Alex's Adventures in Numberland - Alex Bellos [115]
The decimal system throws up an excellent example of a Zeno-inspired paradox. What is the largest number less than 1? It is not 0.9, since 0.99 is larger and still less than 1. It is not 0.99 since 0.999 is larger still and also less than 1. The only possible candidate is the recurring decimal 0.9999…where the ‘…’ means that the nines go on for ever. Yet this is where we come to the paradox. It cannot be 0.9999…since the number 0.9999…is identical to 1!
Think of it this way. If 0.9999…is a different number from 1, then there must be space between them on the number line. So it must be possible to squeeze a number in the gap that is larger than 0.9999…and smaller than 1. Yet what number could this be? You cannot get closer to 1 than 0.9999…. So, if 0.9999…and 1 cannot be different, they must be the same. Counter-intuitive though it is, 0.9999…= 1.
So what is the largest number less than one? The only satisfactory conclusion to the paradox is that the largest number less than 1 doesn’t exist. (Likewise, there is no largest number less than 2, or less than 3, or indeed less than any number at all.)
The paradox of Achilles’ race against the tortoise was resolved by writing the durations of his dashes as a sum with an infinite amount of terms, which is also known as an infinite series. Whenever the terms of a sequence are added together it is called a series. There are both finite and infinite series. For example, if you add up the sequence of the first five natural numbers, you get the finite series:
1 + 2 + 3 + 4 + 5 = 15
Obviously we can work out this sum in our heads, but when a series has many more terms, the challenge is to find a shortcut. One famous example was worked out by the German mathematician Carl Friedrich Gauss when he was a young boy. As the story goes, a schoolteacher is said to have asked him to calculate the sum of the series of the first hundred natural numbers:
1 + 2 + 3 +…+ 98 + 99 + 100
To the teacher’s disbelief, Gauss replied almost instantly: ‘5050.’ The prodigy had worked out the following formula. If you pair off numbers judiciously, by taking the first with the last, the second with the second-last, and so on, then the series can be rewritten as:
(1 + 100) + (2 + 99) + (3 + 98) +…+ (50 + 51)
which is:
101 + 101 + 101 + 101 +…+ 101
There are fifty terms, each with a value 101, so the sum is 50×101 = 5050. We can generalize this to get the result that for any number n, the sum of the first n numbers is n + 1 added times in a row, which is . In the above case n is 100, so the sum s = 5050.
When you add up the terms in a finite series you always get a finite number, that’s obvious. However, when you add up the terms of an infinite series there are two possible scenarios. The limit, which is the number that the sum approaches as more and more terms are added, is either a finite number or it is infinite. If the limit is finite, the series is called convergent. If not, the series is called divergent.
For example, we have already seen that the series
is convergent, and converges on 2. We have also seen that there are many infinite series that converge on pi.
On the other hand, the series
1 + 2 + 3 + 4 + 5 +…
is divergent, heading off towards infinity.
The Greeks may have been wary of infinity, but by the seventeenth century mathematicians were happy to take it on. An understanding of infinite series was required for Isaac Newton to invent calculus, which was one of the most significant developments in mathematics.
When I studied maths one of my favourite exercises was being presented with an infinite series and being asked to work out whether it converged or diverged. I always found it incredible that the difference between convergence and divergence was so brutal – the difference between a finite number and infinity is infinity – and yet the elements that decided which path the series took often seemed so insignificant.
Take a look at the harmonic series:
The nominator of every term is one, and the denominators are the natural