history of mathematics - National STEM Centre

Activity 4.2
Activity 4.3
4 More Greek mathematics
Draw a diagram to illustrate this proposition in the case where the cut straight line is
cut into three line segments.
2 With reference to your diagram, what is the proposition saying?
3 Write an algebraic statement that is equivalent to the proposition.
4 Proposition 4 from Book II states:
I
If a straight line be cut at random, the square on the whole is equal to the
squares on the segments and twice the rectangle contained by the
segments.
Illustrate this proposition and write down an equivalent algebraic statement.
The Euclidean algorithm
Book VII of the Elements contains 39 propositions. The first 19 deal with ratios and
proportions of integers. Proposition 1 introduces the procedure now known as
Euclid's algorithm, which plays an important part in the theory of numbers.
Two unequal numbers being set out, and the less being continually
subtracted from the greater, if the number which is left never measures
the one before it until a unit is left, the original numbers will be prime to
one another.
A number which when subtracted several times from another gives a zero remainder
is said to measure it exactly.
1 Study the proposition above, and use it to write an algorithm which takes two
positive whole numbers as input, and, as output, states whether or not the numbers
are prime to one another; that is, whether or not they are co-prime.
2 Program your graphics calculator to carry out the Euclidean algorithm, and
check that it reports correctly whether or not the two input numbers are co-prime.
3 Proposition 2 of Book VII makes use of the Euclidean algorithm.
I
Given two numbers not prime to one another, to find their greatest
common measure.
Write down in your own words what 'greatest common measure' means.
4 Experiment with your graphics calculator program to find out how the Euclidean
algorithm can be used to find the greatest common measure.
A proposition about prime numbers
Later books in the Elements, Books VII io IX, deal with the theory of numbers. In
Book IX, proposition 20 is particularly well-known:
Prime numbers are more than any assigned multitude of prime numbers.
Let A, B, C be the assigned prime numbers: I say that there are more prime
numbers than A, B, C.
43