history of mathematics - National STEM Centre

14 Answers
To get an interpretation of subtraction remember that the
operation of subtraction is the inverse of addition. If you
want to reverse, or undo, the result of adding a number
then you subtract it again. If you model addition as a
movement to the right on the number line, then it is
natural to model subtraction as a movement to the left.
Suppose that you take 2 and subtract 3 from it. You start
at the point corresponding to 2 and move through a
distance of 3 units to the left, ending up at the point
corresponding to -1. Thus your picture accurately models
the arithmetic operation: 2 - 3 = -1.
b You can derive an interpretation of multiplication of
two positive numbers from your model of addition.
Multiplying a positive number a by another positive
number b is equivalent to adding the number b to the
number zero a times. In this picture, to consider the result
of 2 x 3, for example, start at the point 0 and move
through a distance of 3 to the right, ending up at the
interim point 3. Then move from the interim point 3
through another 3 units to the right thus producing the
final result of 6. On the other hand 3x2 involves starting
at 0 and adding 2 in each of three stages.
This model can be extended to consider examples such as
2 x (-3). Starting at 0, you add -3, or equivalently
subtract 3, in each of two stages. This involves two
movements to the left and produces the final point -6.
In this model it is difficult to come up with an
interpretation of an operation such as (-3) x 2. To be
consistent with the picture used so far, you would have to
start at 0 and then add 2 a certain number of times, but
how many times corresponds to -3? Your model breaks
down at this point and it can only come up with an
answer for (-3) x 2, if you assume that it has the same
value as 2 x (-3). But this assumption is based on things
you know about multiplication from another source; it is
not derived from your model. If you only draw on the
properties of the model itself then the model fails at a
particular point.
It is also difficult in this model to give an interpretation
of the result of multiplying two negative numbers.
Other models of operations on the number line do permit
a consistent interpretation of addition, subtraction,
multiplication and division applied to all combinations of
positive and negative numbers. You will meet one in
Chapter 11. Until then you should bear in mind that some
models of numbers only allow a limited interpretation of
operations on the numbers.
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Activity 10.2, page 125
1 You can use subtracted numbers to model practical
problems since they have many practical applications; for
example, the difference in the length of two lines, which
arose quite naturally in geometric arguments proposed by
the Greeks, and the amount of cereal remaining in the
store after some had been removed during the winter to
feed the people in a village.
You can see from these examples that the process of
subtracting one positive number from another is much
more acceptable. The practical application of the process
means that it is not possible to subtract a larger number
from a smaller. You should think of this in the context of
the cereals. If the store contained 2 tons of rice and you
needed 3 and you started to remove 3, then the store
would be empty long before you had finished. Thus the
subtracted number 2-3 would not have had any
practical solution. Using subtracted numbers does not
imply the need for negative numbers; subtracted numbers
which in modern terms would give rise to a negative
result were not considered possible.
Activity 10.3, page 126
\ You can set out the problem as
-5jc + 6y + 8z = 290
where x, y and z are the prices of cows, sheep and pigs
respectively. First divide the second equation through by
3, and then re-order the equations to give
jc-3;y + z = 0
-5x + 6y + 8z = 290
You can now 'eliminate' x from the second and third
equations using the first one. This gives
You can solve these for y and z, and then substitute these
values into the first of the re-ordered equations to obtain
a value for x.
The prices of the various livestock are as follows: cows
70 pieces, sheep 40 pieces and pigs 50 pieces.