Irrational number


In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length, no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Among irrational numbers are the ratio Pi| of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.
Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of starts with 3.14159, but no finite number of digits can represent exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics.
Irrational numbers can also be expressed as non-terminating continued fractions, and in many other ways.
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.

History

Ancient Greece

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean, who probably discovered them while identifying sides of the pentagram.
The Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was 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:''b.
  • Assume a'', b, and c are in the smallest possible terms .
  • By the Pythagorean theorem: c2 = a2+b2 = b2+b2 = 2b2..
  • Since c2 = 2b2, c2 is divisible by 2, and therefore even.
  • Since c2 is even, c must be even.
  • Since c is even, dividing c by 2 yields an integer. Let y be this integer.
  • Squaring both sides of c = 2y yields c2 = 2, or c2 = 4y2.
  • Substituting 4y2 for c2 in the first equation gives us 4y2= 2b2.
  • Dividing by 2 yields 2y2 = b2.
  • Since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even.
  • 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.
Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans 'for having produced an element in the universe which denied the... doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.' Another legend states that Hippasus was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that numbers and geometry were inseparable; a foundation of their theory.
The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. This was brought to light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for "whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects", but Zeno found that in fact " in general are not discrete collections of units; this is why ratios of incommensurable appear....uantities are, in other words, continuous". What this means is that contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. In fact, these divisions of quantity must necessarily be infinite. For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on. 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 paradoxes, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno's paradoxes accurately demonstrated the deficiencies of contemporary mathematical conceptions, they were not regarded as proof of the alternative. 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 Cnidus, who 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. A magnitude "...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5". Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. 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 out of the equation, he avoided the trap of having to express an irrational number as a number. "Eudoxus' theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios". This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. It was not until Eudoxus developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created.
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 numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases, algebraic conceptions were reformulated into geometric terms. This may account for why we still conceive of x2 and x3 as x squared and x cubed instead of x to the second power and x to the third power. Also crucial to Zeno's work with incommensurable magnitudes was the fundamental focus on deductive reasoning that resulted from the foundational shattering of earlier Greek mathematics. 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. Out of this necessity, Eudoxus developed his method of exhaustion, a kind of reductio ad absurdum that "...established the deductive organization on the basis of explicit axioms..." as well as "...reinforced the earlier decision to rely on deductive reasoning for proof". This method of exhaustion is the first step in the creation of calculus.
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 17.

India

Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the Vedic period in India. There are references to such calculations in the Samhitas, Brahmanas, and the Shulba Sutras.
It is suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava believed that the square roots of numbers such as 2 and 61 could not be exactly determined. Historian Carl Benjamin Boyer, however, writes that "such claims are not well substantiated and unlikely to be true".
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.
Mathematicians like Brahmagupta and Bhāskara I made contributions in this area as did other mathematicians who followed. In the 12th century Bhāskara II evaluated some of these formulas and critiqued them, identifying their limitations.
During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as π and certain irrational values of trigonometric functions. Jyeṣṭhadeva provided proofs for these infinite series in the Yuktibhāṣā.