Choose any three by three square of dates on a calendar page.
Circle any number on the top row, put a line through the other
numbers that are in the same row and column as your circled number.
Repeat this for a number of your choice from the second row. You
should now have just one number left on the bottom row, circle it.
Find the total for the three numbers circled. Compare this total
with the number in the centre of the square. What do you find? Can
you explain why this happens?
Can you go through this maze so that the numbers you pass add to exactly 100?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Published December 2011,February 2011.
During the school year you are
often asked, in different areas of the curriculum, to research the
life of and write a report about somebody who has greatly
influenced life and learning . You may be asked to describe the
work of a famous mathematician and how that influences what you
learn in mathematics today. You might be required in social studies
to choose a person who has had a huge impact on many cultures and
many people throughout time. It could be that for a science project
you investigate the life of a scientist, such as an astronomer, who
made important discoveries about what we know today about our earth
or other planets. Or perhaps the music teacher will set you an
assignment to discover a character who made a musical breakthrough.
For any one of these assignments you could select the same person
to study, Pythagoras .
Usually, when we hear the name Pythagoras, or more formally
Pythagoras of Samos, we think of right-angled triangles or the
hypotenuse and maybe squares and things! But there was a lot more
to Pythagoras than his famous theorem.
He was well known in his day, enough that statues of him were
sculptured and drawings and paintings made. If you look for
information about him at a maths history site, like the wonderfully
informative one produced by
St Andrew's University , you will discover that after Nash,
Einstein and Newton, Pythagoras' is the most requested biography of
a mathematician. He lived from about 580 BC to about 500 BC in
Greece, we can't be sure exactly, but it was a long time ago and we
are as fascinated by him today as people in his time were.
Little is known of Pythagoras' childhood. The only description
of how he looked that is probably true is the description of a
noticeable birthmark on his thigh! Information differs, some
sources say that he had two brothers, although others state it was
three. What they agree on is that he was well educated; he was a
fine musician, he played the lyre and used music to help people who
were ill; he learned poetry and was able to recite famous and
popular Greek writers like Homer. While he was a young student,
three teachers who were philosophers greatly influenced
When he was between 18 and 20
years old, Pythagoras left Greece and went to a town called
Miletus, which is in the country we now call Turkey, and visited an
old man named Thales.
Thales made a big impression on him and advised Pythagoras to
travel to Egypt. While he was there he visited many temples and
took part in discussions with the priests and learned from some of
Thales' pupils about geometry and cosmology. Because of the people
he met and the experiences he had, Pythagoras became a philosopher
like his teachers, but went on to make important discoveries in
mathematics, astronomy, and the theory of music.
In 525 BC, after a battle between the king of Persia and the
Pythagoras was taken prisoner and taken to Babylon where he
continued his quest for learning new things. He was instructed in
the sacred rites of the Babylonians and
learnt about their mystical worship of the gods. He perfected his
skills in music and arithmetic as well as the other mathematical
sciences taught by the Babylonians.
Although he was a very important figure in the development of
mathematical ideas, we don't know much about Pythagoras' actual
mathematical achievements. Unlike many other Greek mathematicians,
none of his writings exist to provide evidence about his interests.
Fortunately, many other people wrote about him and his work and
from them we know that Pythagoras was really into numbers in a
big way! In fact, he thought numbers had personalities - he
regarded each one as either masculine or feminine, perfect or
incomplete, beautiful or ugly. This number interest probably came
about after his capture by the Babylonians.
Pythagoras was the first to
distinguish between odd and even numbers and he came up with the
idea of prime numbers and showed how they were different from what
we call composite numbers. But ten was the
very best number of all: it was the sum of the first four integers
one, two, three, and four [1 + 2 + 3 + 4 = 10]. If each of these
were represented by a dot notation in a series, one above the
other, they formed a perfect triangle. We can add perfect, abundant
and deficient as well as triangular numbers to Pythagoras' amazing
He worked also on the ratio between numbers and then went on
todiscoversome numbers that didn't have ratios, these he called
irrational numbers. Of course, we can't forget that he proved a
Babylonian idea that was already a thousand years old, that parts
of triangles are related in size. This has come to be known as Pythagoras'
The theorem proves that when the two shorter sides of a right
triangle are squared and then added together, the answer is the
same as the square of the longest side - or the hypotenuse. Try it
and see if you agree!
There are only five regular solid
shapes , and it is thought that Pythagoras knew how to
construct the first three but unlikely that he was able to
construct the other two. If you are not sure what 'regular solid'
means you can easily find
out . It seems strange today, but being able to construct three
solids was quite an achievement in Pythagoras' day and people even
wrote about it! For instructions how to construct all five visit a
What clearly made Pythagoras such a great thinker was the fact
that he was a great observer. Pythagoras observed that when a
string is vibrating, the longer or shorter the string is makes a
difference in the pitch of the note. Take a string and twang it.
,try with a string that is just half the length of the first. A
string exactly half its length will produce the same note, only one
octave higher. If you try with a string twice the length of the
original string, the note produced will be one octave lower.
Strings whose lengths form ratios equal to, or are in proportion
to, whole numbers produce harmonious sounds. This idea led to the
development of such things as the thread-board on a guitar.
Pythagoras made remarkable contributions to the mathematical
theory of music. I wonder how many guitar players know that
mathematics is behind their music!
As an astronomer
, Pythagoras was convinced that the Earth was a sphere but thought
that it was at the centre of the Universe. Once he was certain that
the Earth was a sphere, he was able to deduce that the sun and all
the other planets were also spherical and that they moved around
each other in circles.
Pythagoras obviously had a lot of knowledge to share. He didn't
keep it to himself but started a school and a society for other men
interested in the same things as he was. The school was called the
'semicircle'. I wonder what shape their seating arrangement was!
Pythagoras' followers called themselves the Pythagoreans.
Pythagoreans had beliefs and discoveries of their own and, today,
we can't tell the difference between their work and Pythagoras'.
What the Pythagoreans all had in common is the thought that
everything in the cosmos is about numbers.
If you were a Pythagorean, you had to be prepared to follow
strict rules. An interesting thing about the school was that you
were automatically a Pythagorean too. That meant if you were a
student you'd have to obey all the men's rules also.
Have you noticed the word 'men' is used. One rule was, no
females allowed! Another rule was silence. This meant that you
could not share any of your ideas with others outside the group and
that you could only listen to your fellow Pythagoreans. Because
they said nothing about what they studied, it was seen as a secret
society and people were quite suspicious and afraid of them.
Another rule was no eating meat. His followers also believed in
reincarnation. Many of these rules and beliefs came from customs
Pythagoras had come across in Egypt: for example, the secrets kept
by the Egyptian priests, the fact they never ate beans and their
refusal to wear any clothes made from animal skins.
Of course, Pythagoras will continue to be remembered mostly for
his famous geometry theorem. But we should also remember that his
belief that everything can be understood through and explained by
mathematics was extremely important to the development of science
and mathematics throughout the ages. He influenced the world by his
All is number ...