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YOU are better than YOU think. Show
yourself how:
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<| (o) (o) |>
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
attention. |
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
| | 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)
|
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.
|
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.
|
| |
www.whyslopes.com
Number Theory
Start of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
Number Theory
Continued
Decimal Place Value
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots & Primes
Long Division Continued
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II
Related Site Pages:
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck
Food for thought: Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice..
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