Applying the Fundamental Concepts of Algebra
An understanding of the fundamental concepts of algebra and of how those fundamental concepts may be applied is necessary in many professional and most technical careers. For engineers and scientists it is an essential requirement.
The fundamental concepts of algebra are described in the preceding section of this article. How these concepts may be applied to aid in the solution of various types of mathematical problems is explained here.
USING REAL NUMBERS
An example of the use of real numbers is in the measurement of temperatures. If it is a very cold day, it may not be enough to tell someone that the temperature is 5 degrees; you may have to indicate whether it is 5 degrees "above zero" or 5 degrees "below zero." You may use the real numbers +5 [say "positive 5"] or -5 [say "negative 5"] to indicate the temperature. The degree temperatures "above zero" are measured by positive real numbers, and the degree temperatures "below zero" are measured by negative real numbers.
It may be helpful to picture the set of real numbers as the set of points on a line:
A diagram such as this is often called a picture of the number line. The point labeled '0' is called the origin (see Numeration Systems and Numbers).
The number-of-arithmetic 5 is the arithmetic value of the real numbers +5 and -5. The numbers-of-arithmetic are used as measures of magnitude; the real numbers are used as measures of magnitude and direction.
As another example of how real numbers are used, consider the measurement of distances above and below sea level. The elevation of Mount McKinley is 20,320 feet above sea level, measured by the real number +20,320. Death Valley has an elevation of 282 feet below sea level, measured by the real number -282. The elevation at sea level is 0; distances above sea level are measured by positive real numbers, and distances below sea level are measured by negative real numbers.
OPERATIONS ON REAL NUMBERS
In the preceding section of this article, it was mentioned that the sum of a pair of real numbers is the real number that is the measure of the resultant of the corresponding pair of directed changes. To gain further insight into addition of real numbers it may be convenient to refer to a picture of the number line.
For example:
Notice that we may apply our knowledge of addition of numbers-of-arithmetic when we wish to add a pair of positive numbers [or a pair of negative numbers]; for the arithmetic value of the sum of a pair of positive numbers [or of a pair of negative numbers] is the sum of their arithmetic values:
3 + 2 = 5
If we wish to find the sum of a positive number and a negative number, for example:
or the sum of a negative number and a positive number, for example:
we may apply our knowledge of subtraction of numbers-of-arithmetic:
3 - 2 = 1
In the preceding section of this article, subtraction of real numbers was defined in terms of addition and oppositing. For example:
+3 - +2
is simply a shorthand notation for
+3 + - +2
Since
- +2 = -2
it follows that
+3 - +2 = +3 + - +2 = +3 + -2 = +1
Thus, the result of subtracting +2 from +3 is +1. Similarly, the result of subtracting +3 from +2 is -1:
+2 - +3 = +2 + - +3 = +2 + -3 = -1
Multiplication of real numbers is similarly related to multiplication of numbers-of-arithmetic:
Notice that the arithmetic value of the product of a pair of real numbers is the product of their arithmetic values:
2 - 3 = 6
Notice also that the product of a positive number by a positive number [or of a negative number by a negative number] is a positive number. The product of a positive number by a negative number [or of a negative number by a positive number] is a negative number.
In the preceding section of this article, division of a real number by a nonzero number was defined in terms of multiplication and reciprocation. For example:
+6 / +3
is simply a shorthand notation for
+6 - +1/+3
Let us see what it means to divide +6 by +3. We wish to find the real number that satisfies the open sentence
A real number satisfies this open sentence if and only if it satisfies the following open sentence:
So a real number satisfies the open sentence
if and only if it satisfies the open sentence
We notice that +2 is the real number which satisfies the last open sentence
So it follows that
or, equivalently, that
Here are some other examples to illustrate division of real numbers:
Notice the similarity between division of real numbers and division of numbers-of-arithmetic. For example:
Notice also that the result of dividing a positive number by a positive number [or of dividing a negative number by a negative number] is a positive number. The result of dividing a positive number by a negative number [or of dividing a negative number by a positive number] is a negative number.
FORMULAS, FUNCTIONS, AND GRAPHS
A fruit dealer sells apples priced at 12 cents each. He may find it convenient to make a list of the cost of various quantities of apples:
He may, of course, extend this list as far as necessary.
