Algebraic expressions. Order of operations - A complete course in algebra

ALGEBRA IS A WRITTEN SKILL. That implies that the writer solves a problem clearly, efficiently, with the least amount of writing. Like any skill—driving a car, baking cookies, playing the guitar—it requires 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 stand for numbers. We imitate the rules of arithmetic with letters, because we mean that the rule will be true for any numbers.

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
and write their sum over the common denominator."

The symbols for numbers, after all—1, 2, 3— are nothing but written marks. And so are letters. As the student will see, algebra depends on the forms that the symbols take. What you see above is called the formal rule for adding fractions.

The numbers are the numerical symbols, while the letters are called literal symbols.

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1)	Addition: a + b. The operation sign is + , and is called the plus sign. Read a + b as "a plus b."

1)	For example, if a represents 3, and b represents 4, then a + b represents 7.

2)	Subtraction: a − b. The operation sign is − , and is called the minus sign. Read a − b as "a minus b."

1)	If a represents 8, for example, and b represents 2, then a − b represents 6.

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

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.

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

Note: When there is no operation sign between 3 and (4 + 5), it means multiplication.

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).

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.

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?

There are three terms. 2a is the first term. It has two factors:
2 and a.
4ab is the second term. It has three factors: 4, a, and b.
And 5a(b + c) is all one term. It also has three factors: 5, a, and
(b + c). The parentheses mean that we should treat whatever is enclosed as one number.

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.

That small 4 is called an exponent. It indicates the number of times to repeat a as a factor.

The student must take care not to confuse 3a, which means 3 times a, with a³, which means a times a times a.

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. The multiplication cross × is used only in scientific notation -- therefore the student will never see the following:

Such a problem would be purely academic, which is to say, it is an exercise for its own sake. It is of no practical value. It leads nowhere.

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.

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.

-- could we treat 60 + 3 as one number. In the absence of parentheses, the problem means to subtract 60 from 100, then add 3:

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:

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:

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.

ALGEBRAIC EXPRESSIONS