The Oracle

by Nicholas Mee and David Benjamin on October 1, 2017

Plato believed we have an innate knowledge of geometry and much else besides. While this might be true up to a point, his argument is not totally convincing. Plato presents the idea in his dialogue The Meno, which takes the form of a discussion between the philosopher Socrates and a slave-owning aristocrat called Meno.

Socrates demonstrates the intuitive nature of geometry by guiding one of Meno’s slaves through an exercise in which he constructs a square with twice the area of a given square. Although Meno’s slave has no knowledge of geometry he is led to construct the figure shown here. Starting with the top left square, we see that doubling the length of each edge produces a square with four times the area of the original. However, if the diagonal of the square is taken as the base of a new square, the constructed square has twice the area of the original square. (It is composed of four identical triangles, whereas the original square is composed of two such triangles.) If the original square has edges of length 1 and an area of 1, then the area of the constructed square is 2, so the length of each of its blue edges is \sqrt{2}. The diagonal of a unit square therefore has length \sqrt{2}.

Priestess of Delphi (1891) by John Collier, showing the Pythia on her tripod amidst the intoxicating vapours.

The Delian Problem

Delphi was the most important religious site in Ancient Greece, regarded as the omphalos or navel of the world. The site was infused with a sickly-sweet smell, said to be due to the decaying remains of the giant snake Python slain by Apollo. The High Priestess of the Temple of Apollo, known as the Pythia, was inspired by the spirit of Apollo and her oracular pronouncements gave the most revered advice in the ancient world.

We now know that Delphi sits on a geological fault and intoxicating gases, such as ethylene, seep out through cracks in the ground, which helps explain the importance of the site and the origin of the oracle.

The Roman historian Plutarch records that the citizens of Delos consulted the oracle of Delphi about an intractable political issue. The oracle’s response was that the citizens should double the size of the cubic altar of Apollo. The Delians were puzzled by this answer, so they consulted Plato who interpreted it as the geometrical problem of constructing a cube of twice the volume of a given cube. Plato suggested that the oracle was advising the Delians to dedicate themselves to geometry and give up their political infighting.

The remains of the temple complex at Delphi.

The geometrical problem of doubling the cube, also known as the Delian problem, is equivalent to constructing an edge whose length is the cube root of two \sqrt[\leftroot{-1}\uproot{1}3]{2}. Given an edge of this length, constructing further perpendicular edges of the same length to form a cube is readily accomplished. Pierre Wantzel proved in 1837, however, that, unlike doubling the square, this problem is impossible within the restrictive rules of classical geometry which only permit the use of a straight edge and compass.

Square roots, cube roots and higher roots are sometimes known as surds, which is an abbreviation for absurd numbers. The origin of this term will soon become clear. First we will take a look at some fractions.

Plenty of Room at the Bottom

Between any two fractions there is always room for more. Take 1/5 and 2/5. If we express these fractions as 10/50 and 20/50, we can easily squeeze in 11/50, 12/50 and so on. Again, between 11/50 = 110/500 and 12/50 = 120/500 there is plenty of room for 111/500, 112/500 and so on. Clearly, we could continue in this way indefinitely, which implies that an unlimited number of fractions reside between 0 and 1.

Mathematicians often refer to numbers that are the ratio of two whole numbers as rational numbers. These include ordinary fractions such as 121/500, improper fractions such as 601/500 and whole numbers such as 10/2 = 5. But surely, you might say, as there is an infinite supply of such numbers, all numbers must be rational numbers. How could it be otherwise?

Drowning in a Sea of Numbers

Hippasus of Metapontum was a member of the Pythagorean brotherhood over 2400 years ago and he is sometimes credited with the startling revelation that some numbers cannot be expressed as the ratio of two whole numbers. These monsters are known as irrational numbers. Hippasus’ discovery rocked the Pythagorean world as it cast doubt on the stability of the entire number system. According to legend, Hippasus was drowned at sea as punishment for breaking the society’s code of secrecy by communicating his discovery to the outside world. People took their mathematics very seriously in those days.