Notice that to find the cost of any quantity of apples he may use the formula
C = .12 - N
If he substitutes a numeral for 'N', he can find the corresponding cost by using this formula. For example, to find the cost of 17 apples he substitutes '17' for 'N':
C = .12 - 17
If he multiplies .12 by 17, he finds that the cost of 17 apples is $2.04.
We assume that the dealer is not interested in knowing the cost of fractional parts of an apple. Thus the values of 'N' are simply the whole numbers
0, 1, 2, 3, . . .
and the values of 'C' are simply the multiples of .12.
It is also useful to consider the set of ordered pairs:
(0,0), (1, .12), (2, .24), . . .
The first member of an ordered pair is called the first component, and the second member of an ordered pair is called the second component. For example, the first component of the ordered pair (3, .36) is 3, and its second component is .36. Notice that in the set of ordered pairs which we are considering, no two ordered pairs have the same first component. [For example, the two ordered pairs (1, .12) and (2, .24) have different first components.]
A set of ordered pairs that satisfies the condition that no two ordered pairs in the set have the same first component is called a function. Thus, the set of ordered pairs given above is a function.
The set of ordered pairs given above may be described as the set of all ordered pairs (n, c), where the first component is a whole number, such that
c = n - .12
The domain of a function is the set of first components of the set of ordered pairs of which the function consists. The range of a function is the set of second components. In the example considered above, the domain of the function is the set of whole numbers, and the range of the function is the set of multiples of .12.
Instead of listing the costs of various quantities of apples, the dealer may make a graph:
The dot that is above '1' and to the right of '.12' illustrates the point that corresponds to the ordered pair (1, .12). The dot that is above '2' and to the right of '.24' illustrates the point corresponding to the ordered pair (2, .24). By referring to the graph, the dealer can see at a glance that the cost of 4 apples is 48 cents.
The domain of the function whose graph is pictured above consists of only the whole numbers. For this reason, the graph of this function consists of only a sequence of points.
Linear Functions
Consider the set of all ordered pairs of real numbers (x, y) which satisfy the condition that the sum of the first and second components is +5.
x + y = +5
Some ordered pairs that belong to this function are:
(+5, 0), (3 1/2, 1 1/2), (+6, -1)
This set of ordered pairs may also be described as the set of all ordered pairs of real numbers (x, y) such that
y = -1 - x + +5
This set of ordered pairs of real numbers is a function whose domain and range is the set of all real numbers. The graph of this function is a line:
The points on the graph that correspond to the ordered pairs (+2, +3) and (+7, -2) are marked on the graph. You may find it worthwhile to locate the points on the graph that correspond to the following ordered pairs:
(0, +5), (+1.5, +3.5), (-1, +6)
Since the graph of the function is a line, the function is called a linear function, and the equation
y = -1 - x + +5
is called a linear equation.
Since the graph of a linear function is a straight line, many geometric problems that involve lines and line segments may be solved by algebraic methods. The properties of linear functions are studied in analytic geometry and calculus.
Another example of a linear function is the set of all ordered pairs of real numbers (x, y) such that
y = x + +1
Some ordered pairs that belong to this function are:
(0, +1), (-1, 0), (+2.5, +3.5), (-2.5, -1.5)
The graph of this function is a line:
Here are other examples of linear equations:
y = x + -6 y = +5x + -2 y = -4x + +1
In fact, if 'a' and 'b' are real numbers, and 'a' <> 0,
y = ax + b
is a linear equation. The corresponding function is a linear function, and its graph is a straight line.
Quadratic Functions
The formula for computing the area of a rectangle is
This formula tells you that the area of a rectangle may be found by multiplying its length by its width. For example, the area of a rectangle 5 inches long and 2 inches wide may be found by substituting '5' for 'L' and '2' for 'W' in the formula and then multiplying:
A = 5 - 2 = 10
Hence the area of the given rectangle is 10 square inches.
If the rectangle is such that the length is the same as the width, the rectangle is called a square, and the formula for its area is
The numeral '2' is called an exponent. The expression 'W2' is a shorthand notation for
W - W
The set of all ordered pairs of real numbers (x, y) such that
y = x2
is a function. Some ordered pairs that belong to this function are:
(0,0) (+1, +1), (-1, +1), (+2, +4), (-2, +4)
The graph of this function is not a line; the function is not a linear function. it is an example of a quadratic function. Its graph is a parabola:
Another example of a quadratic function is the set of all ordered pairs of real numbers (x, y) such that
y = +2x2 + +5x + -1
If 'a', 'b', and 'c' are real numbers, and 'a' <> 0, then the set of all ordered pairs (x, y) such that
y = +ax2 + bx + c
is called a quadratic function. The properties of quadratic functions are studied in analytic geometry and calculus.
