Alex’s Adventures in Numberland (34 page)

Multiplying out:

The father of algebra died aged 84.

We can now return to the trick at the start of the chapter. I asked you to name a three-digit number for which the first and last digits differed by at least two. I then asked you to reverse that number to give you a second number. After that, I asked you to subtract the smaller number from the larger number. So, if you chose 614, the reverse is 416. Then, 614 – 416 = 198. I then asked you to add this intermediary result to its reverse. In the above case, this is 198 + 891.

As before, the answer is 1089. It always will be, and algebra tells you why. First, though, we need to find a way of describing our protagonist, the three-digit number in which the first and last digits differ by at least two.

Consider the number 614. This is equal to 600 + 10 + 4. In fact, any three-digit number written
abc
can be written 100
a
+ 10
b
+
c
(note:
abc
in this case is
not
a
×
b
^×
c
). So, let’s call our initial number
abc
, where
a
,
b
and
c
are single digits. For the sake of convenience, make
a
bigger than
c
.

The reverse of
abc
is
cba
which can be expanded as 100
c
+ 10
b
+
a
.

We are required to subtract
cba
from
abc
to give an intermediary result. So
abc
–
cba
is:

(100
a
+ 10
b
+
c
) – (100
c
+ 10
b
+
a
)

The two
b
terms cancel each other out, leaving an intermediary result of:

99
a
c
, or

99(
a
–
c
)

At a basic level algebra doesn’t involve any special insight, but rather the application of certain rules. The aim is to apply these rules until the expression is as simple as possible.

The term 99(
a
–
c
) is as neatly arranged as it can be.

Since the first and last digits in
abc
differ by at least 2, then
a
–
c
is either 2, 3, 4, 5, 6, 7 or 8.

So, 99(
a
–
c
) is one of the following: 198, 297, 396, 495, 594, 693 or 792. Whatever three-figure number we started with, once we have subtracted it from its reverse, we have an intermediary result that is one of the above eight numbers.

The final stage is to add this intermediary number to its reverse.

Let’s repeat what we did before and apply it to the intermediary number. We’ll call our intermediary number
def
, which is 100
d
+ 10
e
+
f
. We want to add
def
to
fed
, its reverse. Looking closely at the list of possible intermediary numbers above, we see that the middle number,
e
, is always 9. And also that the first and third numbers always add up to 9, in other words
d
+
f
= 9. So,
def
+
fed
is:

100
d
+ 10
e
+
f
+ 100
f
+ 10
e
+
d

Or:

100(
d
+
f
) + 20
e
+
d
+
f

Which is:

(100 × 9) + (20 × 9) + 9

Or:

900 + 180 + 9

Hey presto! The total is 1089, and the riddle is laid bare.

The surprise of the 1089 trick is that from a randomly chosen number we can always produce a fixed number. Algebra lets us see beyond the legerdemain, providing a way to go from the concrete to the abstract – from tracking the behaviour of a specific number to tracking the behaviour of
any
number. It is an indispensable tool, and not just for maths. The rest of science also relies on the language of equations.

In 1621, a Latin translation of Diophantus’s masterpiece
Arithmetica
was published in France. The new edition rekindled interest in ancient problem-solving techniques, which, combined with better numerical and symbolic notation, ushered in a new era of mathematical thought. Less convoluted notation allowed greater clarity in descrig problems. Pierre de Fermat, a civil servant and judge living in Toulouse, was an enthusiastic amateur mathematician who filled his own copy of
Arithmetica
with numerical musings. Next to a section dealing with Pythagorean triples – any set of natural numbers
a
,
b
and
c
such that
a
²
+
b
²
=
c
²
, for example 3, 4 and 5 – Fermat scribbled some notes in the margin. He had noticed that it was impossible to find values for
a
,
b
and
c
such that
a
³
+
b
³
=
c
³
. He was also unable to find values for
a
,
b
and
c
such that
a
⁴
+
b
⁴
=
c
⁴
. Fermat wrote in his
Arithmetica
that for any number
n
greater than 2, there were no possible values
a
,
b
and
c
that satisfied the equation
a
ⁿ
+
b
ⁿ
=
c
ⁿ
. ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain,’ he wrote.

