Irrational numbers are numbers that are neither whole numbers (like 2, 0, or -3) nor ratios of whole numbers. Irrational numbers are real numbers in the sense that they appear in measurements of geometric objects--for example, the number pi (), which is the ratio of the circumference of a circle to the length of its diameter, is an irrational number. However, irrational numbers cannot be represented as decimals, unlike rational numbers, which can be expressed either as finite decimals or as infinite decimals that eventually follow a repeating pattern. For instance, the decimal 6.412121212... is equal to 6348/990. By contrast, irrational numbers have infinitely long decimal expansions that never form a repeating pattern. Thus, the number pi can never be written down exactly in decimal form, it can only be approximated, by decimals such as 3.14159.
Irrational numbers were discovered in the school of Pythagoras, a great Greek mathematician who founded a Brotherhood of mathematicians and philosophers in the Italian port town of Cortona in the 6th century B.C. Pythagoras and his followers were not looking for the irrational numbers; on the contrary, they did not expect such numbers to exist. Preceding the discovery of the irrational numbers, Pythagoras had found many natural phenomena that were described by rational numbers. He had realized, for example, that in musical instruments, strings whose lengths are related by rational ratios have harmonious pitches. This observation led him to believe that the harmony of the world was closely related to that of the rational numbers, and that every natural phenomenon could be expressed in terms of these numbers--hence the name "rational." Great was his dismay, therefore, when he realized that one of the simplest geometric quantities of all, the sidelength of a square, could not be compared to the length of the square's diagonal by a rational ratio.
The Brotherhood had previously discovered the famous Pythagorean theorem, which states that in a right triangle, the sum of the square powers of the lengths of the legs is equal to the square power of the length of the hypotenuse. The diagonal of a square divides it into two right triangles. Suppose that the sides of the square are one unit long, and let's call the length of the diagonal d. The Pythagorean theorem says that 12 + 12 = d2, or d2=2. Thus, d is a number whose square is equal to 2. According to Pythagoras' credo, this number should be rational. But if it is a rational number, which one is it?
The square root of 2 is in fact not a rational number, as was discovered by the Pythagorean Brotherhood. According to one legend, this idea was so shocking to Pythagoras that he put the discoverer to death. Another version says that Pythagoras declared the irrationality of the square root of 2 to be a secret, not to be revealed to anyone outside the Brotherhood. When one of the members dared to defy Pythagoras' decree and inform the outside world of the discovery, he was killed.
The proof that the Pythagoreans discovered has not survived, and the earliest written proof that has been found is in Euclid's Elements, written more than 2 centuries after the time of Pythagoras. The proof that Euclid gives is one of the most famous in mathematics, renowned for its simplicity and elegance. Euclid follows a mathematical strategy called reductio ad absurdum: to show that something is true, assume that the opposite is true, and follow the logical consequences of that assumption until you reach an absurdity.
To use this technique to show that the square root of 2 is irrational, Euclid starts by supposing that the opposite is true: that the square root of 2 is a rational number. He writes the square root of 2 as a ratio a/b of whole numbers a and b, in lowest terms (a fraction a/b is in lowest terms if a and b have no factors in common; so 2/3 is in lowest terms, but 6/9 is not, because 6 and 9 are both divisible by 3). Since a/b is supposed to be the square root of 2, we have (a/b)2=2, or a2/b2=2. Multiplying both sides of this equation by b2 gives the new equation a2=2b2. This means that a2 is an even number, since it is 2 times a number. Now, if a2 is even then a itself must be even, since only even numbers have even squares. Thus a is equal to 2 times another number c: a=2c.
If we substitute 2c for a in the equation relating a and b, we get (2c)2=2b2, or 4c2=2b2. Dividing both sides of the equation by 2 yields 2c2=b2. This tells us that b2 is also an even number, since it is 2 times a number. And since b2 is even, b is even as well.
We have come to the conclusion that both a and b are even numbers. But a and b were supposed to be in lowest terms, and now we have discovered that they are both divisible by 2. This is a contradiction. So starting with the assumption that the square root of 2 is rational, we have arrived at an absurdity. Hence the square root of 2 must be irrational.
Since the discovery of the irrational numbers, mathematicians have gone on to prove that pi is irrational, and likewise that e, the base of the natural logarithm, is irrational. In fact, it has been established that there are infinitely many irrational numbers. After the mathematician Georg Cantor developed a way to compare the size of infinitely large sets, mathematicians proved that there are more irrational numbers than rational numbers, even though both sets are infinite: it is possible to "count" the rational numbers, but not the irrational numbers. What's more, almost all numbers are irrational--in other words, if you pick a real number completely randomly, you are virtually guaranteed that the number will be irrational, not rational.
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