history of mathematics - National STEM Centre

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of multiplication by negative numbers when the
inequality must be reversed. So if a < b and we multiply
both sides of the inequality by c < 0, then aobc. You
know that you have to reverse the inequality when you
multiply by a negative number. This means that there is
no contradiction in the rules of multiplication when they
are applied to negative numbers as Arnauld suggests.
Activity 10.11, page 131
1 a Lists of assumptions
Euler
• assumes that the outcome of multiplying together two
numbers is not affected by the order in which we take
the two numbers
• models a negative number as a debt and uses obvious
properties of debts to infer properties of negative
numbers
• assumes that the product of two negative numbers
must be either a positive or a negative number, that is,
it cannot be something new
• assumes that if you take a number and in turn
multiply it by different numbers then the outcome
cannot be the same in both cases
• assumes that when multiplying two numbers together
it is possible to calculate the size (absolute value) of
the product by multiplying together the sizes
(absolute values) of the individual numbers.
Saunderson
• assumes that you can establish that numbers are in
arithmetic progression if there are three or more of
them to consider
• assumes that if you know the first two terms of an
arithmetic progression, then the third can easily be
found
• assumes that if all the terms in an arithmetic
progression are multiplied by the same factor then the
resultant terms (taken in the same order) will also
form an arithmetic progression
• assumes that, even though you might start with an
arithmetic progression with decreasing terms, having
multiplied each by the same factor then the outcome
may be a progression with increasing terms, or vice
versa.
b Arnauld's objections to negative numbers are
summarised below to inform the debate. Arnauld's
objection relates fundamentally to ratios of numbers. He
believes that 1 is to a what b is to ab. In other words he
believes that 1 : a is equivalent to b : ab. Examples are:
1 : 3 is equivalent to 4:12;
180
1 : \ is equivalent to -5-: -^.
But he does not believe that 1: - 4 is equivalent to
-5:20.
The reason he gives is that 1 and - 4 cannot be in the
same ratio as -5 and 20 because 1 > -4 whereas
-5 < 20. On that basis alone he disbelieves the rule for
multiplying negative numbers together.
How might a debate between Euler and Arnauld have
been played out on the basis of the extracts? It all
depends on the different assumptions that each makes
about operations of numbers.
Presumably Arnauld would be prepared to accept that
1: - 3 is equivalent to 4 : - 12 since his ordering
principle applies. In a debate Arnauld could be persuaded
to accept that —b multiplied by a (where a and b are both
positive) produces a negative result. However he would
not accept that 1 : 4 is equivalent to -3: -12 since this
violates the ordering principle, and thus presumably
would not accept that a multiplied by -b is valid.
This logically means that Arnauld cannot accept that a
times —b is the same as —b times a.
On the other hand Euler assumes that the order in which
you multiply a positive by a negative number does not
affect the outcome. A debate between Euler and Arnauld
could centre on this difference of opinion with one trying
to convince the other of his view. Euler could be made to
justify his assumption. This might be the chink in Euler's
or Arnauld's armour.
Saunderson and Arnauld have more obvious differences
of opinion. Saunderson considers numbers in arithmetic
progression; for example, the decreasing sequence 3, 0,
-3. By multiplying each of the numbers in the sequence
by - 4 say, he assumes we get another sequence which
starts with -12,0. The third member of the sequence
must be 12 (using the arithmetic sequence properties) and
this led him to deduce that -3 multiplied by -4 is 12.
Arnauld would clearly object to this on the grounds that
Saunderson had started with a decreasing sequence and
yet ended up after multiplication with an increasing
sequence. The ordering principle had been violated.
At the end of the day, the three mathematicians make
different assumptions; perhaps Euler and Saunderson are
prepared to go further than Arnauld and that is where the
fundamental difference lies.