Algebra - Beginnings
Algebra Is Easy
Algebra has been around for thousands of years. It's likely that mastadons still roamed North America when
algebra first began to appear. It was NOT invented by balloon-headed space aliens. It was created by normal
human beings with average-size heads. So if you're thinking that algebra is difficult, think again. Literally billions
of people have learned algebra, and so can you.
Stated simply, algebra has three corner-stones:
- Symbols
- Equality
- A Defined Rule System
That's it. That's all there is to it. If you truly understand these three things, then you will understand algebra.
Misperception
Many people mistakenly believe that algebra is difficult. This misperception arises from two factors:
- Myth (word-of-mouth)
- History (an absence of computers)
The "algebra is difficult" myth is simply passed down through successive generations of students. "Wow, algebra
was really tough." "Wow, algebra was really tough." (Repeat ad nauseam.)
Historically, computers are a new invention. Consequently, and from a pedigogical perspective, the solution of
immense and complicated algebraic constructs has focused on solution by independent humans. The pedigogical question was
always "Can YOU solve this set of equations?" Further, this question was presented regardless of the equations'
usefullness. Advancing students were simply expected to solve more and more complex equation sets
in the same manner that one might be expected to stack additional BBs in an unsupported vertical column.
A difficult exercise, yes. Useful? No.
However, given the current ubiquity of computers, the question should be
rephrased as "Can YOU arrange this set of equations so a COMPUTER can solve them AND give you results that are
useful?"
Algebra then (correctly) becomes a much smaller and less threatening study.
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Symbols
In algebra, symbols are used to represent numbers. Symbols can also be used to represent other symbols; but eventually
this chain of symbols always comes back to numbers.
For example, we can use "A" as a symbol representing the number 4. We would then say that A is 4.
Equality
Equality begins with the equal (=) symbol. Rather than saying "A is 4," we can write the equation
Other examples are:
A Basic Rule System
Any language is accompanied by a set of rules. Certain words mean certain things, and the ordering of words
affects their meaining. For example, "Bob punched Tom" means one thing, and "Tom punched Bob" means something else.
Similarly, the equation
means one thing, and
means something else. Relative to the four basic mathematical operators, "+", "-", "÷" (or "/") and "×",
OUR RULE SYSTEM states that ÷ and × operations are performed first, proceeding from left to right.
Then + and - operations are performed,
again proceeding from left to right. Using this rule system, the right side of the first equation above
becomes 14, while the second equation becomes 11.
Note that this rule system is simply a convenience that helps us communicate with each other.
And, more recently, it helps us communicate with computers.
Basic Rule System
Practice
(coming soon)
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Symbols
Symbols for Numbers
Numerical symbols can be likened to verbal metaphors. For example, if I see a blue painting, and I point at it
and say "Blue," then my spoken word is a verbal metaphor for the color I see. The single word "blue" implies the
metaphorical equation "That color equals 'blue'". If I point at Bob at say "Bob is pig-headed", then I've
created the verbal equation "Bob equals 'difficult to get along with'".
I can also contruct metaphors that "point at" numbers, rather than colors. For example, if I see four pennies,
and I point and say "Four", then my spoken word is a numerical metaphor for the number of pennies I see. Likewise, I can
write the number "4" as a metaphor for the number of pennies I see. Or, in a bush culture I might represent the
number of pennies by scratching "Ook"
in the sand. However, the important point to note is that it really doesn't matter what symbol I use,
provided only that it is known (either generally or specifically) what the symbol represents. If I tell you that when
I say "Ook" I actually mean "4", then you'll know what is meant when I say "Ook". Similarly, if I tell you that
a written "A" actually means "4", then you'll know my intention when I write "A".
It is important to note that many people have trouble understanding symbols. They see a symbol and they panic.
However, to calm the panic, they need only remind themselves that the a symbol represents something in the same
manner that the sentence "It is raining" represents something. They need only determine what the symbol represents.
To fully appreciate the simplicity of symbols at their most basic level, consider the puppy named "Max".
At first, Max will have no idea what you mean when you say "Max". However, as time goes by and you repeat his name
often enough, he will come to realize that when you say "Max", you are referring to him. "Max" becomes a symbolic
representation of himself, or from his perspective,
Why is this important? Only because it demonstrates that dogs understand symbols. And if dogs understand symbols,
then humans should have little trouble accomplishing similar feats of intellect.
Additional Symbolic Equations
(coming soon)
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Symbols for Operations
Symbols can be used to imply processes. For example, when you see a "+" symbol between a 2 and a 5, you know that
it is intended that you add the 2 and 5 together to produce 7. You also know what is intended when you see the
other basic operators, "-", "×", and "÷" (or "/").
Naturally, other operators exist in common algebra. For example, the operation "5 × 5" is referred to
as "five squared." This is represented in print as "52". In this context the superscript 2 is a symbol instructing
you to multiply the preceding number by itself.
