history of mathematics - National STEM Centre

Practice exercises
Chapter 2, page 151
1 a 42.42638889, 1.414212963
b V2
c The answer needs to be a diagram like Figure 2.11.
d Input N, A
repeat
display A
until satisfied
Output The square root of N.
e 2, 2.75, 2.647727273, 2.645752048, 2.645751311,
2.645751311
2 Babylonians used the result in specific cases.
Pythagoras's theorem involves proof, and there is no
evidence that the Babylonians proved the result.
3 1 ; 0,45 and 0; 59, 15,33,20
4 a How much grain is left over if as many men as
possible are given 7 sila from a silo of grain.
b 40,0 times 8,0 to give the number of sila in a silo.
c To divide by 7, multiply by the reciprocal of 7.
d By multiplying
e The reciprocal of 7 has been taken as 0; 8,33 instead
of 0; 8,34, 17,8
Chapter 3, page 152
I a Let AB intersect / at the point K. Draw a circle
centre K, radius KA, to cut / at C, and a circle centre K,
radius KB, to cut / at D. Then KA + KB = KC + KD so
AB = CD.
b Let AB produced intersect / at the point K. Draw
circles as for part a. Then KA - KB = KC - KD so
AB = CD.
c Take any point C on / and draw CA. Construct an
equilateral triangle on CA, with the third point called E.
Draw a circle centre A, radius AB, to cut EA at the point
14 Answers
F. Draw a circle centre E, radius EF, to cut CE at the
point G. Draw a circle centre C, radius CG, to cut / at the
point D. Then CD = AB.
2 Construct an equilateral triangle ABC on AB. Draw a
circle, centre B, with radius BP, to cut BC at D. Draw a
circle, centre C, with radius CD to cut AC at E. Draw a
circle, centre A, with radius AE, to cut AB at Q. Then
AQ = BP.
3 First construct a rectangle twice the area by
constructing a rectangle with a common base and the
same height as the triangle. Then bisect the base of the
rectangle to give a new rectangle with the correct area.
Call this rectangle GHIJ, with HI as one of the shorter
sides. Draw an arc of a circle, centre H, with radius HI.
Extend GH to intersect this arc outside GH at N. Find the
mid-point of GN and construct a semi-circle with this
point as centre. Construct a perpendicular to GN at H.
Call the point of intersection of the perpendicular with
the semi-circle K and construct the required square on
HK.
Chapter 4, page 153
1 a Diagram of the following form.
b c d
b a(b + c + d) = ab + ac + ad
area of large rectangle = sum of areas of small rectangles.
2 Using the algorithm on a graphics calculator, the
greatest common divisor of 2689 and 4001 is 1.
3 If a measures b then ka = b for some integer k. Hence
k 2 a 2 = b , so a 2 measures b 2 .
4 Archimedes: spirals, measurement of a circle, or
approximating pi, quadrature of a parabola, trisecting
angles, measurement of area and volume
Chapter 5, page 154
1 a x 2 + IQx = 39
185