Functions of Higher Degree
Consider the function consisting of all ordered pairs of real numbers (x, y) such that
y = x3 [say "y is equal to x cubed"]
The expression 'x3' is a shorthand notation for
x - x - x
Some ordered pairs that belong to this function are:
(0, 0), (+1, +1), (-1, -1), (+2, +8), (-2, -8)
This function is an example of a third degree function:
The function consisting of the set of all ordered pairs of real numbers (x, y) such that
y = x4 [say "y is equal to x to the fourth"]
is an example of a fourth degree function. The expression 'x4' is, of course, a shorthand notation for
x - x - x - x
Some ordered pairs that belong to this function are:
(0, 0), (+1, +1), (-1, +1), (+2, +16), (-2, +16)
You might find it interesting to draw a graph of this function.
Properties of Exponents
In the preceding section of this article, the basic principles for addition and multiplication of real numbers were listed. From these basic principles, other principles were derived. We now note some properties of exponents that follow easily from the basic principles.
Notice, for example, that
22 - 23 = (2 - 2) - (2 - 2 - 2)
= 4 - 8
= 32
and that 25 = 2 - 2 - 2 - 2 - 2 = 32
In fact, it is easy to prove the following principle:
x x2 - x3 = x5
This principle tells you that for each real number 'x', the product of 'x2' by 'x3' is 'x5'. [Notice that 2 + 3 = 5.]
Note also, for example, that
These examples suggest the following principles that are easily proved to be consequences of the basic principles:
[Notice that 5 - 2 = 3.]
The principles suggest methods you may use to solve problems like the following without using pencil and paper:
You may, of course, check the answers by computing. For example, in the last problem we have:
Products and Factoring
You already know that, for each real number 'a',
2 - a + 3 - a = 5 - a
This follows from the distributive principle (see section on fundamental concepts)
2 - a + 3 - a = (2 + 3) - a
and the fact that 2 + 3 = 5.
Consider now the indicated product:
(a + 2) - (a + 3)
From the basic principles it follows that:
(a + 2) - (a + 3) = (a + 2) - a + (a + 2) - 3
= (a2 + 2 - a) + (a - 3 + 2 - 3)
= (a2 + 2 - a) + (3 - a + 6)
= a2 +( 2 - a + 3 - a) + 6
= a2 + 5 - a + 6
Thus,
(a + 2) - (a + 3) = a2 + 5 - a + 6
We say that the expression a2 + 5 - a + 6 is the expanded form of the indicated product
(a + 2) - (a + 3)
When we transform an expression into an indicated product, we say that the expression has been factored. We see that the expression
a2 + 5 - a + 6
may be factored, and that its factors are
(a + 2) and (a + 3)
Notice that for each pair of real numbers 'a' and 'b':
(a + b) - (a + b) = (a + b) - a + (a + b) - b
= (a2 + b - a) + (a - b + b2)
= a2 + (a - b + a - b )+ b2
= a2 + 2 - a - b + b2
Thus,
(a + b) - (a + b) = a2 + 2 - a - b + b2
and we see that for each pair of real numbers 'a' and 'b', the expression
a2 + 2 - a - b + b2
may be factored.
For each pair of real numbers 'a' and 'b', the expression
a2 - b2
may also be factored. This is demonstrated as follows:
(a - b) - (a + b) = (a - b) - a + (a - b) - b
= (a2 - b - a) + (a - b - b2)
= (a2 + - a - b) + (a - b - b2)
= a2 + (- a - b + a - b) - b2
= a2 + 0 - b2
= a2 - b2
Thus,
(a - b) - (a + b) = a2 - b2
A knowledge of factoring may be helpful when we wish to simplify a given expression or take a shortcut in computing. The following examples illustrate this:
The ability to factor is also important in finding the solutions of quadratic equations. This will be shown under the subhead "Quadratic Equations."
