The square root of 2
The mathematical truth upset the ancient Pythagoreans, writes Paul Davies.
The Pythagorean philosophers of ancient Greece had a deep regard for numbers. Arithmetic and geometry were guarded as a form of secret knowledge to be disclosed only to a privileged few.
They believed the Universe itself was a manifestation of numbers and geometry, so that by exploring mathematics they were gaining insights into a deeper layer of cosmic reality. So it came as a profound shock when the Pythagoreans discovered that some numbers lay outside their tidy intellectual scheme.
As young children, we first learn to count the whole numbers, 1, 2, 3 … which mathematicians call the “natural” numbers. We’re then taught fractions, which are expressed as the ratio of natural numbers, such as 2/5 or 1/3. After that come decimals.
Every fraction, we learn, may be expressed as a decimal – for example, 2/5 = 0.4 and 1/3 = 0.33333… (where “…” indicates that the succession of 3s goes on forever). But is the reverse true? Can every decimal be expressed as a fraction?
To be sure, finite-length decimals can always be expressed as fractions, for example, 0.43857 = 43857/100000. What about infinite-length decimals, however? Well, repeating decimal expansions can be expressed as fractions, e.g. 0.33333… = 1/3 and 0.285714285714285714… = 2/7. But suppose the decimal expansion doesn’t repeat? There is, after all, an infinity of such numbers!
The early Pythagoreans were convinced that every conceivable number could in principle be written in fractional form, as the ratio of two natural numbers. Since there is an infinite supply of natural numbers, they thought, there must be enough to do the job. The discovery that this was an erroneous belief, possibly by the geometer Hippasus in the 5th century BCE, was shocking news. According to legend, Hippasus was hurled off a boat and drowned to prevent the truth becoming widely known, such was its threat to the Pythagorean concept of order in the Universe.
Even today, numbers that cannot be expressed as a ratio of natural numbers are called irrational numbers, even though they make perfect sense to modern mathematicians.
It is actually easy to see why some numbers are irrational. A famous example is the square root of 2, which is roughly 1.4142, and denoted √2.
If √2 were rational, it must be expressible in the form a/b, where a and b are natural numbers (that is, whole numbers). We can write this in equation form, √2 = a/b, then give it a quick mathematical kick of the tyres.
For starters, we know at least one of the two unknown numbers, a and b, must be odd. If both were even, we could divide top and bottom by 2, and reduce the fraction to a ratio of smaller numbers (like 2/8, for example, which reduces to 1/4).
Now let’s view the equation from a couple of different angles. If we square both sides of the equation √2 = a/b we get
2 = a2/b2 (1)
which may be written as
a2 = 2b2. (2)
We can immediately conclude that a2 is an even number. Why? Because it is twice b2. Multiply any natural number by two (and we know b is a natural number), and the answer is even. So a2 must be even.
Now if a2 is even, then so is a (the square of an odd number is always odd). Since we already specified that a and b cannot both be even, we can deduce that b must be odd. So far, so good. Alarm bells start ringing, however, when we note that if a is even it could always be expressed as 2c, where c is another natural number. Substituting this into equation (1), we get
2 = 4c2/b2
or, rearranging the equation and dividing by 2,
b2 = 2c2.
By the same reasoning that follows equation (2), we conclude that b must be an even number. But we already determined that b is odd, so we reach the absurd conclusion that b is both even and odd – clearly impossible. The flawed reasoning stems from the starting assumption that √2 can be expressed as a ratio of whole numbers. It cannot; it is “irrational”.
It turns out that almost all numbers are irrational, including some famous ones like π and φ, the Golden Ratio (Cosmos 65, page 120).
There is a limitless number of both rational and irrational numbers, yet there are somehow more irrational numbers than rational – they form a bigger class of infinity.
Not appreciating this fact often misled the ancient Greeks and led to all sorts of paradoxes until the subject of infinity was eventually sorted out in the 19th century.
Today we can see that irrational numbers are not a disaster, merely an extension of the number system, just as fractions were introduced as an extension of the natural numbers. In later centuries, the number system was extended in other ways too – but that’s a subject for a future column.