10. How to Multiply Signed Numbers
A. Products of Signed Numbers (Multiplication Rule)
August 11, 2008 addition
So far we have met and define products of
-
unsigned fractions
-
unsigned decimals (finite and infinite)
But the product or multiplication of signed numbers has yet to
be defined.
We start by giving a sign multiplication rule
(+)(+) = +
(+)(-) = -
(-)(+) = -
(-)(-) = +
Next if A and B are real numbers, we let their product
AB = (sign A)(sign B) [(length of A)] [(Length of B)]
AB = [(sign A)(sign B)] [(magnitude of A)(magnitude of
B)]
Call this the multiply the signs, multiply the lengths,
product rule. Examples follow to illustrate the rule
(+5)(+6) = [(+)(+)][5*6]
= + 30
(+5)(-3) = [(+)(-)][5*3]
= - 15
(-8)(+½) = [(-)(+)][8*½]
= - 4
(-2)(+4.5) = [(-)(+)][2*4.5]
= - 9
(-6)(-40) = [(-)(-)][6*40]
= + 240
The rule or law for sign multiplication implies
(+1)(+1) = +1
(+1)(-1) = -1
(-1)(+1) = -1
(-1)(-1) = +1
Cosmetic Option: Instead of writing sign(-10) = -,
we may write sign(-10) = -1. Likewise, instead of writing sign(+10) = +,
we may write sign(+10) = +1 or 1. Then we may write -10 = 10 (-1) or (-1)10 and
we may write +8 = 8 (+1) or (+1)8. This option replaces the sign + and - by the
factors +1 and -1.
B. Repeated Multiplication of unit vectors by real numbers:
August 11, 2008 addition
Let u be a nonzero vector.
u:
(o)==>
Then we may let v = +3u
+3u:
(o)==>==>==>
v: (o)========>
and w = +2 v = +2(+3u).
2(3u):
==>==>==>==>==>==>
2v: ========>========>
w: ========>========>
Thus w = 6 u or +2(+3u)=
[(+2)(+3)] u
Next x = -2(+3u) gives x = -(2*3)u =
-6 u
==>==>==> 3u
========>3u
-2(3u) <========<========
Hence -2(+3u)= [(-2)(+3)] u
Likewise, y = +2(-3u) gives y = -(2*3)u
= -6 u
==>==>==> 3u
-3u <========
2(-3u) <========<========
Hence +2(-3u)= [(+2)(-3)] u
Lastly, z = -2(-3u) gives z = (2*3)u
= -6 u
==>==>==> 3u
-3u <========
2(-3u) <========<========
========>========> -2(-3u)
Hence -2(-3u)= [(-2)(-3)] u
The foregoing suggests if A and B are real numbers, and u
is a vector, then A(Bu) = [AB]u. The latter
equality, an associative law for multiplication of vectors by
signed numbers, provides motivation for multiply the
signs, multiply the lengths, product rule for real numbers.
Suppose vector k is employed as a ( unit) vector. Let m
= q k for some real number q and let p be a real number as
well. Then we may form the product p m and p m = c k
for some unique real number c by measurement assumption. Now the
associative for the multiplication of vectors by signed numbers implies
p m = p ( q k ) = (pq) k
Thus c = pq since the multiplier c that makes c k
= p m is unique.
Advanced Material:
Before August 11, 2008, the following method was
indicated for defining the product of real numbers p and q. It is no
longer required in the development. But it points to an alternative
viewpoint that was explored.
Changing the Coordinate Scale
A unit vector k may be replaced by positive real
multiple m
= q k. When q is a a natural number or fraction, we may argue that
A = s m implies A = s(q k) . So each multiple s of m
correspond to a multiple sq of k
Changing the Coordinate Scale and Direction.
Defining Products of Real Numbers
from a change of units
(or from successive multiplication of vectors by signed
numbers)
Saying how to compute a number defines it.
Suppose vector k is employed as a (unit) vector. Let m
= q k for some real number q and let p be a real number as
well. Then we may form the product p m and p m = c k
for some unique real number c by measurement assumption.
Product Definition: Let the product of p and q, denoted by pq,
be given by c if ck =
p(qk) for the unit vector k. (Saying how to compute
pq defines it)
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This computational method for pq raises a few questions.
- Is it consistent with the previously encountered method for multiplying pairs of fractions and unsigned real numbers with infinite decimal
expansions?. The answer needs to be Yes
- Does the product depend on the choice of unit length k? The
answers needs to be No.
Showing the answers are as required is also left for the reader for the
time being. In what follows, we assume pq is defined for pairs of real numbers pq.
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Exercise: Show the foregoing product definition leads to the law of
signs:
(positive)(positive) gives a positive
(positive)(negative) gives a negative
(negative)(positive) gives a negative
(negative)(negative) gives a positive
Exercise: Each real number can written as a sign (direction)
prefixed to an unsigned real number - the magnitude of the real number
Show the product can be given by multiplying signs and magnitudes
separately. From this show, earlier obtain properties of fractions and real
numbers imply multiplication is commutative and associative. One step in this
process is to observe or to show that the multiplication of signs is
commutative and associative.
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