Fermat never produced a proof – marvellous or otherwise – of his proposition even when unconstrained by narrow margins. His jottings in
Arithmetica
may have been an indication that he had a proof, or he may have believed he had a proof, or he may have been trying to be provocative. In any case, his cheeky sentence was fantastic bait to generations of mathematicians. The proposition became known as Fermat’s Last Theorem and was the most famous unsolved problem in maths until the Briton Andrew Wiles cracked it in 1995. Algebra can be very humbling in this way – ease in stating a problem has no correlation with ease in solving it. Wiles’s proof is so complicated that it is probably understood by no more than a couple of hundred people.

Improvements in mathematical notation enabled the discovery of new concepts. The
logarithm
was a massively important invention in the early seventeenth century, thought up by the Scottish mathematician John Napier, the Laird of Merchiston, who was, in fact, much more famous in his lifetime for his work on theology. Napier wrote a best-selling Protestant polemic in which he claimed that the Pope was the Antichrist and predicted that the Day of Judgement would come between 1688 and 1700. In the evening he liked to wear a long robe and pace outside his tower chamber, which added to his reputation as a necromancer. He also experimented with fertilizers on his vast estate near Edinburgh, and came up with ideas for military hardware, such as a chariot with a ‘moving mouth of mettle’ that would ‘scatter destruction on all sides’ and a machine for ‘sayling under water, with divers and other strategems for harming of the enemyes’ – precursors of the tank and the submarine. As a mathematician, he popularized the use of the decimal point, as well as coming up with the idea of logarithms, coining the term from the Greek
logos
, ratio, and
arithmos
, number.

Don’t be put off by the following definition:
the logarithm, or log, of a number is the exponent when that number is expressed as a power of 10
. Logarithms are more easily understood when expressed algebraically: if
a
= 10
^b
, then the log of
a>
is
b
.

Finding the log of a number is self-evident if the number is a multiple of 10. But what if you’re trying to find the log of a number that isn’t a multiple of 10? For example, what is the logarithm of 6? The log of 6 is the number
a
such that when 10 is multiplied by itself
a
times you get 6. However, it seems completely nonsensical to say that you can multiply 10 by itself a certain number of times to get 6. How can you multiply 10 by itself a fraction of times? Of course, the concept
is
nonsensical when we imagine what this might mean in the real world, but the power and beauty of mathematics is that we do not need to be concerned with any meaning beyond the algebraic definition.

The log of 6 is 0.778 to three decimal places. In other words, when we multiply 10 by itself 0.778 times, we get 6.

Here is a list of the logarithms of the numbers from 1 to 10, each to three decimal places.

log 1 = 0

log 2 = 0.301

log 3 = 0.477

log 4 = 0.602

log 5 = 0.699

log 6 = 0.778

log 7 = 0.845

log 8 = 0.903

log 9 = 0.954

log 10 = 1

So, what’s the point of logarithms? Logarithms turn the more difficult operation of multiplication into the simpler process of addition. More precisely, the multiplication of two numbers is equivalent to the addition of their logs. If X × Y = Z, then log X + log Y = log Z.

We can check this equation using the table above.

3×3 = 9

log 3 + log 3 = log 9

0.477 + 0.477 = 0.954

Again,

2×4 = 8

log 2 + log 4 = log 8

0.301 + 0.602 = 0.903

The following method can therefore be used in order to multiply two numbers together: convert them into logs, add them to get a third log, and then convert this log back into a number. For example, what is 2×3? We find the logs of 2 and 3, which are 0.301 and 0.477, and add them, which is 0.788. From the list above, 0.788 is log 6. So, the answer is 6.

Now, let’s multiply 89 by 62.

First, we need to find their logs, which we can do by putting the number into a calculator or Google. Until the late twentieth century, however, the only way of doing this was done by consulting log tables. The log of 89 is 1.949 to three decimal places. The log of 62 is 1.792.

So, the sum of the logs is 1.949 + 1.792 = 3.741.

The number whose log is 3.741 is 5518. This is again found by using the log tables.

So, 89×62 = 5518.