11. Introducing Addition of Signed Numbers
and Addition Properties
A. How to add Signed 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)
B. 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
C. 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.
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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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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Calculus Starter Lessons
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Learning to do and high marks if it comes to easy is often
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