Equality Axioms and their Meaning
Expressing the same number in different ways
Counts and measures may be given in many ways. A person fond of decimals may write 0.75 while a person fond of fractions
may write $\frac 34$ and another fond of percentages may write 75%. Each way
is correct. The surrounding context may decide among them.
As a further example, the number 7 may be given, written or obtained in many different ways.
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VII
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7.0
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5 + 2
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(1+1+1) + (1+1+1) + 1
Fractions and mixed numbers may also be given, written or obtained in
different ways. Abraham Lincoln use the phrase Four score and ten years ago
insteads of saying 70 years ago. So numbers can be given and presented
as arithmetic expressions:
\[ 4 \times 20 + 10 \quad 1 +\frac34 \quad \frac 74 \]
\[ 1.75 = 1 +\frac{75}{100}= \frac{175}{100} \]
In practice, the different ways
to give, express or write a numbers do not affect the value of results
based on the
numbers, albeit writing results in different forms will affect their appearance.
Using Letters to denote Numbers
Imagine we have a situation in which a letter x stands for one of three
values, say 4, 8 and 11. Those numbers might count the number of pennies
in your pocket. Imagine another letter y also has one of the three values
4, 8 and 11.Those numbers might count the number of pennies in my pocket.
Then we keep the counts x and y separate, or we might add them or compare
them. Their sum would be written as x + y. If I have less pennies than
you, we might write x < y.
What does $x=y$ mean?
There is also the possibility that we have the same number pennies. In
that case we would write x =y. In that case, we would have a situation in
which two different counts or measures, here x and y, are equal. In the
case of equality, ten times the number x of your pennies would the same
as 10 times the number x of my pennies.
Changing Calculations
In arithmetic expressions, numbers may be rewritten without changing
their value. Arithmetic expressions give numbers. Arithmetic expressions
also describe calculations that might be done. Formulas and computation
rules also describe calculations that might be done. Evaluation turns
the might into actual.
We assume when two different numerals, two different letters, two
different arithmetic expressions, two different computation rules,
two different algebraic expressions give or denote the same number,
then each can replace the another without changing the result of actual
or potential calculation.
When two calculations give the same result, one can be done or written instead of
the other.
This is the replacement principle. The axioms or properties
of real numbes describe when different computation rules will give the same
result). They thus describe when one calculation can be replaced by another.
Working with Equal Counts, Numbers and Measures
[Equality Implication Assumption - Axiom]: If a, b, c and d are real
numbers with \[ a =b \quad \mbox{ and } \quad c=d \] then \[ a+c =b+d
\quad \mbox{ and } \quad a\times c=b \times d \] and
\[ a-c =b-d
\quad \mbox{ and } \quad a\div c=b \div d \] Furthermore if $f(a,c)$ is computation rule for which $f(a,c)$ is
defined, we assume $f(a,c) = f(b,d).$
In practice, a, b, c and d may be given by arithmetic expressions or
algebraic calculations that have or will have the same value. We may use
this equality patterns to say or imply that further equalities hold.
[Special Case] If a, b, and c are real
numbers with \[ a =b \] then \[ a+c =b+c
\quad \mbox{ and } \quad a\times c=b \times c \] and
\[ a-c =b-c
\quad \mbox{ and } \quad a\div c=b \div c \]
Examples
To solve the equation \[5x+{}^-7 = 13\] the equality implication pattern
implies adding 7 to both sides gives the equality \[(5x+{}^-7)+7 = 13+7\]
Now the associative law for addition gives \[5x+({}^-7+7) = 20\] Whence $
{}^-7+7 = 0, $ the reason for adding 7 to both sides, gives \[5x = 20\]
Now the equality implication pattern implies \[\frac15 \times 5x = \frac
15 \times 20\] Hence \[x = 40\] since $\frac15 (5x) = 1x = x $ by the
associative law for multiplication. The foregoing represents a chain of
reasoning or figuring for arriving at a solution.
The foregoing provides a long and too detailed solution. A shorter
presentation of the type you may want to do in class follows.
\begin{eqnarray*} \mbox{Given } &:& 5x+7 = 13
\\ \mbox{Add } {}^-7
&:& 5x = 20 \\ \mbox{Divide by 5} &:& x = 4
\end{eqnarray*} Here adding ${}^-7$ gives the same result as subtracting 7 while division by 5 gives the same result as multiplying
by $\frac15$ - a fifth. We may describe the operations as additions, subtractions, multiplications and divisions as we like -
whatever takes less writing.
For most, mathematics in an automatic just-do-it manner is quick,
efficient and enough. As student, you many begin with that manner, but
mathematics can also be done in more legalistic,less relaxed and more
rigourous manner by explaining how and why each step is justified or
implied by an axiom. That kind of detailed, step-by-step reasoning
appears in undergraduate and graduate college courses for students
studying in pure mathematics. In that process, no step is to done and
recorded until the reason for it is identified. In the frontier or
non-routine research areas of pure and applied mathemmtics, great care in
reasoning may avoid mistakes. But for routine problems, where the
mathematics follow paths known to work, steps have to be done and
recorded carefully in a mistake free manner. But the justification for
the steps need not be recorded, except in situation where seeing and
learning the justifications is an objective.
Euclidean Ideal
Euclidean geometry provided a model for careful rule based reason. In the
ideal form, all words or terms are clearly defined and all conclusions follow
from the logical direct and indirect use of assumed rules and patterns - the axioms.
In applying the rule and patterns, rigour or careful reason requires all
conditions and implications be clearly recorded and expressed in
chains of reason that the author and others may see for confirmation,
refinement or identification of error.
In the Euclidean ideal for reason, all steps in the logic are
should be fully observable
for the sake of verification. This ideal is very demanding. In practice, outside
of research and studies in pure mathematics,
explaining and writing each and every step of step in arithmetic, algebra,
geometry and so on would not keep the attention of the author and any
audience. The ideal, it has been tried, requires too many steps and too much
detail to provide students and people applying with an quick and effective
command. Mastery of mathematics requires a compromise. Mathematics needs
to be done in a more relaxed manner. In that some students may assume
a teacher has been hired to provide correct methods, and not worry further.
In contrast, other more careful students, may learn how to do and record
steps in reason, algebra and arithmetic quickly while being aware of
how extra details or detours could be written to express and record
a fuller version of the steps. The advantage of being aware depends on
circumstance. For example the value of
\[ (-1) \frac34 \times 67.23+ \left(\frac 34 +2.345 \right)\times 67.23 +(-1) \times 2.345 \times 2.345
\]
can be obtained without doing any arithmetic by applying the
properties of arithmetic given described algebraically in Read Number axioms.
So greater awareness from time to time has benefits, small and large.
One who masters in full axioms, the rule and patterns of mathmatics
and logic, and understand their origins and limitation, will be better placed
to check and extend the reasoning and justifaction when and
where formulas and logic appear. There-in lies an end and value
for studies in mathematics itself, and in technical subjects where
mathematics is employed in practical, in more theorectical or
speculative or unreasonable ways. Depth of knowledge may help you see which is which,
skill and time permitting. Good luck.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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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
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the choice is theirs. But in retrospect, the selection does not
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Calculus Starter Lessons
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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