Irrational Number

In mathematics, an irrational number is any real number which cannot be expressed as a fraction a/b, where a and b are integers, with b non-zero, and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are precisely those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational. Perhaps the best-known irrational numbers are π, e and √2.When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common. A measure of a line segment I in this sense is a line segment J that "measures" I in the sense that some whole number of copies of J laid end-to-end occupy the same length as I.

Square roots

The square root of 2 was the first number to be proved irrational and that article contains a number of proofs. The golden ratio is the next most famous quadratic irrational and there is a simple proof of its irrationality in its article. The square roots of all non-square natural numbers are irrational and a proof may be found in quadratic irrationals.

The irrationality of the square root of 2 may be proved by assuming it is rational and inferring a contradiction, called an argument by reductio ad absurdum. The following argument appeals twice to the fact that the square of an odd integer is always odd.

General roots

The proof above for the square root of two can be generalized using the fundamental theorem of arithmetic which was proved by Gauss in 1798. This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime in the denominator which does not divide into the numerator whatever power each is raised to. Therefore if an integer is not an exact kth power of another integer then its kth root is irrational.

If √2 is rational it has the form m/n for integers m, n not both even. Then m2 = 2n2, hence m is even, say m = 2p. Thus 4p2 = 2n2 so 2p2 = n2, hence n is also even, a contradiction.

Logarithms

Perhaps the numbers most easily proved to be irrational are certain logarithms. Here is a proof by reductio ad absurdum that log2 3 is irrational. Notice that log2 3 ≈ 1.58 > 0.

Assume log2 3 is rational. For some positive integers m and n, we have





It follows that

However, the number 2 raised to any positive integer power must be even (because it will be divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer can not be both odd and even at the same time: we have a contradiction. The only assumption we made was that log2 3 is rational (and so expressible as a quotient of integers m/n with n ≠ 0). The contradiction means that this assumption must be false, i.e. log2 3 is irrational, and can never be expressed as a quotient of integers m/n with n ≠ 0.

Cases such as log10 2 can be treated similarly.

All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction.

An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:

π = 3.141592…
√2 = 1.414213…

Although irrational numbers are not often used in daily life, they do exist on the number line. In fact, between 0 and 1 on the number line, there are an infinite number of irrational numbers!

Irrational Numbers

The ancient Greeks discovered that some naturally occurring geometric lengths, such as the diagonal of a unit square,cannot be expressed as rational numbers.This discovery led eventually to the notion of irrational numbers(numbers that cannot be expressed as a ratio of integers).

Any number that cannot be expressed as a fraction a/b is called an irrational number.

The development of geometry revealed the need for more types of numbers; the length of the diagonal of a square with sides one unit long cannot be expressed as a rational number.

Similarly,the ratio of the circumference to the diameter of a circle is not a rational number. These and other needs led to the introduction of the irrational numbers.

Some examples of irrational numbers are: square roots of whole numbers that are not perfect squares; for example,decimal numbers that don't repeat or terminate. e = 2.718281828459... and π = 3.1415926535... are irrational numbers,

and their decimal expansions are necessarily nonterminating and non periodic.

Some irrational numbers can be expressed as square roots; they can be also written as decimals but the decimal is always infinite and never repeats.

The word "irrational" suggests that these numbers were "wrong" in some way. In fact, many early mathematicians like Pythagoras were unwilling to accept that such numbers could exist. Nowadays, irrational numbers are accepted as perfectly "proper."

form of a fraction. ¾, for example, is a rational number, which can also be expressed as .75. When a number is irrational, it cannot be written out as a fraction with integers and the number will be impossible to record in decimal form. Pi is a famous example of an irrational number; while it is often simplified to 3.14 for the purpose of rough calculations, pi cannot actually be fully written out in decimal form because the decimal is endless.

Some other examples of irrational numbers include the square root of two, Euler's number, and the golden ratio. For the purpose of simplicity, some irrational numbers are written out as symbols, as in the case of “e” for Euler's number, and sometimes they will be represented in partial decimal form. When an irrational number is presented in decimal form, ellipses are usually used after the last number in the decimal to indicate that it continues, as in 3.14... for pi.

Reference:
http://en.wikipedia.org/wiki/Irrational_number
http://www.wisegeek.com/what-are-irrational-numbers.htm
http://encyclozine.com/science/mathematics/number