Rational number

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Rational number

History[ edit ] Set of real numbers Rwhich include the rationals Qwhich include the integers Zwhich include the natural numbers N.

Ancient Greece[ edit ] The first proof of the existence of irrational numbers is usually attributed to a Pythagorean possibly Hippasus of Metapontum[5] who probably discovered them while identifying sides of the pentagram.

However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction.

He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible.

His reasoning is as follows: Start with an isosceles right triangle with side lengths of integers a, b, and c. The ratio of the hypotenuse to a leg is represented by c: Assume a, b, and c are in the smallest possible terms i.

By the Pythagorean theorem: Since c2 is even, c must be even. Since c is even, dividing c by 2 yields an integer. Since b2 is even, b must be even. We have just shown that both b and c must be even. Hence they have a common factor of 2.

However this contradicts the assumption that they have no common factors. This contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Hippasus, however, was not lauded for his efforts: Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable—a foundation of their theory.

The discovery of incommensurable ratios was indicative of another problem facing the Greeks: Brought into light by Zeno of Eleawho questioned the conception that quantities are discrete and composed of a finite number of units of a given size.

That in fact, these divisions of quantity must necessarily be infinite. For example, consider a line segment: This process can continue infinitely, for there is always another half to be split.

The more times the segment is halved, the closer the unit of measure comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating four paradoxeswhich demonstrated the contradictions inherent in the mathematical thought of the time.

In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur. The next step was taken by Eudoxus of Cniduswho formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities.

Central to his idea was the distinction between magnitude and number. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5.

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Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios.

By taking quantitative values numbers out of the equation, he avoided the trap of having to express an irrational number as a number. As a result of the distinction between number and magnitude, geometry became the only method that could take into account incommensurable ratios.

Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from those numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases algebraic conceptions were reformulated into geometrical terms.

This may account for why we still conceive of x2 or x3 as x squared and x cubed instead of x second power and x third power. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory.

Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because the algebra he used could not be applied to the square root of It is suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava c.

Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square roots.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.

The square root of 2, or the (1/2)th power of 2, written in mathematics as √ 2 or 2 1 ⁄ 2, is the positive algebraic number that, when multiplied by itself, gives the number caninariojana.comcally, it is called the principal square root of 2, to distinguish it from the negative number with the same property..

Geometrically the square root of 2 is the length of a diagonal across a square with sides. In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.

Since q may be equal to 1, every integer is a rational number. An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q.

Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational. There is no standard notation for the set of irrational numbers, but the notations Q^_, R-Q, or R\Q, where the bar, .

Rational number

A rational number is a number that can be in the form p/q where p and q are integers and q is not equal to zero. Improve your math knowledge with free questions in "Rational numbers: find the sign" and thousands of other math skills.

Rational and irrational numbers explained with examples and non examples