2\cdot2\cdot2\cdot2\cdot2\\\\ 64=2\cdot2\cdot2\cdot2\cdot2\cdot2 \end{array}$$

Goldbach's conjecture states that any even integer $x\geq4$ can can written as the sum of two primes.

The Twin Prime Conjecture states that there are infinitely sets of prime numbers $x_1,x_2$ such that $x_1+2=x_2$. These are called twin primes.

The opposite of prime numbers are composite numbers. A composite number is a positive integer which can be written as the product of two integers $y_1\geq2,y_2\geq2$.