It will be instructive for you to try to factor the following expressions by transforming each expression into an indicated product:
(a) 3 - 3 + 3 - 4
(b) a2 - a - b
(c) a2 - 2 - a - b + b2
Answers:
(a) 3 - 3 + 3 - 4 = 3 - (3 + 4)
(b) a2 - a - b = a - (a - b)
(c) a2 - 2 - a - b + b2 = (a - b) - (a - b)
Quadratic Equations
Consider the equation
x2 - +25 = 0
This equation is an example of a quadratic equation. The real numbers +5 and -5 both satisfy the above open sentence:
The solution set of the given equation consists of simply the real numbers +5 and -5. We say that +5 and -5 are the two roots of the given quadratic equation.
When we are trying to find the roots of a quadratic equation, we often make use of an important principle of real numbers. This principle, which is derived from the basic principles listed in the preceding section, states that the product of a pair of real numbers is 0 if and only if one of the numbers is 0. That is,
x y xy = 0
if and only if
x = 0 or y = 0
Let us see how we may use this principle to find the roots of the quadratic equation
x2 - +16 = 0
We first notice that the expression
x2 - +16
may be factored:
x2 - +16 = x2 - +42 = (x - +4) - (x + +4)
Thus, we may transform the given equation
x2 - +16 = 0
into the equivalent equation
(x - +4) - (x + +4) = 0
A real number satisfies the last equation if and only if it satisfies the given equation.
We know that
(x - +4) - (x + +4) = 0
if and only if
x - +4 = 0 or x + +4 = 0
Thus, a real number satisfies the equation
(x - +4) - (x + +4) = 0
If and only if the real number satisfies one of the following equations:
x - +4 = 0 or x + +4 = 0
The solution set of the equation
x - +4 = 0
consists of simply the number +4, since
The solution set of the equation
x + +4 = 0
consists of simply the number -4, since
Thus, the solution set of the equation
(x - +4) - (x + +4) = 0
consists of the numbers +4 and -4. Hence the roots of the given quadratic equation
x2 - +16 = 0
are the numbers +4 and -4. We may check this result:
Here is another example that shows how factoring aids us in finding the roots of a quadratic equation:
Example: Find the roots of the equation
x2 - +3x = 0
Solution: The expression x2 - 3x may be factored
x2 - +3x = (x - +3) - x
Hence the given equation may be transformed into the equivalent equation
(x - +3) - x = 0
A real number satisfies this equation if and only if it satisfies one of the following equations:
x - +3 = 0 or x = 0
The only real number that satisfies the equation
x - +3 = 0
is +3, and the only real number that satisfies the equation
x = 0
is 0. Thus, the roots of the given equation are +3 and 0.
SOLVING PROBLEMS
The following are some typical problems that, in a natural way, lead us to algebraic equations. In each problem, the solution to the problem is found by finding the solution set of an equation. When we find the solution set of an equation, we say that we have solved the equation.
Example 1: A number has such a property that when 6 is subtracted from twice the number, the result is 16. What is the number?
Solution: The required number must satisfy the open sentence
or, equivalently, the required number must satisfy each of the following open sentences:
Since the only number that satisfies the last equation is 11, 11 is the required number. We may check our result:
Check:
Example 2: A number has such a property that when 10 is subtracted from twice the number, the result is the same as when 6 is added to the number. What is the number?
Solution: A number has the required property if and only if it satisfies the open sentence
2 - x - 10 = x + 6
or, equivalently, a number has the required property if and only if it satisfies each of the following open sentences:
(2 - x - 10) + 10 = (x + 6) + 10
2x = x + 16
- x + 2x = - x + (x + 16)
(- x + x) + x = (- x + x)+ 16
x = 16
The only number that satisfies the last open sentence is 16. Therefore, 16 is the required number.
Check:
Example 3: Suppose that Tom has $6.00 more than Bill and that together they have a total of $12.00. How much money does Tom have? How much money does Bill have?
Solution: Suppose that Tom has 't' dollars and that Bill has 'b' dollars. Since Tom has $6.00 more than Bill, then
(1) t = 6 + b
Since Tom and Bill have together a total of $12.00, then
(2) t + b = 12
Equation (1) tells us that
t is the same as 6 + b
We may therefore replace 't' by '6 + b' in equation (2):
(6 + b) + b = 12
Thus,
6 + 2b = 12
2b = 6
b = 3
Therefore Bill has $3.00. If follows that
t = 6 + b = 6 + 3 = 9
and so Tom has $9.00.
Check:
(1) 9 = 6 + 3
(2) 9 + 3 = 12