Note that the superscript 2 has a well-known mathematical meaning. It is "locked down" in the
same manner as the "+" symbol. However, we need not confine ourselves to such well known operations. Instead, we
can construct our own. And conveniently, our standard rule system gives us a way to do this. For example, we can write
to imply that we perform an operation (f) on the number 7 to discover the value of A. We might define f as
the number × the number + 3
In this case, the number is 7, and consequently our operation (f) tells us that A is 52.
In another simple example we might write
and define g as
the number + the number / 3
In this case, the number is 9, and our operation (g) tells us that B is 12 (remember to perform the division
before the addition).
For future reference, please note that this type of custom operation is referred to as a "function".
Simple Function
Practice
(coming soon)
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Not Quite as Simple Function
Practice
(coming soon)
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Symbols: Division
Symbols for Division Operations
In beginning math nomenclature, the symbol "÷" is used to instruct you to divide the preceding amount by
the trailing amount, or:
In algebra, two other symbolic arrangements may be used for division. These include (a) replacing the "÷"
symbol with a "/", and (b) writing the first quantity above the second, with a horizontal line separating them, as
in
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In another example, we can write the equation
Note: OUR RULE SYSTEM defines a different interpretation for this division arrangement. In this case,
our BASIC RULE SYSTEM is applied first to the top portion. Then it is applied to the bottom portion.
And ONLY THEN is the top portion divided by the bottom portion. (The horizontal line division is the very last thing
that we do.) Note that this type of division is also referred to as a "fraction".
"Horizontal Line" Division (Fraction)
Practice
(coming soon)
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Equality: Uses
Equality and Substitution
When two thing are equal, then one can be used to replace the other. For example, if you purchase an item
that costs $20, you can pay with two $10 bills, or you can pay with four $5 bills. As payment methods, the
two groups of bills are equal, and one can be substituted for the other.
Similarly, suppose I have three equations:
then substitution can be used to determine the numerical value of F. The first equation tells us that
we can replace the A in the third equation with the number 4:
The second equation tells us that we can replace the B in the third equation with the number 7:
Then we simply perform the math and discover that F = 18.
Substitution for Progress
When performing substitutions, you must take care to ensure that each substitution takes you nearer to a solution.
For example, consider the three equations:
Using substitution to solve for F (find the value of F), the second equation tells us that we can replace the A
in the third equation with a B:
Did this substitution take us nearer to a solution? Not really, since we don't have an equation stating the value
of B. If instead we replace the B in the third equation with an a, as:
then we are closer to a solution because we can use the first equation to replace the A's with 3's:
Then we perform the math and discover that F equals 6.
Simple Substitution
Practice
(coming soon)
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Equivalent Operations
If we have two things or quantities that are equal, and if we perform equal operations on each, then the two
things or quantities will still be equal. For example, if we have two long-haired hippies, and if we shave each
one bald, then we will have two bald hippies. (The two remain equal to each other.)
However, if we shave one bald and tattoo
the other, then they will no longer be equal. One will be a bald hippie, and the other will be a long-haired hippie
with a tattoo. Because we performed different operations on each, we invalidated our hippie
equation, and we can no longer say that they are equal. Hence:
Bald-headed
Hippie |
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Long-haired Hippie
with Tattoo |
(The symbol "!=" means "is not equal to".)
Mathematically, if we have an equation, and if we perform identical operations to the quantities on each side of the
equal sign, then the equation will remain valid (the quantites on each side will remain the same as the other).
For example, if I have the equation
and if I add 2 to each side, then I arrive at the equation
Because we know that D = 5, we can see that the two sides of the equation remain equal.
Equivalent Operations
Practice
(coming soon)
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Division by Self Equals One
We know that any number divided by itself is 1. For example,
Simlarly, any symbol or group of symbols, divided by itself, is 1. For example,
This fact is often very handy, as will be seen below.
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Rule System
Rule System: Parentheses
Pairs of parentheses are another feature in our algebraic rule system. Our rule system states that we must
process the items inside the parentheses first. That is, we must apply the BASIC RULE SYSTEM
within the parentheses before moving outside of the parentheses. For example, solving this equation
using our basic rule system tells us that H is 13. However, if we insert a pair of parentheses we can modify
the meaning of the equation:
Processing the content inside the parentheses first gives us
and H is then 18. As you can see, the parentheses are used to group a set of numbers and/or symbols.
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Parentheses Within Parentheses
When prentheses are inside other parentheses, our basic rule system is applied to the innermost
parentheses first, and then proceeds toward the outer parentheses. This arrangement follows directly from the
statement "Process the items inside the parentheses first", which implies that inner parentheses must
be processed first.
For example, if we begin with the equation:
M = 2 × (3 + 7 × (4 + 1))
we first arrive at
then
then
and finally, M = 76.
Parentheses
Practice
(coming soon)
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