Reds: Real number sets  Blues: Imaginary number sets  Purples: Complex number sets 
Natural, NNatural numbers are the counting numbers {1, 2, 3, ...} (positive integers) or the whole numbers {0, 1, 2, 3, ...} (the nonnegative integers). Mathematicians use the term "natural" in both cases. 
Integer, ZIntegers are the natural numbers and their negatives {... −3, −2, −1, 0, 1, 2, 3, ...}. (Z is from German Zahl, "number".) 
Rational, QRational 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, A_{R}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, RReal 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 nonalgebraic (transcendental). π = 3.14159... and e = 2.71828... are transcendental. A transcendental number can be defined by an infinite series. 
N ⊂ Z ⊂ Q ⊂ A_{R} ⊂ R
ImaginaryImaginary 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, AThe 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 ±2i, and the roots of x^{2} −4x +7 = 0 are 2±i√3. 
Complex, CComplex 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. 
z = x + iy, i = √−1
N ⊂ Z ⊂ Q ⊂ A_{R} ⊂ R ⊂ C
Natural N 
Integer Z 
Rational Q 
Real R 
Algebraic A 
Complex C 


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 
The complex numbers are the algebraic completion of the real numbers. This may explain why they appear so often in the laws of nature.
The integers, rational numbers, and algebraic numbers are countably infinite, meaning there is a onetoone correspondence with the counting numbers. The real numbers and complex numbers are uncountably infinite, as Cantor proved.
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