The Natural Numbers

The natural (or counting) numbers are 1,2,3,4,5, etc. There are infinitelymany natural numbers. The set of natural numbers, 1,2,3,4,5,... ,is sometimes written N for short.

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The whole numbers are the natural numbers together with 0.

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

The sum ofany two natural numbers is also a natural number (for example, 4+2000=2004), and the product of any two natural numbersis a natural number (4×2000=8000). Thisis 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.


The set of integers is sometimeswritten J or Z for short.

Thesum, 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 arethose numbers which can be expressed as a ratio betweentwo integers. For example, the fractions 13 and −11118 are bothrational numbers. All the integers are included in the rational numbers,since any integer z can be written as the ratio z1.

All decimals which terminate are rational numbers (since 8.27 can be written as 827100.) Decimalswhich have a repeating pattern after some point are also rationals:for example,

The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, theirsum, 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. Theancient Greeks discovered that not all numbers are rational; thereare equations that cannot be solved using ratios of integers.

The first such equationto be studied was 2=x2. Whatnumber times itself equals 2?

2 isabout 1.414, because 1.4142=1.999396, which is close to2. But you"ll never hit exactly by squaring a fraction (or terminatingdecimal). The square root of 2 is an irrational number, meaning itsdecimal equivalent goes on forever, with no repeating pattern:


Other famous irrationalnumbers are the golden ratio, a number with greatimportance to biology:

π (pi), theratio of the circumference of a circle to its diameter:


and e,the most important number in calculus:


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

TheReal 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 andrationals is sometimes called ℵ0(alef-naught),and the uncountable infinity of the realsis called ℵ1(alef-one).

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

TheComplex Numbers

The complex numbersare the set a and b are real numbers, where i is the imaginary unit, −1. (click here formore 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 sometimeswritten as C for short. The set of complex numbersis important because for any polynomial p(x) with real number coefficients, all the solutions of p(x)=0 will be in C.

See more: What Is The Greatest Common Factor Of 30 And 75, Gcf Of 30 And 75


There are even "bigger" setsof numbers used by mathematicians. The quaternions,discovered by William H. Hamilton in 1845, form a number system with threedifferent imaginary units!