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Number Systems: Naturals, Integers, Rationals, Irrationals, Reals, and Beyond

The Natural Numbers

The natural (or counting) numbers are $1,2,3,4,5,$ etc. There are infinitely many natural numbers. The set of natural numbers, $\left\{1,2,3,4,5,\mathrm{...}\right\}$, is sometimes written $N$ for short.

The whole numbers are the natural numbers together with $0$.

(Note: a few textbooks disagree and say the natural numbers include $0$.)

The sum of any two natural numbers is also a natural number (for example, $4+2000=2004$), and the product of any two natural numbers is a natural number ($4\times 2000=8000$). This is not true for subtraction and division, though.

The Integers

The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.

$\left\{\mathrm{...},-5,-4,-3,-2,-1,0,1,2,3,4,5,\mathrm{...}\right\}$

The set of integers is sometimes written $J$ or $Z$ for short.

The sum, product, and difference of any two integers is also an integer. But this is not true for division... just try $1÷2$.

The Rational Numbers

The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions $\frac{1}{3}$ and $-\frac{1111}{8}$ are both rational numbers. All the integers are included in the rational numbers, since any integer $z$ can be written as the ratio $\frac{z}{1}$.

All decimals which terminate are rational numbers (since $8.27$ can be written as $\frac{827}{100}$.) Decimals which have a repeating pattern after some point are also rationals: for example,

$\mathrm{0.0833333....}=\frac{1}{12}$.

The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by $0$).

The Irrational Numbers

An irrational number is a number that cannot be written as a ratio (or fraction).  In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.

The first such equation to be studied was $2=x^2$. What number times itself equals $2$?

$\sqrt{2}$ is about $1.414$, because ${1.414}^2=1.999396$, which is close to $2$. But you'll never hit exactly by squaring a fraction (or terminating decimal). The square root of $2$ is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern:

$\sqrt{2}=\mathrm{1.41421356237309...}$

Other famous irrational numbers are the golden ratio, a number with great importance to biology:

$\frac{1+\sqrt{5}}{2}=\mathrm{1.61803398874989...}$

$\pi$ (pi), the ratio of the circumference of a circle to its diameter:

$\pi =\mathrm{3.14159265358979...}$

and $e$, the most important number in calculus:

$e=\mathrm{2.71828182845904...}$

Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation (like $\sqrt{2}$ and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. $\pi$ and $e$ are both transcendental.

The Real Numbers

The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers.  The real numbers are “all the numbers” on the number line.  There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers.  But, it can be proved that the infinity of the real numbers is a bigger infinity.

The "smaller", or countable infinity of the integers and rationals is sometimes called ${\aleph }_0$(alef-naught), and the uncountable infinity of the reals is called ${\aleph }_1$(alef-one).

There are even "bigger" infinities, but you should take a set theory class for that!

The Complex Numbers

The complex numbers are the set {$a+bi$ | $a$ and $b$ are real numbers}, where $i$ is the imaginary unit, $\sqrt{-1}$. (click here for more on imaginary numbers and operations with complex numbers).

The complex numbers include the set of real numbers.  The real numbers, in the complex system, are written in the form $a+0i=a$. a real number.

This set is sometimes written as $C$ for short. The set of complex numbers is important because for any polynomial $p\left(x\right)$ with real number coefficients, all the solutions of $p\left(x\right)=0$ will be in $C$.

Beyond...

There are even "bigger" sets of numbers used by mathematicians. The quaternions, discovered by William H. Hamilton in $1845$, form a number system with three different imaginary units!