Equalities and Inequalities
Equality
In talking about a set S of numbers, we may say let a and b denote
elements of S. In the case where S = {4, 6}, you might think if a stands
for 4 then b has to stand for 6, or vice-versa. But we follow a different
convention. When we let a and b denote elements of S, the plural here is
our notation and perhaps not in the element or elements they represent.
Thus when say let a and b denote elements of this set S = {4, 6}, in the
case where a stands for 4, the other letter b may stand for 4 or 6.
Suppose we have two paths. Without knowing their lengths, we may say the
length of the first path is P and the length of the second path is Q.
Both lengths will be positive numbers. Without measuring the paths, we do
not know if one is longer than the other. Thus the values of P and Q may
be equal or not.
More generally, we speak about letters a and b denoting real numbers, we
mean each denotes a real number, or equivalently, that the value of each
is a real number. But we do require the value of each to be different.
Properties of Equality
Let a, b, c and d denote real numbers. Read the equal sign = as meaning
has the same value or equivalent value.
Axiom 1 - Reflexive Axiom: If a = b then b = a.
Axiom 2 - Transitive Axiom: If a = b and b = c then a = c.
Axiom 3 - Function Substitution Axiom: if a = b and f is computation
rule, with f(a) defined then f(b)=f(a)
First Consequences: If a = b then a+ c = b+c ac = bc
Second Consequences: If a = b and c =d then a+ c = b+d and ac = bd
Note the inclusion here of these axioms is informal and different -
nervously so - from what appears elsewhere. The textbooks I have read
instead of saying let a, b, c and d denote real numbers would say let
them be real numbers. In high school and college textbooks, those I have
seen, there has been a habit and tradition of talking about numbers but
not about denoting them, nor about the shorthand role of lettes and
symbols. The modern mathematics courses designs of the period 1960-80s
valued logic, rigour and precision. But over time, logic and rigour left
the classroom - faded out. Talking about letters denoting numbers instead
of being numbers is an issue that can be discussed in school systems
where mathematics instruction attempts or still attempts to provide a
rigourous account of the subject.
Inequalities
In site introduction introduction of real numbers, the latter may be
zero, and if not, the latter will have a sign and an unsigned part. In
consequence, all real numbers are either zero, negative or positive. On
horizontal coordinate line or number line, the positive numbers are to
right of a zero mark and the negative ones are to the left. For vertical
coordinate or number lines, we could have above and below instead.
Let us begin with a few examples.
The number 5 is more than
-
2 more than the number 3
-
4 more than the number 1
-
4.5 more the number 0.5
-
4.9 more than number 0.1
If we talking about temperature, then 5 degrees above zero would 5
degrees higher or more than zero and 10 degrees higher than -5 degrees.
Definition: Suppose a and b denote real numbers. Then we say a is more
than b when and only when there is a positive number c such that a = c+
b. In the latter case we write say a is more than b and $a\gt b$ as
shorthand for that, with the sign $\gt$ read as the more than sign. The
more than here is a comparison not of magnitude, but of position. For
example -5 = -15 +10 is 10 more than -15
Definition: Suppose a and b denote real numbers. We say b is less than
a and write $ b \lt a$ when and only when a is more than b.
The foregoing implies $ b \lt a$ when and only when there is positive
number c such b + c = a.
Properties of Inequalities
If a, b and d are real numbers with $a \gt \gt b$ then
$a + d \gt b +d $
$a - d \gt b -d $
$d \gt 0$ implies and $ad \gt bd$
$d \lt 0$ implies and $ad \lt bd$ (Direction Reversal)
$d \gt 0$ implies $a \div d \gt b \div d $
$d \lt 0$ implies $a \div d \lt b \div d $ (Direction Reversal)
Proofs of the Properties
The property $a = c+ b$ for some positive number c implies $a+d = (c+ b)
+ d = c+(b+d)$ and hence $a + d \gt b +d $
The first property is proven.
The differences $a-d = a+(-d)$ and $b-d = b+(-d)$. Therefore $a = c+ b$
for some positive number c implies implies $a+(-d)> b+(-d)$ or
equivalent $a - d \gt b -d $.
the second property is proven.
The property $a = c+ b$ for some positive number c implies $da = d(c+ b)
= dc+ db$ Now d positive implies dc positive by the law of signs, and
hence da is more than db by dc. That implies the third propertry. Now d
negative implies dc is negative, and $da = dc + db$ implies $db = da
+^-(dc)$ is more than da by the positive amount $^-dc$. Hence $db \gt da$
or equivalently $ad \lt bd$
Division by d gives the same result as multiplication by
\[d^{-1}
=\mbox{sign}(d)\times \frac1 {|d|} =\mbox{sign}(d)\times \frac 1 {\mbox{length}(d)}
\] The latter has the same sign as d. So the last two
properties follow from the previous two.
Note: the properties that $a-d = a + (-d)$ and \[d^{-1}
=\mbox{sign}(d)\times [1 \div |d|] =\mbox{sign}(d)\times [1 \div
(\mbox{length}(d)]\] follows from the earlier development of real numbers
in site material.
|
|
Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
|
|