Common Number Sets

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Natural Numbers
The whole numbers from 1 upwards. (Or from 0 upwards in some fields of mathematics).
The set is {1,2,3,...} or {0,1,2,3,...}

(Q is from the Italian "Quoziente" meaning Quotient, the result of dividing one number by another.)

Irrational Numbers
Any real number that is not a Rational Number.

Integers
The whole numbers, {1,2,3,...} negative whole numbers {..., -3,-2,-1} and zero {0}. So the set is {..., -3, -2, -1, 0, 1, 2, 3, ...}

Algebraic Numbers
Any number that is a solution to a polynomial equation with rational coefficients.
Includes all Rational Numbers, and some Irrational Numbers.

(Z is from the German "Zahlen" meaning numbers, because I is used for the set of imaginary numbers).

Transcendental Numbers
Any number that is not an Algebraic Number
Examples of transcendental numbers include π and e.

Rational Numbers
The numbers you can make by dividing one integer by another (but not dividing by zero). In other words fractions.
Q is for "quotient" (because R is used for the set of real numbers).
Examples: 3/2 (=1,5), 8/4 (=2), 136/100 (=1,36), -1/1000 (=-0,001)

Real Numbers
All Rational and Irrational numbers. They can also be positive, negative or zero.
Includes the Algebraic Numbers and Transcendental Numbers.
Also see Real Number Properties

(Q is from the Italian "Quoziente" meaning Quotient, the result of dividing one number by another.)

Irrational Numbers
Any real number that is not a Rational Number.

Algebraic Numbers
Any number that is a solution to a polynomial equation with rational coefficients.
Includes all Rational Numbers, and some Irrational Numbers.

Transcendental Numbers
Any number that is not an Algebraic Number
Examples of transcendental numbers include π and e.

Real Numbers
All Rational and Irrational numbers. They can also be positive, negative or zero.
Includes the Algebraic Numbers and Transcendental Numbers.
Also see Real Number Properties

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A simple way to think about the Real Numbers is: any point anywhere on the number line (not just the whole numbers).
Examples: 1,5, -12,3, 99, √2, π
They are called "Real" numbers because they are not Imaginary Numbers. 

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The values a and b can be zero, so the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers.
Examples: 1 + i, 2 - 6i, -5,2i, 4

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Imaginary Numbers
Numbers that when squared give a negative result.
If you square a real number you always get a positive, or zero, result. For example 2×2=4, and (-2)×(-2)=4 also, so "imaginary" numbers can seem impossible, but they are still useful!
Examples: √(-9) (=3i), 6i, -5,2i
The "unit" imaginary numbers is √(-1) (the square root of minus one), and its symbol is i, or sometimes j.
i2 = -1

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Complex Numbers
A combination of a real and an imaginary number in the form a + bi, where a and b are real, and i is imaginary.

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The values a and b can be zero, so the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers.
Examples: 1 + i, 2 - 6i, -5,2i, 4

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Illustration
Natural numbers are a subset of Integers
Integers are a subset of Rational Numbers
Rational Numbers are a subset of the Real Numbers
Combinations of Real and Imaginary numbers make up the Complex Numbers.

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Number Sets In Use
Here are some algebraic equations, and the number set needed to solve them:

Other Sets
We can take an existing set symbol and place in the top right corner:
a little + to mean positive, or
a little * to mean non zero, like this:

By Muchait DEGIRMENCI
Student




Kadri Saman MTSO Vocational and Technical Anatolian High School

By Muchait DEGIRMENCI
Student




Kadri Saman MTSO Vocational and Technical Anatolian High School