Unveiling The Mysterious World Of Irrational Numbers In Mathematics

irrational numbers

Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating.

In mathematics, an irrational number is a real number that cannot be expressed as a ratio of two integers. This means that its decimal representation goes on forever without repeating or ending in a pattern. Examples of irrational numbers include √2, π, and e.

One way to prove that a number is irrational is by contradiction. Suppose that a number can be expressed as a ratio of two integers, such as a/b, where a and b have no common factors (i.e. they are coprime). Then, we can square both sides of the equation to get a^2/b^2 = n, where n is a rational number. Rearranging the equation, we get a^2 = nb^2, which means that a^2 is divisible by b^2. Since a and b have no common factors, this implies that a is divisible by b, which contradicts our assumption that a and b are coprime. Therefore, the original number (i.e. the square root of n) must be irrational.

Irrational numbers have some interesting properties. For example, any irrational number added to a rational number is still irrational. Also, the sum, difference, or product of two irrational numbers may be rational or irrational. However, the product or quotient of a rational and an irrational number is always irrational.

Irrational numbers can be found in many areas of mathematics, and they have important applications in physics, engineering, and computer science. They are also associated with some famous mathematical problems, such as the squaring of the circle and the duplication of the cube, which were shown to be impossible using only a ruler and compass.

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