The natural Numbers

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

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The whole numbers room the natural numbers in addition to 0.

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

The sum ofany two natural numbers is additionally a herbal number (for example, 4+2000=2004), and the product of any kind of two natural numbersis a herbal number (4×2000=8000). Thisis no true because that subtraction and division, though.

The Integers

The integers space the collection of actual numbers consisting of the herbal numbers, your additive inverses and zero.


The set of integers is sometimeswritten J or Z because that short.

Thesum, product, and also difference of any kind of two integers is additionally an integer. Yet this is not true because that division... Just shot 1÷2.

The rational Numbers

The rational numbers arethose number which deserve to be expressed together a proportion betweentwo integers. For example, the fractions 13 and also −11118 space bothrational numbers. Every the integers are contained in the rational numbers,since any kind of integer z deserve to be composed as the proportion z1.

All decimal which terminate are rational numbers (since 8.27 have the right to be written as 827100.) Decimalswhich have a repeating sample after some allude are likewise rationals:for example,

The collection of rational numbers is closeup of the door under every four basic operations, that is, given any kind of two rational numbers, theirsum, difference, product, and also quotient is also a reasonable number(as lengthy as we don"t division by 0).

The Irrational Numbers

An irrational number is a number that cannot be created as a proportion (or fraction). In decimal form, it never ever ends or repeats. Theancient Greeks discovered that no all numbers room rational; thereare equations that cannot be solved using ratios that integers.

The an initial such equationto it is in studied to be 2=x2. Whatnumber times itself amounts to 2?

2 isabout 1.414, since 1.4142=1.999396, i m sorry is near to2. However you"ll never ever hit exactly by squaring a fraction (or terminatingdecimal). The square source of 2 is one irrational number, meaning itsdecimal identical goes ~ above forever, v no repeating pattern:


Other renowned irrationalnumbers room the gold ratio, a number through greatimportance to biology:

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


and e,the most vital number in calculus:


Irrational numbers deserve to be additional subdivided into algebraic numbers, which room the solutions of some polynomial equation (like 2 and also the golden ratio), and also transcendental numbers, which space not the remedies of any type of polynomial equation. π and also e are both transcendental.

TheReal Numbers

The genuine numbers is the collection of number containing all of the rational number and every one of the irrational numbers. The real numbers space “all the numbers” top top the number line. There room infinitely many real numbers just as there space infinitely numerous numbers in each of the various other sets the numbers. But, it can be verified that the infinity the the genuine numbers is a bigger infinity.

The "smaller",or countable infinity that the integers andrationals is sometimes dubbed ℵ0(alef-naught),and the uncountable infinity that the realsis dubbed ℵ1(alef-one).

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

TheComplex Numbers

The facility numbersare the set a+bi , where i is the imagine unit, −1. (click right here formore on imagine numbers and also operations with facility numbers).

The complex numbers incorporate the set of genuine numbers. The real numbers, in the complex system, room written in the type a+0i=a. A real number.

This collection is sometimeswritten together C because that short. The set of complex numbersis important because for any polynomial p(x) with actual number coefficients, all the remedies of p(x)=0 will be in C.

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There are also "bigger" setsof numbers offered by mathematicians. The quaternions,discovered by william H. Hamilton in 1845, kind a number system with threedifferent imaginary units!