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Irrational Numbers

An irrational number is a real number that cannot be expressed in the form a b , when a and b are integers ( b 0 ).  In decimal form, it never terminates (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 ?

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:

2 = 1.41421356237309...

Historical Note:

According to legend, the ancient Greek mathematician who proved that 2 could NOT be written as a ratio of integers p q made his colleagues so angry that they threw him off a boat and drowned him!

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

1 + 5 2 = 0.61803398874989...

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

π = 3.14159265358979...

and e , the most important number in calculus :

e = 2.71828182845904...

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.

The Venn diagram below shows the relationships of the various sets of numbers.

Venn diagram showing subset relationships in real numbers, rational numbers, integers, natural numbers