Natural, N
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, Z
Integers are the natural numbers and their negatives {... −3, −2, −1, 0, 1, 2, 3, ...}. (Z is from German
Zahl, "number".)
Rational, Q
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, AR
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, R
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 Venn Diagram
N ⊂
Z ⊂
Q ⊂
AR ⊂
R
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, A
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 ±2
i, and the roots of
x2 −4
x +7 = 0 are 2±
i√3.
Complex, C
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.