From 0 to Infinity in 26 Centuries (15 page)

A dodecahedron (12-sided polyhedron) has 20 vertices, 30 edges and 12 faces: 20 – 30 + 12 = 2

A truncated icosahedron (the combination of hexagons and pentagons used to make a football) has 60 vertices, 90 edges and 32 faces: 60 – 90 + 32 = 2

Euler also teamed up with Daniel Bernoulli (1700–82, son of Johann, nephew of Jacob) to work in applied mathematics. They considered the forces acting on beams in
buildings and how the forces would make the beams bend – a very useful tool in engineering applications.

C
ARL
G
AUSS
(1777–1855)

Carl Gauss was born into a poor family in Germany in 1777 but it soon became apparent that he possessed an extraordinary intellect and a special ability in mathematics in
particular.

According to legend, when Gauss was at school he continually annoyed his maths teacher by completing his work far faster than the rest of his class. Exasperated, Gauss’ teacher finally
told him to add together all the numbers from 1 to 100, thinking that might give him some peace.

Gauss immediately stated the correct answer: 5050.

Gauss was not a lightning calculator; he had instantaneously seen a shortcut. If you repeat the series, but backward, it can be seen that all the terms add up to 101:

Gauss then quickly worked out that 100 terms of 101 gives 10100, but as this is twice the sum we actually want, he halved it to give the answer 5050.

Back to the beginning

Gauss furthered Euler’s initial work in a branch of number theory called
modular arithmetic
, in which numbers are allowed up to a certain value, after which they
wrap-around and start again. The twenty-four-hour clock is an example of modular arithmetic – after 23:59 we start again at 00:00.

Much like normal arithmetic, in modular arithmetic you need to define what your highest number can be. In normal arithmetic we work in tens, but our highest digit is one less than this: 9. If we
are working in modulo 8 we can only use the digits from 0 to 7. This means that 8 would be 0 in modulo 8 because we start again from zero after we reach 7. Likewise, 15 would be 7 in modulo 8
because 15 = 8 + 7 but the 8 counts as 0. Mathematicians would write this as:

In certain circumstances, if you divide two numbers you may be more interested in the
remainder
(what’s left over) than the
quotient
(the answer to the
division). This is where modular arithmetic can be useful, because a number’s value in a particular modulo is the same as the remainder if you were dividing it. For example:

If you wanted to check whether a number was prime, you could see whether the number was ever equal to zero in successive modulos, which is something that computers are good
at.

A magnificent spread

Gauss’ work naturally moved into prime numbers, which remain one of the greatest mysteries in mathematics. Gauss made a conjecture, now called the
prime number
theorem
(it is a theorem because it has since been proven, (see
here
), about the way in which prime numbers are spread out. Although we do not have a formula for making prime numbers, Gauss
noticed that the higher up through the numbers you go, on average the more spread out the prime numbers become. He wrote:

The symbol ≈ means ‘is roughly equal to’ and the symbol ln means ‘natural logarithm’. Therefore:

This shows that, although we made x ten times larger, there are fewer than eight times as many primes. This trend continues as we make x bigger, so primes become fewer the
higher we count.

Uneven distribution

Gauss also made an important contribution to statistics by being the first person to introduce the
normal distribution
. This bell curve applies to all manner of
real-world situations such as animals’ heights and weights, marks in examinations, measurements made in scientific experiments and so on.

If you measured the height of every thirteen-year-old boy in the country, you could work out the average or
mean
height (worked out by adding up all of the data and dividing by how many
data there were). You could then look at the percentage of the boys in a certain height bracket and you would find that most of them were within a certain distance from this mean. As you move away
from the mean, either higher or lower, you find that there are fewer and fewer boys. Thinking in terms of percentages like this is the same as thinking in terms of probabilities, and so the normal
distribution is said to be a
probability density function
:

The shape of the graph shows what we know to be true. Think about your own friends. Unless you hang out with professional basketball players, most of your friends are clustered
around an average or normal height for their gender, and you probably know far fewer very tall and very short people.

The idea of IQ (intelligence quotient) is an example of a score that has been standardized using the normal distribution. An IQ score of 100 is the mean, and a 15-mark interval is called the
standard deviation
, which is a measure of how spread out the marks are. As a result of the equation of the line used for the normal distribution, it turns out that over 68% of scores are
within one standard deviation of the mean, so nearly 70% of people will have an IQ of between 85 and 115. Over 95% of people are within two standard deviations, scores of 70 to 130. Over 99.7% of
people are within three standard deviations, scores of 55 to 145. Mensa, a society for people with high IQs, has an entrance test intended to select people who have an IQ higher than 98% of the
population, which corresponds to an IQ of just under 131.