history of mathematics - National STEM Centre

Activity 3.7, page 38
1 A point is that which has no part. A line is breadthless
length.
3 It is not clear what 'lying evenly' means.
4 A circle is a closed curve. It contains a point O inside
it such that for all points X on the circle the lengths OX
are equal to each other. O is called the centre of the
circle. A diameter of the circle is any line through the
centre which has its ends on the circle. The diameter also
bisects the circle.
Activity 3.8, page 39
1 Postulate 3 says that if you are given the centre of a
circle and its radius, you can draw the circle. Postulate 4
says that any right angle is equal to any other right angle.
2 If a = b , then
a- x = b — x.
= b + x.Ifa = b, then
3 Postulate 5 gives a condition for two lines to meet,
and where they should meet. But what happens if the
'angles are two right angles'?
Activity 4.1, page 42
1 A P R B
D Q S C
2 ADx(AP + PR + RB) =
3 a(b
4
AD x AP + AD x PR + AD x RB
ab
ab
(a + b) 2 =a 2 +b 2 +2ab
Activity 4.2, page 43
l
Euclid
Input A,B
{A, E positive integers}
14 Answers
repeat
while A > B
A-B-+A
endwhile
if A = B
then
if 5 = 1
then
print "A, B co-prime"
else
print "A, B not co-prime"
stop
endif
else
endif
until 0*0
Output The highest common factor of A, B.
3 The 'greatest common measure' is the highest
common factor.
4 In the algorithm in question 1, replace the lines
following if A = B by
then
display B
stop
and the output is the highest common factor of A and B.
Activity 4.3, page 43
1 Suppose that the number of prime numbers is finite,
and can be written a, b, ... , c where c is the largest. Now
consider the number / = ab . . . c + 1 . Either/ is prime or
it is not.
Suppose first that/is prime. Then an additional prime/,
bigger than c has been found.
Suppose now that/is not prime. Then it is divisible by a
prime number g. But g is not any of the primes
a, b, ... ,c, for, if it is, it divides the product ab ... c , and
also ab ...c + l, and hence it must also divide 1 , which is
nonsense. Therefore g is not one of a, b, . . . , c, but it is
prime. So an additional prime has been found.
Therefore the primes can be written a,b,...,c,g, which
is more than were originally supposed.
Activity 4.4, page 45
1 Two lengths are commensurable if there is a unit such
that each length is an exact multiple of that unit. If no
such unit exists, the lengths are incommensurable.
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