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YOU are better than YOU think. Show yourself
how:
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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.
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.
| |
Suppose m and n are whole numbers. We will study the solution of the
equation
yn = xm
when one and hence both numbers x and y are non-zero.
Exercise: What can be said in the case where n and m are nonzero integers?
The sign function
Each real number q is positive, zero or negative. We compute and hence
define sign(q) as follows.
| sign(q) = |
{ |
+1 |
if q > 0 |
| 0 |
if q = 0 |
| -1 |
if q < 0 |
So sign (5) = +1 and sign(-3) = -1 and sign(0) = 0.
Real Number Multiplication Revisited.
Multiple Distance to Origin & Multiple Signs
Now the product of a pair of real numbers a and b can be computed as
follows.
ab = [sign(a)sign(b)] |a|*|b|
By mathematical induction we can show
tk = [sign(t)]k |t|k
and
for all whole numbers k. For t non-zero, We can also show sign(t)k
= 1 when k = 2s is even for all real numbers t. for all real t, we can
show sign(t)k = sign(t) when k = 2s+ 1 is odd.
Examples:
- sign(5)4 = 1,
- sign(-3)2 = 1;
- sign(5)3 = sign(5) and
- sign(-2)7 = sign(-2)
Sign Analysis of the equation yn = xm
Applying the sign function to both sides of the equation yn
= xm forces
|y|n = |x|m
Applying the sign function to both sides of the equation yn
= xm gives
sign(y)n |y|n = sign(x)m |x|m
So sign(y)n = sign(x)m
Now y = sign(y)|y|. The equation
|y|n = |x|m
implies
n ln |y| = ln |y|n = ln |x|m = m ln
|x|
Therefore n ln |y| = = m ln |x| and
Hence
That says how to compute |y|.
Now we sign(y) from the equation
sign(y)n = sign(x)m
- n-odd case: If n is odd, sign(y) = sign(y)n
= sign(x)m for all real numbers x. Here sign(x)m
will be 1 if m is even and sign(x) if m is odd.
- n-even case: If n is even, 1 = sign(y)n So the equation
can only be satisfied when 1 = sign(x)m That is when
x is positive with no restriction on m or when x is negative and m is even.
Note: the equation 1 = sign(y)n allows y to be postive or
negative. So if y is a solution, so is -y, and when n is even, the
equation yn = xm with x non zero has two
solutions (a positive and negative) or no solutions.
Conclusion I
For x < 0, the equation yn = xm
has the positive solution y = 0
When x > 0 or m even the equation yn = xm
has the positive solution
and if n is even, the equation also has the negative solution In the
latter case, the positive solution is called the principal root.
| y |
= (-1) exp( |
m
n |
|
ln |x| ) |
For x < 0 and m odd, the equation yn = xm
for n even has no real solutions when n is even, and for n odd, it has
the solution
Conclusion II.
When m and n are both odd, and x is non-zero, the equation yn
= xm has the solution y = 0
| y |
= sign(x) exp( |
m
n |
|
ln |x| ) |
To see why, note the case x > 0 gives
while the case x < 0 gives
in agreement with conclusion I.
|
What is xb when b = |
m
n |
? |
|
Answer: When m and n are odd, and x is non-zero
| xb |
= sign(x) exp( |
m
n |
|
ln |x| ) |
When n is even, and x > 0
Remark: For x < 0 and m odd, the equation yn
= xm for n even has no real solutions when n is even, but the
equation
y2n = x2m
has two solutions, the principal positive solution
and its negative.
| y |
= (-1)exp( |
m
n |
|
ln |x| ) |
Animated Example
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More Calculus
Calculus Videos
0. First Calculus Preview
0. Triangle Inequality
0 Inequalities
0. Solving Inequalities
1. Distance+Midpt Formulas
1. Function Domains
1.Polynomial Domain+Range
1. Fn: Linear Combinations
1. Fn Composition II
1. Fn Composition I
1. Solving y**n = x**m
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
3. Derivative Motivation
3. Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x) (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3. Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
4. Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5 Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere
To Learn More, visit Volumes 2 and 3.
Advanced Topics
Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity
To Learn More: Visit Real Analysis.
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