## 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 $y_1\geq2,y_2\geq2$.