8 Addition of Signed Numbers
and Addition Properties
8A. How to add Signed (Real) Numbers
Saying how to compute a sum defines it.
Saying how to compute a expression, defines it. So far we have said how to
add
- fractions
- unsigned real numbers p with finite and/or infinite decimal
expansions,
- collinear (or horizontal) vectors with the same or opposite directions
- signed multiples of a nonzero vector k.
but we have not said how to add signed real numbers in general. We do
that now in a way suggest by and consistent with the previous method for adding
signed multiples of a nonzero vector k.
Preliminary Step - Define Magnitude and sign
The magnitude of a signed number is given by removing its sign prefix.
The result is an unsigned number. Thus the magnitude (or absolute value)
of the signed number -10 is 10; the magnitude of the signed number +8.5 is
8.5; and the magnitude of -5 is 5; and the magnitude of 0 (zero) is 0
(zero). Here +0 = -0 = 0 all have the same value.
We could say length instead of magnitude. The actual length of a
multiple -10k of a unit vector k would be 10 units, while the
length relative to the unit vector k would be 10.
The sign of a real number is given by the value of the prefix use to
indicate the sign. So the sign of -10 is -; the sign of +8.5 and
8.5 is +. The sign of 0 need not be defined, but it can be taken to be
+.
Addition of Real Numbers:
The sum of two real numbers A and B is given as follows
- If A and B have the same sign then
A+B = (common sign)( Magnitude(A) + Magnitude(B))
= (common sign)(sum of the
addend's magnitudes)
Here the magnitudes are unsigned real numbers given by decimal or fractions
etc.
- If A and B have opposite signs and are equal in magnitude (length)
then A and B are additive inverses, and
A+B = 0
- If A and B have opposite signs and unequal in magnitude (length)
then
A+B = (sign of Biggest)( Biggest - Smallest)
= (sign of longest)
(Longest - Shortest)
8B. Distributive identity (law) for multiplication of vectors
Let A
and B be real numbers. Let k be unit length for a real number line.
Then u = A k and v = B k are lengths of
directed line segments which may be added head of u to tail of v.
The resultant vector w = u + v is a multiple C of k.
That is
A k + B k = C k
for some real number C, unique due to the measurement
assumption. It is clear that C = A+ B. That
implies the distributive law identity (law) for multiplication of vectors by
signed numbers:
(A+
B) k = A k + B k
8C. Addition of Real Numbers is Commutative
From the commutativity of vector addition show above, we observe A k + B
k = B k + A k, and so conclude C = A+ B = B +
A. That is, addition is commutative.
8D. Addition of Real Numbers is Associative
The associatively of in-place head-to-tail vector addition, and the
"rigid body" preservation of lengths and direction under translations implies
(A k + B k) + C k
= A k + (B k + C k)
Therefore
[(A+B)+C] k
= (A k + B k) + C k
= A k + ( B k + C k)
= [A+{B+C)] k
The foregoing implies
[(A+B)+C] k = [A+{B+C)] k
Whence the associative law
[(A+B)+C] = [A+{B+C)]
holds by comparison of coefficients that is justified by the unique measurement assumption.
The following argument appeared earlier. It is no longer
required except to indicate a different development was considered.
8. Addition of Signed Real Numbers
version prior to 2008, Aug 11 - no longer required,
preserved to indicate exploration of a different path.
Defining A Sum of Real Numbers from Vector Addition (or
displacements)
Saying how to compute a number defines it.
Saying how to compute a expression, defines it. So far we have said how
to add fractions and further unsigned real numbers p with infinite decimal
expansions, but we have not said how to add real numbers in general.
Let A and B be real numbers. Let k be unit length for a real number
line. Then u = A k and v = B k are
lengths of directed line segments which may be added head of u to
tail of v. The resultant vector w = u + v is a
multiple C of k. That is
A k + B k = C k
for some real number C, unique due to the measurement
assumption. We put A+ B = C. The
foregoing defines A+ B given a unit vector k.
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This computational method for A+B raises a few
questions.
- Is it consistent with the previously encountered method
for adding fractions and unsigned real numbers with infinite
decimal expansions. The answer needs to be Yes
- Does the real number C depend on the choice of unit
length k? The answers needs to be No
Through examination of five cases the required answers can be
verified.
- A and B are both positive,
- A and B have opposites signs with A positive and B
negative,
- A and B have opposites signs with A negative and
B positive,
- A and B are both negative.
- At least one of A and B is zero.
Details of the five cases and their examination is left for
the reader for the time being. In what follows, we assume A+B is
defined for pairs of real numbers A and B.
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