Number Sets

Note: The definitions below are fine for the New Zealand and Cambridge syllabuses. There are some variations around the world.

Natural Numbers

The symbol for the set of natural numbers is blackboard bold N: `bbb{N}`.

Natural numbers are the set of numbers from 1 upward. These are the numbers naturally used for counting. Hence, they are also called counting numbers.

`bbb{N} = {1, 2, 3, 4, 5, ...}`

Whole Numbers

The symbol for the set of whole numbers is blackboard bold W: `bbb{W}`.

`bbb{W} = {0, 1, 2, 3, 4, 5, ...}`

The set of whole numbers are the natural numbers including zero.

` bbb{W} = bbb{N} cup {0}`

The natural numbers are a subset of whole numbers.

`bbb{N} sub bbb{W}`

Integers

The symbol for the integers set is blackboard bold Z: `bbb{Z}` (from zahlen, the German word for numbers).

The set of integers includes all whole numbers, `bbb{W}`, and negative versions of all of them as well.

`bbb{Z} = {... -3, -2, -1, 0, 1, 2, 3, ...}`

The whole numbers are a subset of integers.

`bbb{N} sub bbb{W} sub bbb{Z}`

Rational Numbers

The symbol for the rational numbers is blackboard bold Q: `bbb{Q}` (from quotient, the result of a division).

The set of rational numbers includes any number that can be expressed as a fraction of one integer divided by another integer. The set it thus often simply called fractions.

`bbb{Q} = {3/1, 1/2, 2/3, 3/-1, 73/3, -43/-44, 1/-137, ...}`

Integers is a subset of rational numbers.

`bbb{N} sub bbb{W} sub bbb{Z} sub bbb{Q}`

When written as a decimal, some fractions are very easy to write because they terminate (their decimal places finish), while some never end, repeating one or more digits.

¹/₂ = 0.5

¹/₇ = 0.142857142857...

¹/₃ = 0.333...

²/₃ = 0.666...

³/₃ = 0.999...

∴ 0.999... ≡ 1

(The symbol ∴ means "therefore" and the symbol ≡ means "equivalent" or "is the same as", not just "has an equal value to".)

For more on rational numbers see the Fractions Overview page.

Irrational Numbers

There is no fixed symbol for irrational numbers. `bbb{R\\Q}` (meaning the set of real numbers "set minus" the set of rational numbers) is probably the most common and the most likely to be understood.

Several famous constants (and their negative versions) are irrational numbers.

`bbb{Q} = {-1/sqrt2, sqrt3/2, sqrt2, -phi, sqrt3, e, -pi, e^pi, ...}`

Irrational numbers are numbers which when written as a decimal, never end and never repeat. They can therefore not be expressed as fractions of integers, so rational numbers and irrational numbers are separate sets. Their intersection is the empty set.

`bbb{Q} cap (bbb{R\\Q}) = {} = O/`

This is the first number set in this list which does not include all of the preceding set; it instead includes none of the preceding sets.

`bbb{Q} cancel(sub) (bbb{R\\Q})`

`bbb{N} cancel(sub) (bbb{R\\Q})`

Real Numbers

The symbol for the set of real numbers is blackboard bold R: `bbb{R}`.

The set of real numbers is the combined sets of rational numbers and irrational numbers together, which are complementary sets (in the space of the real numbers).

` bbb{R} = bbb{Q} cup (bbb{R\\Q})`

The rational numbers and irrational numbers are therefore both subsets of real numbers.

`bbb{N} sub bbb{W} sub bbb{Z} sub bbb{Q} sub bbb{R}`

`(bbb{R\\Q}) sub bbb{R}`

This set includes any number which can be placed on a number line.

Complex Numbers

The symbol for the complex numbers is blackboard bold C: `bbb{C}`.

This set of numbers starts to get particularly interesting. Complex numbers are "complex" because they have a real component and an imaginary component, so that a complex number is written with two terms as a + bi, where a and b are real numbers and i is the imaginary unit.

`bbb{C} = {a+bi | a, b in bbb{R}; i^2=-1}`

Note that there are two solutions to `i^2=-1`, which are `i` and `-i`. We have no way of telling them apart, but it doesn't actually matter as long as we're consistent with which one we use.

Because of the two components in a complex number, two number lines are required to place them. With the number lines at right angles to each other, they make the real and imaginary axes of an Argand diagram (which looks similar to the x and y axes of a Cartesian plane).

Because the real number line is part of the set of complex numbers, real numbers are a subset of complex numbers.

`bbb{N} sub bbb{W} sub bbb{Z} sub bbb{Q} sub bbb{R} sub bbb{C}`

See more about complex numbers on the Complex Numbers page.