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YOU are better than YOU think. Show
yourself how:
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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.
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.
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.
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Remainder Arithmetic for Real Numbers
Remainder or modulo arithmetic is useful in
understanding polar coordinates, modulo 2p
or 360 degrees, and the periodicity of unit-circle trig
functions and the complex-valued cis(q) =
cos(q) + i sin(q)
= exp(iq) function.
For any pair of real numbers d > 0 and n >
0, there is a natural numbers q > 0 and a real number r such
that
0 < r < d
and n = qd +r Here the quotient
q = the number of whole times that the divisor d goes into the dividend
n, and r = the remainder.
Here q and r may be computed via long
division exactly or in principle via an infinite sequence of decimal
approximations - their decimal expansions. The case where d is a
decimal fraction may be somewhat different (less involved) than the
case where d is given by an infinite decimal expansion. We will skip
the details.
Two natural numbers n and m are said to be equivalent or
equal modulo d, when there remainders on division by d are equal. In
this case, we write n = m, modulo d. Equality
modulo a whole number or divisor d is
-
reflexive, that is, each number n =
itself, modulo d, or equivalent n = n, modulo d, for each natural
number n.
-
symmetric, that is, n = m
modulo d when and only when m = n modulo d, and
-
transitive, that is, if n =
m modulo d, and m = t modulo d then n =
t modulo d.
A whole number n is divisible by the divisor d
when and only when n = qd for some whole number q. That is when
and only when n = 0, modulo d and when and only when n is a whole
or natural number multiple of the divisor d. The number 0 is
a multiple of all divisors d. Observe, if n > m then n =
m, modulo d when and only when n - m is a multiple of d
while if n < m then n = m, modulo d when and only
when m -n is a multiple of d.
Remainder Calculations for real numbers are based on the
following properties or theorems.
Theorem: Suppose m, n, u and v are real
numbers. Suppose d > 0 is a real number. If m =
n, modolo d and u = v, modulo d then (i) m + u
= n +v modulo d, and (ii) mu= nv, modulo d.
Proof: First, m = n, modulo
d, implies m = a d +r and n = b d +r for some real numbers
a, b and a common real remainder r with 0 < r < d.
Likewise, u = v, modulo d, implies u = A d + s and v = B d
+s for some real numbers A, B and a common real remainder r with 0 <
s < d.
Arguments for (i): Suppose (m+ u) >
(n+v) then
(m+ u) - (n+v)
= (ad +r + Ad+s) - (bd+r + Bd+s)
= (a+A)d + (r+s) - [(b+B)d + (r+s)]
= (a+A)d-(b+B)d
= [(a+A)-(b+b)]d
is a multiple of d, and hence (i) m + u =
n +v modulo d holds when (m+ u) > (n+v). The case
where (n+v) > (m+u) follows similarly.
Arguments for (ii): Suppose m u >
nv then
mu - nv
= (ad +r)(Ad+s) - (bd+r)(Bd+s)
= aAd2 + asd+ Ard+ rs - [bBd2 + bsd+ Brd+
rs]
= [{(aA)-(bB)}d + (as-bs)]d
is a multiple of d, and hence (i) m
u = n v modulo d holds when m u > nv. The case
where nv > mu follows similarly.
Calculator Usage: For every divisor d > 0 and
every number N, there is a unique integer q such that qd <
N < (q+1)d so that r = N-qd satisfies 0 < r < d.
With the aid of a calculator, if N is positive, the whole number part of
the decimal representation of the computed value of N/d gives q
> 0. But if N is negative, the whole number part of the decimal
representation of the computed value of N/d gives q+1 <
0, and q is one less than the whole number part of N/d.
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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
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Long Division Continued
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
How to Add Reals
How to Multiply Reals
Distributive Law for Reals
Remainder Arithmetic II
Related Site Pages:
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