ALGEBRA IS A METHOD OF WRITTEN CALCULATIONS that help us reason about numbers. At the very outset, the student should realize that algebra is a skill. And like any skill -- driving a car, baking cookies, playing the guitar -- it requires practice. A lot of practice. Written practice. That said, let us begin.
The first thing to note is that in algebra we use letters as well as numbers. But the letters represent numbers. We imitate the rules of arithmetic with letters, because we mean that the rule will be true for any numbers.
Here, for example, is the rule for adding fractions:
The letters a and b mean: The numbers that are in the numerators. The letter c means: The number in the denominator. The rule means:
"Whatever those numbers are, add the numerators
Algebra is telling us how to do any problem that looks like that. That is one reason why we use letters.
(The symbols for numbers, after all, are nothing but written marks. And so are letters As the student will see, algebra depends only on the patterns that the symbols make.)
The numbers are the numerical symbols, while the letters are called literal symbols.
Question 1. What are the four operations of arithmetic, and
what are their operation signs?
To see the answer, pass your mouse over the colored area.
3) Multiplication: a· b. Read a· b as "a times b."
The multiplication sign in algebra is a centered dot. We do not use the multiplication cross ×, because we do not want to confuse it with the letter x.
And so if a represents 2, and b represents 5, then
a· b = 2· 5 = 10.
"2 times 5 equals 10."
Do not confuse the centered dot -- 2·5, which in the United States means multiplication -- with the decimal point: 2.5.
However, we often omit the multiplication dot and simply write ab. Read "a, b." In other words, when there is no operation sign between two letters, or between a letter and a number, it always means multiplication. 2x means 2 times x.
In algebra, we use the horizontal division bar. If a represents 10, for example and b represents 2, then
"10 divided by 2 is 5."
Note: In algebra we call a + b a "sum" even though we do not name an answer. As the student will see, we name something in algebra simply by how it looks. In fact, you will see that you do algebra with your eyes, and then what you write on the paper, follows.
This sign = of course is the equal sign, and we read this --
a = b
-- as "a equals (or is equal to) b."
That means that the number on the left that a represents, is equal to the number on the right that b represents. If we write
a + b = c,
and if a represents 5, and b represents 6, then c must represent 11.
Question 2. What is the function of parentheses () in algebra?
3 + (4 + 5) 3(4 + 5)
Parentheses signify that we should treat what they enclose
3 + (4 + 5) = 3 + 9 = 12. 3(4 + 5) = 3· 9 = 27.
Note: When there is no operation sign between 3 and (4 + 5), it means multiplication.
Problem 1. In algebra, how do we write
a) 5 times 6? 5· 6
b) x times y? xy
d) x plus 5 plus x minus 2? (x + 5) + (x − 2)
e) x plus 5 times x minus 2? (x + 5)(x − 2)
Problem 2. Distinguish the following:
a) 8 − (3 + 2) b) 8 − 3 + 2
a) 8 − (3 + 2) = 8 − 5 = 3.
b) 8 − 3 + 2 = 5 + 2 = 7.
In a), we treat 3 + 2 as one number. In b), we do not. We are to first subtract 3 and then add 2. (But see the order of operations below.)
There is a common misconception that parentheses always signify multiplication. In Lesson 3, in fact, we will see that we use parentheses to separate the operation sign from the algebraic sign. 8 + (−2).
Question 3. Terms versus factors.
When numbers are added or subtracted, they are called terms.
When numbers are multiplied, they are called factors.
Here is a sum of four terms: a − b + c − d.
In algebra we speak of a "sum" of terms, even though there are subtractions. In other words, anything that looks like what you see above, we call a sum.
Here is a product of four factors: abcd.
The word factor always signifies multiplication.
And again, we speak of the "product" abcd, even though we do not name an answer.
Problem 3. In the following expression, how many terms are there? And each term has how many factors?
