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1 ## ALGEBRAIC EXPRESSIONSThe four operations and their signs 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 Here, for example, is the algebraic rule for adding fractions:
The letters "Whatever those numbers are, add the numerators Algebra is telling us how to do any problem that 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 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: 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 And so if
"2 times 5 equals 10." Do not confuse the centered dot -- 2 However, we often omit the multiplication dot and simply write
In algebra, we use the horizontal division bar. If
"10 divided by 2 is 5."
Similarly, we call This sign = of course is the equal sign, and we read this --
-- as " That means that the number on the left that
and if 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
Problem 1. In algebra, how do we write a) 5 times 6? 5 b)
d) ( e) ( 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: In algebra we speak of a "sum" of terms even though there are subtractions. In other words, anything that Here is a The word And again, we speak of the "product"
Problem 3. In the following expression, how many 2
There are three terms. 2 Powers and exponents When all the factors are equal --
2 Now, rather than write
That small 4 is called an exponent. It indicates the number of times to repeat 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) 5 b) 2 c) 10 d) 12 However, it is the style in algebra not to write the exponent 1.
The student must take care not to confuse 3
Question 4. When there are several operations, 8 + 4(2 + 3) 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. 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. It 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
Example 1. 8 + 4(2 + 3) First, we will evaluate the parentheses, that is, we will replace 2 + 3 with 5: = 8 + 4 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 Now multiply: = 8 + 100 − 7 Finally, add = 108 − 7 = 101. While if we subtract first: 8 + 100 − 7 = 8 + 93 = 101. Example 2. 100 − 60 + 3. First: 100 − 60 + 3 does 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
See: Skill in Arithmetic, Property 3 of Division.
Example 4. ½(3 + 4)12 = ½ The ½ (See Lesson 27 of Arithmetic, Question 1.)
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.
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