Rational And Irrational Numbers Rational And Irrational Numbers

Rational Number A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. The number 8 is a rational number because it can be written as the fraction 8/1. Likewise, 3/4 is a rational number because it can be written as a fraction. Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction. Every whole number is a rational number, because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3,867 can be written as 3,867/1.

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Examples of rational numbers include , 0, 1, 1/2, 22/7, 12345/67, and so on. Farey sequences provide a way of systematically enumerating all rational numbers. The set of rational numbers is denoted Rationals in Mathematica, and a number can be tested to see if it is rational using the command Element[x, Rationals]. The elementary algebraic operations for combining rational numbers are exactly the same as for combining fractions. It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable. Irrational Number 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… = 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! Read More About Rational Numbers Worksheets

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Example proofs Square roots The square root of 2 was the first number 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 numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals. 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 that 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.

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