2a + 4ab + 5a(b + c)
There are three terms. 2a is the first term. It has two factors:
Powers and exponents
When all the factors are equal -- 2· 2· 2· 2 -- we call the product a power of that factor. Thus, a· a is called the second power of a, or "a squared." a· a· a is the third power of a, or "a cubed." aaaa is a to the fourth power, and so on. We say that a itself is the first power of a.
Now, rather than write aaaa, we write a just once and place a small 4:
a4 ("a to the 4th")
That small 4 is called an exponent. It indicates the number of times to repeat a as a factor.
83 ("8 to the third power" or simply "8 to the third") means 8· 8· 8.
Problem 4. Name the first five powers of 2. 2, 4, 8, 16, 32.
Problem 5. Read, then calculate each of the following.
a) 52 "5 to the second power" or "5 squared" = 25.
b) 23 "2 to the third power" or "2 cubed" = 8.
c) 104 "10 to the fourth" = 10,000.
d) 121 "12 to the first" = 12.
However, it is the style in algebra not to write the exponent 1.
a = a1 =1a.
The student must take care not to confuse 3a, which means 3 times a, with a3, which means a times a times a.
Question 4. When there are several operations,
8 + 4(2 + 3)2 − 7,
what is the order of operations?
Before answering, let us note that since skill in science is the reason students are required to learn algebra; and since orders of operations appear only in certain forms, then in these pages we present only those forms that the student is ever likely to encounter in the actual practice of algebra. The division sign ÷ is never used in scientific formulas, only the division bar. And the multiplication cross × is used only in scientific notation -- therefore the student will never see the following:
3 + 6 × (5 + 3) ÷ 3 − 8.
Such a problem would be purely academic, which is to say, it is an exercise for its own sake, and is of no practical value. It leads nowhere.
The order of operations is as follows:
In Examples 1 and 2 below, we will see in what sense we may add or subtract. And in Example 3 we will encounter multiply or divide.
Note: To "evaluate" means to name and write a number.
Example 1. 8 + 4(2 + 3)2 − 7
First, we will evaluate the parentheses, that is, we will replace 2 + 3 with 5:
= 8 + 4· 52 − 7
Since there is now just one number, 5, it is not necessary to write parentheses.
Notice that we transformed one element, the parentheses, and rewrote all the rest.
Next, evaluate the exponents:
= 8 + 4· 25 − 7
= 8 + 100 − 7
Finally, add or subtract, it will not matter. If we add first:
= 108 − 7 = 101.
While if we subtract first:
8 + 100 − 7 = 8 + 93 = 101.
Example 2. 100 − 60 + 3.
100 − 60 + 3 does not mean 100 − 63.
Only if there were parentheses --
100 − (60 + 3)
-- could we treat 60 + 3 as one number. In the absence of parentheses, the problem means to subtract 60 from 100, then add 3:
100 − 60 + 3 = 40 + 3 = 43.
In fact, it will not matter whether we add first or subtract first,
100 − 60 + 3 = 103 − 60 = 43.
When we come to signed numbers, we will see that
100 − 60 + 3 = 100 + (−60) + 3.
The order in which we "add" those will not matter.
There are no parentheses to evaluate and no exponents. Next in the order is multiply or divide. We may do either -- we will get the same answer. But it is usually more skillful to divide first, because we will then have smaller numbers to multiply. Therefore, we will first divide 35 by 5:
Example 4. ½(3 + 4)12 = ½· 7· 12.
The order of factors does not matter: abc = bac = cab, and so on. Therefore we may first do ½· 12. That is, we may first divide 12 by 2:
½· 7· 12 = 7· 6 = 42.
In any problem with the division bar, before we can divide we must evaluate the top and bottom according to the order of operations. In other words, we must interpret the top and bottom as being in parentheses.
Now we proceed as usual and evaluate the parentheses first. The answer is 4.
Problem 6. Evaluate each of the following according to the order of operations.
Question 5. What do we mean by the value of a letter?
The value of a letter is a number. It is the number that will replace the letter when we do the order of operations.
