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Each rational number can be written in many forms, for example . The simplest form is when and have no common divisors, and every rational number has a simplest form of this type.
The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above 1. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational.
A real number that is not rational is called an irrational number.
In mathematics, the term "rational something" means that the underlying field considered is the field of rational numbers. For example, rational polynomials or rational prime idealIn mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. Prime ideals have a simpler description for commutative rings, so we consider this case separately below. This article ons.
Addition and multiplication of rational numbers are as follows:
Two rational numbers and are equal iffIn mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if . It is often, not always, written italicized: iff''. Although "P iff Q" is most standard, common alternative phrases include "P
Additive and multiplicative inverses exist in the rational numbers.
Any positive rational number can be expressed as a sum of distinct reciprocalIn mathematics, the reciprocal or multiplicative inverse of a number x is the number which, when multiplied by x, yields 1. Zero does not have a reciprocal. Every complex number except zero has a reciprocal that is a complex number. If it is real then sos of positive integers.
For instance,
For any positive rational number, there are infinitely many different such representations. These representations are called Egyptian fractions, because the ancient Egyptians used them. The hieroglyphHieroglyphs are a system of writing used by the Ancient Egyptians, using a combination of logographic, syllabic, and alphabetic elements. Etymology The word hieroglyph comes from the Greek words (hierogluphos) hiero , meaning "sacred", and glyph , meaning used for this is the letter that looks like a mouth, which is transliterated R, so the above fraction would be written as R2R6R21, or, using the hieroglyphs and writing left to rightThe writing systems of some languages, such as Hebrew and Arabic are written from right to left. When Latin-based text is mixed with these languages in the same sentence, each type of text should be written in its own direction. This is known as bi-direct:
½ is one of exactly three exceptions: it is written as shown in the first hieroglyph above. The two other exceptions were the two only non-unit fractions for which there were symbols:
The Egyptians also had a different notation for dyadic fractions. See also Egyptian numerals.
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