history of mathematics - National STEM Centre

Activity 11.1, page 138
1 You may have established the result for the product of
two negative numbers by putting equal to zero the first
number in the pair forming each of the subtracted
numbers. Yet you have no idea whether this rule is valid
or not since you have not proved it as such. Equally you
have no way of knowing whether your equation is valid
with a and c both put as zero; the original argument was a
geometric one with a and c representing lengths of the
sides of a rectangle. It makes no sense geometrically to
put these lengths equal to zero and to retain the existence
of a rectangle.
Activity 11.2, page 138
1 a If you take +2 as representing two steps to the
right, then -(+2) represents two steps to the left since the
minus sign indicates a change of direction.
You can consider -(+2) to be the same as (-1) x (+2).
Hence (-1) x (+2) represents two steps to the left, that is,
the number -2.
If the order in which the multiplication is carried out is
reversed, then the result is the same although the analogy
is slightly different. You start with - 1, a single step to
the left, and then you multiply it by 2. This gives two
steps to the left, that is the number -2.
This extends to any positive number a, so you have
shown that (-1) x (a) = (a) x (-1). Now multiply the
result of the first case by 3; this produces six steps to the
left. Hence (+3) x (-1) x (+2) is -6. But you have
already decided that (+3) x (-1) should be the same as
(-3). So we have shown that (-3)x(+2)=-6. A similar
argument can be applied using the step model to any
negative number —b and positive number a.
By analogy with the step model the product of a negative
number with a positive number (with the negative one as
the first number in the product) is negative .
b A similar reasoning applies to the product of two
negative numbers. In the first instance you start with -2
which represents two steps to the left. So -(-2)
represents a change of direction applied to this step and
gives two steps to the right. Thus the operation
(-1) x (-2) can be considered to be the same as the
number +2. The rest of the explanation follows in an
analogous way to part a.
14 Answers
Activity 11.3, page 139
1 a If i is a line of unit length, inclined at 90° to the
positive number +1, then you can consider multiplying
by i as applying a rotation of 90° anti-clockwise. Since
i = i x i, multiplying by i 2 is the same as multiplying by
i twice, or rotating by 90° twice. This is the same as
rotating by 180° or multiplying by -1.
+1 -1 +i -i
+1
-1
-i
+1 -1 +i -i
-1 +1 -i +i
+i -i -1 +1
-i +i +1 -1
2 a (2 + x)(3 - x) = 6 + 3* - 2x -
= 6 + x - x 2 . By analogy
(2 + i)(3-i) = 6 + 3i-2i-i 2
= 6 + 3i-2i + l
= (6 + l) + (3-2)i
b (2 + i)(3.5-27i) = 34-50.5i,
(2 + i)(0 + 94i) = 188i-94
(3.5 -27i)(0 + 94i) = 2538 + 329i
c (2 + i) + (3.5-27i) = 5.5-26i,
(2 + i) + (0 + 94i) = 2 + 95i
(3. 5 - 27i) + (0 + 94i) = 3. 5 + 67i
Activity 11.4, page 141
1 One counter-example is given in the answer to
question Ic in Activity 10.10, that is, if you have two
positive numbers a and b which satisfy a < b and
multiply both sides of the inequality by c > 0 , then
ac < be . However if you multiply by c < 0 then you have
to reverse the inequality to give ac> be .
A second counter-example is: if a and b are positive
numbers then their sum is greater than each of the
individual numbers; this is not true if a and b are
negative numbers.
Activity 11.5, page 141
1 a The required number under addition is 0 since
a + 0 = 0 + a = a that is, adding zero to any number in
either order leaves that number unchanged. Zero is called
the identity under addition. The number which plays a
similar role in multiplication, that is, the identity under
multiplication, is +1. You can justify this because
axl = 1 xa = a.
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