Question 6. What does it mean to evaluate an expression?
It means to replace each letter with its value, and then do the order of operations.
Example 6. Let x = 10, y = 4, z = 2. Evaluate the following.
In each case, copy the pattern. Copy the + signs and copy the parentheses ( ). When you come to x, replace it with 10. When you come to y, replace it with 4. And when you come to z, replace it with 2.
Problem 7. Let x = 10, y = 4, z = 2, and evaluate the following.
g) x2 − y2 + 3z2 = 100 − 16 + 3· 4 = 100 − 16 + 12 = 84 + 12 =96.
Again, 100 − 16 + 12 does not mean 100 − (16 + 12).
That is 168 divided by 100. See Lesson 4 of Arithmetic, Question 4.
Question 7. Why is a literal symbol -- x, y, z -- called a variable?
Because its value may vary.
A variable, such as x, is a kind of blank or empty symbol. It is therefore available to take any value we might give it: a positive number or, as we shall see, a negative number; a whole number or a fraction.
Numerical symbols -- 2, 3, 4 -- are called constants. The value of those symbols does not vary.
Problem 8. Two variables. Let the value of the variable y depend
y = 2x + 4.
Calculate the value of y that corresponds to each value of x:
When x = 0, y = 2· 0 + 4 = 0 + 4 = 4.
When x = 1, y = 2· 1 + 4 = 2 + 4 = 6.
When x = 2, y = 2· 2 + 4 = 4 + 4 = 8.
When x = 3, y = 2· 3 + 4 = 6 + 4 = 10.
When x = 4, y = 2· 4 + 4 = 8 + 4 = 12.
Real problems in science or in business occur in ordinary language. To do such problems, we typically have to translate them into algebraic language.
Problem 9. Write an algebraic expression that will symbolize each of the following.
a) Six times a certain number. 6n, or 6x, or 6m. Any letter will do.
b) Six more than a certain number. x + 6
c) Six less than a certain number. x − 6
d) Six minus a certain number. 6 − x
e) A number repeated as a factor three times. x· x· x = x3
f) A number repeated as a term three times. x + x + x
g) The sum of three consecutive whole numbers. The idea, for example,
h) Eight less than twice a certain number. 2x − 8
i) One more than three times a certain number. 3x + 1
Now an algebraic expression is not a sentence, it does not have a verb, which is typically the equal sign = . An algebraic statement has an equal sign.
Problem 10. Write each statement algebraically.
a) The sum of two numbers is twenty. x + y = 20.
b) The difference of two numbers is twenty. x − y = 20.
c) The product of two numbers is twenty. xy = 20.
d) Twice the product of two numbers is twenty. 2xy = 20.
e) The quotient of two numbers is equal to the sum of those numbers.
A formula is an algebraic rule for evaluating some quantity. A formula is a statement.
Example 7. Here is the formula for the area A of a rectangle whose base is b and whose height is h.
A = bh.
"The area of a rectangle is equal to the base times the height."
And here is the formula for its perimeter P -- that is, its boundary:
P = 2b + 2h.
"The perimeter of a rectangle is equal to two times the base
For, in a rectangle the opposite sides are equal.
Problem 11. Evaluate the formulas for A and P when b = 10 in, and h = 6 in.
A = bh = 10· 6 = 60 in2.
P = 2b + 2h = 2· 10 + 2· 6 = 20 + 12 = 32 in.
Problem 12. The area A of trapezoid is given by this formula,
A = ½(a + b)h.
Find A when a = 2 cm, b = 5 cm, and h = 4 cm.
A = ½(2 + 5)4 = ½· 7· 4 = 7· 2 = 14 cm2.
When 1 cm is the unit of length, then 1 cm² ("1 square centimeter") is the unit of area.
Problem 13. The formula for changing temperature in degrees Fahrenheit (F) to degrees Celsius (C) is given by this formula:
Find C if F = 68°.
Replace F with 68:
"One ninth of 36 is 4. So five ninths is five times 4: 20."
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