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Study Notes 006:
By William E. Steinman:
Algebra 1:
In my previous Study Notes I have generalized about the problems I am attempting to solve. From now on I will be more specific about the details of the subjects as I take them up. My first step is just a review of the basic operations of algebra.
Algebra is nothing more that a way of making mathematics more powerful
by giving us formal symbolic procedures for problem solving. It is still math
and all the rules and procedures of basic mathematics are applicable. In
algebra, the only difference is that we represent an unknown quantity with some
kind of symbol. In basic math, we also have unknowns, but we do not formally
represent them. When we find the area of a rectangle by multiplying the width
times the length, the unknown is the area.
We are always
solving for unknowns in mathematics. That=s what math is about, solving problems for
unknowns. Even in counting we are really solving for the next number in the
sequence. In algebra, we use alphabetic characters to represent unknown
quantities. That allows us to represent a problem as an equation. Then we can
solve the equation for the unknown by doing regular arithmetic. This shorthand (equation) way of
representing mathematical relationships is the big difference between basic
mathematics and algebra.
In arithmetic, we
found there are just four basic operations to consider; these being, Addition,
Subtraction, Multiplication, and Division. The other things in math, Fractions,
Decimals, and Percents, turn out to be different ways of looking at
mathematical situations but the four basic operations applied to all of them.
In algebra:
When we multiply
a number by a letter the multiplication sign may be omitted.
For example A
times Z can be written as AZ.
The commutative
law of addition is:
Interchanging
addends does not change their sum.
a + b = c and b +
a = c are equivalent statements.
The Commutative
law of multiplication is:
Interchanging
factors does not change their product.
AZ = X and ZA = X
are equivalent statements.
It is important to notice that algebraic expressions need not be
complete equations.
For example 3x is an algebraic expression that means 3 times x.
12/y is an algebraic expression that means 12 divided by y.
Division by zero is undefined or impossible in math. This is still
true in algebra. In the case of 12/y above where y = 0 the solution is
undefined.
Evaluation is a formal math word. It just means finding the answer or
solution.
Normally, when the order of evaluation of an expression may not be
clear, we use parentheses to make it clear.
The rule is, we always evaluate the part of the expression enclosed in parentheses
first.
Most of the time, we can avoid ambiguity through the use of
parentheses.
However, when we encounter an expression without parentheses we have a
procedural rule.
To evaluate an
expression without parentheses, do the multiplication and division in order
from left to right. Then do the addition and subtraction in order from left to
right.
A term is a
number or the product of numbers.
A factor of a term
is any of the numbers multiplied to form the term.
We can have complex factors.
For example, in cd(a + b), (a + b) is a complex factor.
The factors of this expression are c, d, and (a + b).
We can also have more than one term in an expression.
For example, in the expression b + bd - (b - d) the terms are b, bd,
and (b - d).
Any factor or
group of factors of a term is a coefficient of the product of the remaining
terms.
In the expression 6(x + y) 6 is a coefficient of (x + y) and (x + y)
is a coefficient of 6.
We have literal coefficients and numerical coefficients.
In the expression 5xy
5 is the numeric coefficient of xy?
xy is the literal coefficient of 5.
In basic math, we learned to raise a number by a power or multiply it
by itself a number of times. We indicate that operation with a superscript
number.
Formally:
We call the superscript number the exponent.
We call the number being multiplied the base.
The result of the operation is called a power of the base.
For example, 53
125 is the third power of 5.
5 is the base.
3 is the exponent.
All of these concepts are also applicable in algebra.
In certain case we can combine terms.
We can do this if the terms have identical literal factors and
exponents.
For example:
We can combine 12x + 8x into a single term 20x.
Formally 12x + 8x
= 20x