Euler’s number
Why things don’t grow forever
PUT a pound in the bank. If the yearly interest were 100 per cent, then a year later you would have £2. That’s simple enough. But what if instead of calculating the interest at the end of the year, the bank worked it out more regularly? It turns out this question leads us to one of the most subtle numbers in mathematics.
Say your bank paid interest twice a year but halved the rate to 50 per cent. That would take your £1 to £1.50 after 6 months, and at the end of the year you would get another 50 per cent, making £2.25 – a nice gain. If you got interest monthly but scaled down the rate accordingly, you’d end up with £2.61. Do the same thing daily, reducing the interest rate in the same fashion, and you’d get £2.71. The improvements get ever smaller as this process continues, and the most you could have turns out to be about £2.71828.
Advertisement
Read more: Fantastic figures – A tour of the most wonderful numbers
The world of numbers may be abstract, but it contains incredible sights. Come explore the many wonders in numberland
This number is actually a special irrational, which, like π, keeps on going forever after the decimal point. It’s called Euler’s number (or simply e), after the Swiss mathematician Leonhard Euler.
Euler’s number doesn’t just appear when computing compound interest. For instance, mix together the imaginary number i (see “The imaginary number”) and e and, with a little mathematical nous, you can derive one of the most famous equations ever, Euler’s identity: eľ±Ď€ + 1 = 0. Mathematicians hold it in high regard for its beauty, cramming five of the most important numbers into a single, elegant expression.
Euler’s number is also practical. It is crucial to a mathematical technique called Fourier analysis, for example, which is used by researchers who probe crystals by shining X-rays at them. Applying the analysis to the patterns that emerge helps reveal the structure of molecules such as DNA.
But it’s not all so serious. Take the mathematical expression ex and carry out the technique called integration, co-invented by Isaac Newton. Ignoring the usual constant that appears in such a calculation, you get back ex. This standstill only happens with ex or multiples of it.
That leads to one of the best-worst maths jokes ever. Why is ex always stood alone at parties? Because when it tries to integrate nothing happens.
Timothy Revell
Ěý
i
The imaginary number
THE rules of mathematics say that two positive numbers multiply to give a positive, and two negative numbers also multiply to give a positive. So what number could you multiply by itself to give -1? This is not a trick question – it’s just that the answer is imaginary.
The square roots of negative numbers were first called “imaginary” by René Descartes in 1637. But it wasn’t until the 18th century that they came to be represented as multiples of i, the square root of -1.
Imaginary numbers don’t fit on the regular number line, so they are put on a second, independent line, with the two intersecting at zero. The lines can be treated as axes, making imaginary numbers handy for representing things that change in two dimensions. They are regularly used to describe wave functions in quantum mechanics and to define alternating current.
What does maths sound like?
Conjuring an entirely different family of numbers from thin air might seem unjustifiable. But the truth is that “real” and “imaginary” numbers are both abstract concepts. We might be more familiar with 5 than 5i, but neither exists in the real world.
That gives mathematicians a certain creative licence. In 1843, the Irish mathematician William Hamilton invented numbers called quaternions, using additional solutions for the square root of -1 that he called j and k. These form the basis of additional number lines that are used to construct axes capable of encoding rotations in 3D. Computer game design is one area where they have proved useful.
If you follow the same mathematical logic, then there is no reason to stop there. The octonions add an extra four dimensions of imaginary numbers, and the rarely used sedenions give the option of extending the total to 15. Down here, it’s a world of pure imagination.
Gilead Amit
274,207,281-1
The force behind encryption
MULTIPLY 2 by itself just over 74 million times, then subtract 1. This is the largest known prime number – a number that can only be divided by 1 and itself – with more than 22 million digits. It isn’t just any old kind of prime, either. It is a Mersenne prime, one equal to a power of 2, minus 1.
Other numbers in the Mersenne club include 3 and 31, but finding larger ones is no easy task. We have only discovered 49 of them. Despite knowing for thousands of years that there are infinitely many primes, we have no idea if there are infinitely many Mersenne primes.
Without prime numbers like this one, the world would be a very different place. To ensure that all sorts of online transactions are encrypted, so that only the intended recipient can unscramble them, we rely on primes.
The idea is that the receiver multiplies two big primes to create a new number called the public key. Anyone with this key can encrypt messages, but to turn them from gobbledegook to something meaningful requires knowledge of the original two primes.
Multiplying primes together is easy for computers, but if the answer is large enough, the only way to work out the primes that produced it is essentially to try out all the possibilities. That’s practically impossible, making the whole process secure.
We don’t really need to find a 50th Mersenne prime for the sake of encryption. But it’d be nice all the same.
Timothy Revell
Ěý
Graham’s number
The biggest number with a name of its own
MOST numbers have never touched a human mind. There are an infinite number of numbers, after all, so it stands to reason that we have only bothered with the small ones.
But in the 1970s, Ronald Graham, a mathematician now at the University of Califonia, San Diego, was working on a problem that proved to have a truly gargantuan answer. He was trying to solve a problem to do with cubes in higher dimensions, and when he finally got there, the answer involved a number so large we can’t write down its digits – there isn’t enough space in the universe.
Yet there is a way to grasp at Graham’s number. A more concise way of writing 3×3×3 is exponentiation: 33 means “multiply three threes together”, giving 27.
We can go further, using something called Knuth’s up-arrow notation. 3↑3 means the same as 33, but 3↑↑3 starts a rising tower. The two arrows tell us to repeat the exponentiation, giving us 3 to the power of (3 to the power of 3) or 327, which is around 7.6 trillion.
Add a third arrow, 3↑↑↑3, and things take a major uptick, so that you reach an unimaginable stack of exponentiation upon exponentiation. Graham’s number is written as 64 layers of up-arrow notation, with each layer longer than the last. In case you’re wondering, its last digit is 7.
Graham’s number is a whopper, but we can think of bigger ones still.
Take the function TREE(n), which relates to putting a certain number of labels on mathematical objects similar to a family tree, as part of a proof known as Kruskal’s tree theorem. TREE(1) is 1. TREE(2) is 3. TREE(3) is so big it makes Graham’s number seem practically zero.
Another function, called Busy Beaver, grows so fast that it has been mathematically proven to be impossible for any computer program to calculate all but its smallest values.
Busy Beaver was recently used to show that some problems are impossible to solve using the standard axioms of mathematics. But that’s another big problem altogether.
Jacob Aron
This article appeared in print under the headline “Wonders ofĚýnumberland”
