# Numbers

## Real Number Sets

### Natural

Natural numbers are the counting numbers {1, 2, 3, ...} (positive integers) or the whole numbers {0, 1, 2, 3, ...} (the non-negative integers). Mathematicians use the term "natural" in both cases.

### Integer

Integers are the natural numbers and their negatives {... −3, −2, −1, 0, 1, 2, 3, ...}.
(Z is from German *Zahl*, "number".)

### Rational

Rational numbers are the ratios of integers, also called fractions, such as 1/2 = 0.5 or 1/3 = 0.333... Rational decimal expansions end or repeat. (Q is from quotient.)

### Real Algebraic

The real subset of the algebraic numbers: the real roots of polynomials. Real algebraic numbers may be rational or irrational. √2 = 1.41421... is irrational. Irrational decimal expansions neither end nor repeat.

### Real

Real numbers are all the numbers on the continuous number line with no gaps. Every decimal expansion is a real number. Real numbers may be rational or irrational, and algebraic or non-algebraic (transcendental). π = 3.14159... and e = 2.71828... are transcendental. A transcendental number can be defined by an infinite series.

## Real Number Line

## Real Number Set Diagram

## Complex Number Sets

### Imaginary

Imaginary numbers are numbers whose squares are negative. They are the
square root of minus one, *i* = √−1, and all
real number multiples of *i*, such as 2*i* and *i*√2.

### Algebraic

The roots of polynomials, such as *ax*^{3} + *bx*^{2} + *cx* + *d* = 0,
with integer (or rational) coefficients. Algebraic numbers may be real,
imaginary, or complex. For example, the roots of *x*^{2} − 2 = 0
are ±√2, the roots of *x*^{2} + 4 = 0
are ±2*i*, and the roots of *x*^{2} −4*x* +7 = 0
are 2±*i*√3.

### Complex

Complex numbers, such as 2+3*i*, have the form *z* = *x* + *iy*,
where *x* and *y* are real numbers. *x* is called the
real part and *y* is called the imaginary part. The set of complex
numbers includes all the other sets of numbers. The real numbers are
complex numbers with an imaginary part of zero.

## Complex Number Plane

*z* = *x* + *iy*, *i* = √−1

## Complex Number Set Diagram

Note: This is an Euler diagram, not a Venn diagram.

## Properties of the Number Sets

Natural | Integer | Rational | Real | Algebraic | Complex | |
---|---|---|---|---|---|---|

Closed under Addition^{1} |
x | x | x | x | x | x |

Closed under Multiplication^{1} |
x | x | x | x | x | x |

Closed under Subtraction^{1} |
x | x | x | x | x | |

Closed under Division^{1} |
x | x | x | x | ||

Dense^{2} |
x | x | x | x | ||

Complete (Continuous)^{3} |
x | x | ||||

Algebraically Closed^{4} |
x | x |

- Closed under addition (multiplication, subtraction, division) means the sum (product, difference, quotient) of any two numbers in the set is also in the set.
- Dense: Between any two numbers there is another number in the set.
- Continuous with no gaps. Every sequence that keeps getting closer together (Cauchy sequence) will converge to a limit in the set.
- Every polynomial with coefficients in the set has a root in the set.

The complex numbers are the algebraic completion of the real numbers. This may explain why they appear so often in the laws of nature.

## Infinity ∞

The integers, rational numbers, and algebraic numbers are countably infinite, meaning there is a one-to-one correspondence with the counting numbers. The real numbers and complex numbers are uncountably infinite, as Cantor proved.