The whole numbers from 1 upwards. (Or from 0 upwards in some fields of mathematics). Read More ->

The set is 1,2,3,... Or 0,1,2,3,...

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The whole numbers, 1,2,3,... Negative whole numbers ..., -3,-2,-1 & zero 0. So the phối is ..., -3, -2, -1, 0, 1, 2, 3, ...


(Z is from the German "Zahlen" meaning numbers, because I is used for the mix of imaginary numbers). Read More ->

Rational Numbers

The numbers you can make by dividing one integer by another (but not dividing by zero). In other words fractions. Read More ->

Q is for "quotient" (because R is used for the phối of real numbers).

Examples: 3/2 (=1.5), 8/4 (=2), 136/100 (=1.36), -1/1000 (=-0.001)

(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. Read More ->


Algebraic Numbers

Any number that is a solution lớn a polynomial equation with rational coefficients.

Includes all Rational Numbers, và some Irrational Numbers. Read More ->

Transcendental Numbers

Any number that is not an Algebraic Number

Examples of transcendental numbers include π & e. Read More ->

Real Numbers

Any value on the number line:

Can be positive, negative or zero.Can be Rational or Irrational.Can be Algebraic or Transcendental.Can have infinite digits, such as 13 = 0.333...

Also see Real Number Properties

They are called "Real" numbers because they are not Imaginary Numbers. Read More ->


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, & (-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), & its symbol is i, or sometimes j.

i2 = -1


Complex Numbers

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

The values a & b can be zero, so the set of real numbers và the set of imaginary numbers are subsets of the mix of complex numbers.

Examples: 1 + i, 2 - 6i, -5.2i, 4





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 & Imaginary numbers biến hóa the Complex Numbers.

Number Sets In Use

Here are some algebraic equations, and the number set needed lớn solve them:

Equation Solution Number phối Symbol
x − 3 = 0 x = 3 Natural Numbers
x + 7 = 0 x = −7 Integers
4x − 1 = 0 x = ¼ Rational Numbers
x2 − 2 = 0 x = ±√2 Real Numbers
x2 + 1 = 0 x = ±√(−1) Complex Numbers

Other Sets

We can take an existing phối symbol and place in the top right corner:

a little + khổng lồ mean positive, or a little * to lớn mean non zero, lượt thích this:
Set of positive integers 1, 2, 3, ...
Set of nonzero integers ..., -3, -2, -1, 1, 2, 3, ...

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And we can always use set-builder notation.