Number sets

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.

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Real Number Line

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The Cistercian numerals

The Cistercian numerals are a forgotten number system, developed by the Cistercian monastic order in the early XIII century.

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Real Number Set Diagram

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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 2i and i√2.

Algebraic

The roots of polynomials, such as ax³ + bx² + cx + d = 0, with integer (or rational) coefficients. Algebraic numbers may be real, imaginary, or complex. For example, the roots of x² − 2 = 0 are ±√2, the roots of x² + 4 = 0 are ±2i, and the roots of x² −4x +7 = 0 are 2±i√3.

Complex

Complex numbers, such as 2+3i, 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.

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Complex Number Plane

z = x + iy, i = √−1

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Complex Number Set Diagram

Note: This is an Euler diagram.

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Properties of the Number Sets

	Nat­ural	Int­eger	Ra­tion­al
Closed under Addition1	●	●	●
Closed under Multiplication1	●	●	●
Closed under Subtraction1		●	●
Closed under Division1			●
Dense2			●
Complete (Continuous)3
Algebraically Closed4

	Real	Alge­braic	Com­plex
Closed under Addition1	●	●	●
Closed under Multiplication1	●	●	●
Closed under Subtraction1	●	●	●
Closed under Division1	●	●	●
Dense2	●	●	●
Complete (Continuous)3	●		●
Algebraically Closed4		●	●

1. Closed under addition (multiplication, subtraction, division) means the sum (product, difference, quotient) of any two numbers in the set is also in the set.
2. Dense: Between any two numbers there is another number in the set.
3. Continuous with no gaps. Every sequence that keeps getting closer together (Cauchy sequence) will converge to a limit in the set.
4. 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.

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∞ 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.

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