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Adults.
Algebra for Adults:
Part 5, Representing Numbers with Symbols:
December 30, 2002:
We have reviewed math in our previous exercises.
Now, we can begin the transition from math to algebra.
It is not necessary to feel intimidate. Algebra flows out of mathematics
in a very natural progression much like multiplication flows out
of addition. Both are natural extensions.
In arithmetic, we found there were just four basic operations
to consider. These being, Addition, Subtraction, Multiplication,
and Division.
The other things we studied, Fractions, Decimals, and Percents,
turned out to be different ways of looking at mathematical situations
but the four basic operations applied to all of them.
We will find this is still true in algebra. We will still be solving problems (think puzzles) for sums (Addition), differences (Subtraction), products (Multiplication) and quotients (Division). And, for sure, all of the rules, laws, symbols, and operators we used in math will still apply. You will find these listed in the algebra archive.
The big difference in algebra is that we will represent numbers
with symbols, namely characters of the alphabet.
This does two cool things for us.
First, it makes any mathematical statement much shorter that its
verbal equivalent. In turn, that will make the statement easier
to understand and manipulate.
For example:
A verbal statement, might be: three times any number plus the
same number is equal to four times the number.
To make the equivalent statement in algebra we can let the letter
n stand for the number.
Then the algebra statement becomes: 3n + n = 4n.
As you can see, the second statement is a great deal shorter, yet it contains all of the pertinent information.
Notice one other thing we did here.
We left out the multiplication sign.
Instead of 3 x n, we wrote it as 3n.
This is a handy convention for algebra.
When we multiply a number by a letter the multiplication sign
may be omitted.
It is not necessary to follow this convention, it just requires
less writing when we do.
So, these four statements are equivalent.
3n, 3(n), 3 x n, 3 n.
In passing, notice that when we multiply a number by a number
we may not leave out the multiplication sign.
For example:
3 x 6 and 36 are not equivalent.
Let's try another one to make sure we are on to this idea.
The volume of a cube is equal to the length of one side cubed.
We can let the letter v stand for the volume and the letter s
stand for the length of one side.
Then, the equivalent algebra statement becomes: v = s3.
There you have it.
That is all there is to say about using symbols for numbers.
Now, we can pick up on a couple of other things we learned
in math.
I speak of the commutative law of addition and the commutative
law of multiplication.
Remember?
The math nerd word.
Commutative is what we say
when any order is okay.
Formally:
Interchanging addends does not change their sum.
Interchanging factors does not change their product.
Guess what?
As we will continue to repeat, all of the rules of math will still
apply in algebra.
So we can still say 2 + 7 = 9 and 7 + 2 = 9 are equivalent.
Now we can let a represent the 2, b represent the 7, and c represent
the 9.
This gives us a + b = c
Can you see, by the commutative law of addition, that a + b =
c and b + a = c are equivalent statements?
Likewise in multiplication, 2 x 4 = 8 is the same as 4 x 2
= 8.
Substituting a for the 2, b for the 4, and c for the 8 we get
ab = c.
Now, by the commutative law of multiplication, ab = c and ba =
c are equivalent statements.
Cool!
Now we can apply these rules to help us, much as we did in math. One way we used these rules was to check our solutions. If we add a set of numbers to get a sum, we can add the same numbers in reverse order. If we get the same sum both times, we can be quite confident that our solution is correct.
We can also apply this procedure to multiplication by changing the order of the factors. If we still get the same product, we probably have a winner.
In general, these two rules can be applied to manipulate and simplify our algebra statements.
That will do for this exercise.
Here are a couple of problems to work out to make sure you got
it.
Give the equivalent algebra statement for this verbal statement.
Six times any number plus two times the same number is equal to eight times the number.
The volume of a cylinder is given as 3.14 times the radius
of the cylinder squared times the length of the cylinder. Let
v represent the volume, r represent the radius, and l represent
the length. Write the equivalent algebra statement for this verbal
statement.
Back to Algebra for Adults.
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