Hippasus’ argument was not about strange and exotic numbers, it applies to everyday numbers such as \sqrt{2}, the length of the diagonal of a unit square. This is a perfectly good number like any other, so we might suppose it to be an improper fraction with a value somewhere between 1 and 2. However, Hippasus showed this is not the case, with fatal consequences.

A Simple Example

Before looking at a general proof we will take a simple example that illustrates the general picture.

First, if A/B = \sqrt{2}, then A = \sqrt{2} x B and squaring both sides gives A2 = 2 x B2.

Next, the rational number A/B = 99/70 is certainly close to \sqrt{2} and we might suppose that it equals \sqrt{2} exactly. But if this were the case then A2 = 992 = 9801 should equal 2 x B2 = 2 x 702 = 9800, which it clearly doesn’t.

Decomposing A and B into prime factors A = 99 = 3 x 3 x 11 and B = 70 = 2 x 5 x 7, we find

A2 = 992 = 3 x 3 x 3 x 3 x 11 x 11 \approx 2 x 2 x 2 x 5 x 5 x 7 x 7 = 2 x 702 = 2B2,

where \approx means approximately equal to. The only way the two sides of this expression could be equal would be if the two products of primes were equal, which is impossible because the decomposition of a whole number into a product of primes is unique, as discussed in the post Sieving for Numbers.

Truth from Absurdity

Hippasus used a very powerful method that generalises the above discussion and shows that no rational number equals \sqrt{2}. It is a standard tool known as proof by contradiction or more formally reductio ad absurdum and is often deployed by mathematicians when tackling the thorniest of questions. We assume a mathematical statement is true, then examine its consequences and show they are impossible, which implies our initial assertion is false.

Assume \sqrt{2} is a rational number and write \sqrt{2} = A/B, where A and B are two integers with no common factor. (We can always remove common factors.)

We know each integer is composed of a unique product of prime numbers pi, so

A = p1 x p2 x …. x pj and B = pj+1 x pj+2 x …. x pj+k ,

where the primes pi might be any primes and the same prime might occur more than once, but as A and B have no factors in common, none of the primes in the list p1,p2, …. pj, occur in the list pj+1, pj+2, …. pj+k.

Using the expansion of B as a product of primes, we obtain

A2 = 2B2 = 2 x pj+12 x pj+22 x …. x pj+k2 .

We also know that A2 = p12 x p22 x …. x pj2, so we have found two decompositions of the number A2 formed of completely different primes. But this is impossible, as the decomposition of any integer into primes is unique. We must conclude, therefore, that our initial assertion that \sqrt{2} = A/B, where A and B are two integers is false. Whatever rational approximation we make to \sqrt{2}, it is only that, an approximation, there is no rational number that exactly equals \sqrt{2}.

Confronting the Irrational

The proof is easily adapted to \sqrt{3}, as follows: if the rational number C/D = \sqrt{3}, then C2 = 3D2. Expanding C2 and 3D2 as products of primes, we obtain two different decompositions of the same number into primes, which is impossible. This implies that \sqrt{3} is also an irrational number. The same is true for the square root of any whole number, with the obvious exceptions of square numbers. Furthermore, the same proof works for cube roots and higher roots. Such numbers are all either irrational numbers or integers.

There is more. The ratio of the circumference C to the diameter d of a circle is known as \pi , where \pi = C/d. The first few hundred digits in its decimal expansion are shown on the right. Johann Lambert proved in 1761 that \pi is an irrational number, which means that if the diameter of a circle is a whole number, then the circumference must be an irrational number. It also means that the decimal expansion of \pi goes on for ever without repeating. In a sense the digits appear at random.

The Tip of an Irrational Iceberg

But this is just the tip of the iceberg. In fact, the vast majority of numbers are irrational!

Justifying this statement will require a further article.

 

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