From 0 to Infinity in 26 Centuries (14 page)
Read From 0 to Infinity in 26 Centuries Online
Authors: Chris Waring
Arch rivalry
One example of the Bernoulli brothers’ rivalry stemmed from the problem of the
catenary curve
: the shape produced by a rope or chain when it hangs from both ends. A
mathematical equation of this shape had eluded mathematicians up to this point. Jacob proposed the problem in 1691; Johann, with some assistance from Leibniz and the Dutch mathematician Christiaan
Huygens, then went on to solve it. Catenary curves have important applications in bridge building and in architecture because arches that follow an upside-down catenary curve are the strongest.
In everyone’s interest
Jacob also discovered something interesting when he looked at a problem involving
compound interest
. Jacob noticed that if you had £100 in a bank account that paid
10% interest per annum, the way the interest is paid throughout the year affects the total money you will have at the end of the year. Compound interest payments add to the principal sum of money
in a bank account, which increases the interest you earn year after year:
Interest | Interest Calculation | Total |
10% paid at end of year | 100 × 1.1 | £110 |
5% every six months | (100 × 1.05) × 1.05 | £110.25 |
2.5% every 3 months | ((((100 × 1.025) × 1.025) × 1.025) ×1.205) | £110.38 |
Daily interest | | £110.51 |
Admittedly, the changes are not making a vast difference to your balance, a fact that most banks rely on. Of greater significance was Jacob’s investigation into instances
when interest is paid continuously, in tiny amounts, over the entire year. He discovered that if your interest rate is x (e.g. x=0.045 for a rate of 4.5%) at the end of a year you would have
2.718281
x
times what you started with. I have rounded the 2.718281 – it is in fact an irrational number, like π, that goes on for ever without repeating.
Napier made reference to this number in his work with logarithms, and it also plays a very important role in calculus, as exemplified by our next mathematician.
L
EONHARD
E
ULER
(1707–83)
Euler (rhymes with ‘boiler’ rather than ‘ruler’) was a Swiss mathematician who had originally intended on becoming a priest. However, while at university
he met Johann Bernoulli, who recognized Euler’s extraordinary mathematical talent and
managed to persuade Euler senior to allow his son to transfer to studying
maths.
An increase in power
Euler’s contributions to mathematics and science were far-reaching. The number 2.718281..., discovered by Jacob Bernoulli in relation to his compound interest problem,
also turned up in Euler’s work on calculus.
When you integrate to find the area under a graph you need to increase the power of x by one. For example, if your graph is y=x
2
, the integral is 1/3 x
3
– the power
of x has gone up by one. If you’re faced with something slightly more tricky, let’s say y = 1/x
4
, there is a handy rule of powers that can help you:
1/x
n
= x-
n
So y=1/x
4
becomes y=x
-4
, and when you increase the power by one your answer will be something to do with x
-3
, which is 1/x
3
.
But what happens if your graph is y = 1/x?
This is the same as x
-1
, but if you increase the power by one you get x
0
. Anything to the power of zero is 1, implying that, no
matter which section of the graph you look at, the area will be the same. This doesn’t make sense!
Well, it turns out, through a complicated system of algebra developed by Euler, that the area is equal to the natural logarithm of x. A natural logarithm is similar to a normal logarithm, but
its base is the number 2.7818281... There is a sizeable family of equations that can only be integrated or differentiated using natural logs, and the 2.7818281 was known as ‘e’ for
Euler’s number.
Returning to the area under the graph, if you wanted to know the area between x=1 and x=4, you would need to work out:
area = log
e
4 - log
e
1
As ‘log
e
’ turns up so often in calculus, it is denoted by ln and you will find this button on all good scientific calculators.
area = ln 4 - ln 1 = 1.386 (to 3 decimal places)
A bridge too far
Euler’s work on ‘The Seven Bridges of Königsberg’ contributed to methods of simplifying maps. Königsberg was the old Prussian name for the city of
Kaliningrad in that strange bit of Russia that sits between Poland and Lithuania. The city is centred on an island, which straddles a river. Seven bridges connect the two sides of the island at
various locations:
A popular Sunday afternoon pastime for the residents of Königsberg was to attempt to walk over all seven bridges and return to their starting place without having to use
the same bridge twice. Not one person ever managed it, but Euler was the first to tackle the problem mathematically. He redrew the map as a network:
Euler counted the number of routes into and out of each node or intersection on his network. He then reasoned that you would have to walk into and out of each node as you made
your way around the map, so you needed each node to have an even
number of routes. All the nodes in Königsberg have an odd number, so it is impossible to complete the
challenge. Networks where all the nodes have even numbers are known as
eulerian
. A
semi-eulerian
network is one that possesses two nodes with an odd number; if you start at one odd
node and finish at the other you can complete the network without repeating yourself, as with the following famous example:
In My Imagination
Euler was also interested in what are known as complex numbers. These are numbers that are made up of two parts, one real (i.e. any number between plus
and minus infinity) and one imaginary.
Diophantus had the first inklings of imaginary numbers (see
here
), but it wasn’t until the sixteenth century and the arrival of two Italian mathematicians,
Niccolò Fontana Tartaglia and Gerolamo Cardano, that the study of imaginary numbers really took off. Tartaglia and Cardano discovered that some equations only generate an answer if you
are prepared to allow negative numbers to have square roots – which can’t happen with real numbers, because a negative multiplied by a negative gives a positive.
Descartes coined the term ‘imaginary’ – even though such numbers don’t exist, you can permit them to exist in your imagination in order to find
answers to previously unsolvable equations.
The letter i is used to denote the square root of minus 1: √-1
This allows you to reference the square root of any negative number:
√ -49 = √(49 × -1) = √49 × √-1 = 7i
Despite these numbers being imaginary, they have many practical applications, especially in electronics and electrical engineering.
This way of simplifying and thinking about maps and routes had implications for cartography. It’s also an important part of
decision mathematics
, the branch of
mathematics that many businesses rely on for planning delivery routes and other logistical operations.
The many faces of geometry
Euler worked in three-dimensional geometry too. He discovered there is a relationship between the number of corners (or
vertices), edges and faces of a
polyhedron (a three-dimensional shape with flat faces, like a cube or a pyramid):
vertices – edges + faces = 2
Mathematical Perfection
Euler devised an equation now known as
Euler’s identity
(an identity is an equation that is always true no matter what value you use for the
unknown), which is said to be the most beautiful and elegant mathematical equation. It relates to complex numbers, but unfortunately its meaning is beyond the scope of this book. Here is the
equation in all its glory:
e
iπ
+ 1 = 0
Its subjective mathematical beauty arises from the fact that it uses five of the most important numbers in mathematics: e, i, π, 1 and 0.
A cube has 8 vertices, 12 edges and 6 faces: 8 – 12 + 6 = 2
A tetrahedron (triangular-based pyramid) has 4 vertices, 6 edges and 4 faces: 4 – 6 + 4 = 2