Realm of ALGEBRA NOTE: Do not try to convert this book into any other format than adobe accrobat, or you will lose parts of it Isaac Asimov diagrams by Robert Belmore 314873 19 6 1 HOUGHTON MIFFLIN COMPANY BOSTON The Riverside Press Cambridge Some other books by Isaac Asimov Words of Science Realm of Numbers Breakthroughs in Science Realm of Measure Words from the Myths NOTE: Do not try to convert this book into any other format than adobe accrobat, or you will lose parts of it COPYRIGHT © 1961 BY ISAAC ASIMOV ALL RIGHTS RESERVED INCLUDING THE RIGHT TO REPRODUCE THIS BOOK OR PARTS THEREOF IN ANY FORM. LIBRARY OF CONGRESS CATALOG CARD NUMBER: 61-10637 PRINTED IN THE U.S.A. C O N T E N TS 1 The Mighty Question Mark 1 2 Setting Things Equal 11 3 More Old Friends 31 4 Mixing the Operations 52 5 Backwards, Too! 67 6 The Matter of Division 77 7 The Ins and Outs of Parentheses 90 8 The Final Operations 111 9 Equations by Degrees 133 10 Factoring the Quadratic 152 11 Solving the General 169 12 Two at Once 188 13 Putting Algebra to Work 208 1 The Mighty Question Mark SYMBOLS ALGEBRA IS just a variety of arithmetic. Does that startle you? Do you find it hard to believe? Perhaps so, because the way most of us go through our schooling, arithmetic seems an "easy" subject taught in the lower grades, and algebra is a "hard" subject taught in the higher grades. What's more, arithmetic deals with good, honest numbers, while algebra seems to be made up of all sorts of confusing x's and y's. But I still say there's practically no difference between them and I will try to prove that to you. Let's begin by saying that if you had six apples and I gave you five more apples, you would have eleven apples. If you had six books and I gave you five more books, you would have eleven books. If you had six dandelions and I gave you five more dandelions, you would have eleven dandelions. I don't have to go on that way, do I? You can see that if you had six of any sort of thing at all and I gave you five more of that same thing, you 2 ALGEBRA would end with eleven of it altogether. So we can forget about the actual object we're dealing with, whether apples, books, dandelions, or anything else, and just concentrate on the numbers. We can say simply that six and five are eleven, or that six plus five equals eleven. Now people are always dealing with numbers; whether in the work they do, in the hobbies they pursue, or in the games they play. They must always remember, or be able to figure out if they don't remember, that six plus five equals eleven, or that twenty-six plus fifty-eight equals eighty-four, and so on. What's more, they often have to write down such arithmetical statements. But the writing can get tedious, particularly where the numbers grow large and complicated. For that reason, ever since the earliest days of civilization, people have been trying to figure out good short-cuts for writing down numbers. The best system ever invented was developed in India some time in the 800's. In that system, each number from one to nine had its own special mark. The marks we use these days in our country are 1, 2, 3, 4, 5, 6, 7, 8, and 9. In addition, the system includes a mark for zero, which we write as 0. Any mark written down as a short-cut method of representing something is called a "symbol." (The very words you are now reading are symbols The Mighty Question Mark 3 of the various sounds we make when we speak, and the sound we make when we say "horse" is just a symbol of the actual creature itself.) The marks I have written two paragraphs ago are symbols for the first nine numbers and zero, and may be called "numerical symbols." The Arabs picked them up from the mathematicians of India and passed them on to the Europeans in about the tenth century. We still call these numerical symbols the "Arabic numerals," in consequence. All numbers higher than nine can be written by using combinations of these numerical symbols according to a system which I won't explain here because it is so familiar to you.* Thus, the number twenty-three is written 23, while seven hundred and fifty-two is written 752. You can see how handy numerical symbols can be. In fact, they are so handy that you would never see anyone write: "The total is six thousand seven hundred and fifty-two." It would always be written, "The total is 6752." A great deal of space and effort is saved by writing the numerical symbols in place of words, yet you are so accustomed to the * Actually, I have explained the number system in a book I wrote called Realm of Numbers, published in 1959 by Houghton Mifflin Company. You don't have to read it to understand this book, but you might find it useful in explaining some arithmetical points I will have to skip over a little quickly here. 4 ALGEBRA symbols that you read a sentence containing them just as though the words had been spelled out. Nor are the numerals the only symbols used in everyday affairs. In business, it is so usual to have to deal with dollars that a symbol is used to save time and space. It is $, which is called the "dollar sign." People just read it automatically as though it were the word itself so that $7 is always read "seven dollars." There is also i for "cents," % for "per cent," & for "and," and so on. So you see you are completely at home with symbols. There's no reason why we can't use symbols to express almost anything we wish. For instance, in the statement six plus five equals eleven, we can replace six by 6, five by 5, and eleven by 11, but we don't have to stop there. We can have a symbol for "plus" and one for "equals." The symbol for "plus" is + and the symbol for "equals" is =. We therefore write the statement: 6 + 5 = 11. FACING THE UNKNOWN We are so familiar with these symbols and with others, such as — for subtraction, X for multiplica tion, and / for division that we give them no thought. • We learn them early in school and they're with us for life. But then, later in our schooling, when we pick up The Mighty Question Mark 5 new symbols, we are sometimes uneasy about them because they seem strange and unnatural, not like the ones we learned as small children. Yet why shouldn't we learn new symbols to express new ideas? And why should we hesitate to treat the new symbols as boldly and as fearlessly as we treat the old? Let me show you what I mean. When we first start learning arithmetic, what we need most of all is practice, so that we will get used to handling numbers. Consequently, we are constantly pre sented with numerous questions such as: How much is two and two? If you take five from eight, how much do you have left? To write these questions down, it is natural to use symbols. Therefore, on your paper or on the blackboard will be written 2 + 2 = 8 - 5 = and you will have to fill in the answers, which, of course, are 4 and 3 respectively. But there's one symbol missing. What you are really saying is: "Two plus two equals what?"; "Eight minus five equals what?" Well, you have good symbols for "two," "eight," "five," "plus," "minus," and "equals," but you don't have a symbol for 'Vhat?" Why not have 6 ALGEBRA one? Since we are asking a question, we might use a question mark for the purpose. Then, we can write 2 + 2 = ? 8 - 5 = ? The ? is a new symbol that you are not used to and that might make you uneasy just for that reason. However, it is merely a symbol repre senting something. It represents an "unknown." You always know just what 2 means. It always stands for "two." In the same way + always stands for "plus." The symbol ?, as I've used it here, however, can stand for any number. In the first case, it stands for 4; in the second case, it stands for 3. You can't know what it stands for, in any particular case, unless you work out the arithmetical problem. Of course, in the cases given above, you can see the answer at a glance. You can't, though, in more complicated problems. In the problem ? equals a particular number, but you can't tell which one until you work out the division. (I won't keep you'in suspense because this is not a book of problems. In this case, ? stands for 72.) But you might wonder if I'm just making some- The Mighty Question Mark 7 thing out of nothing. Why put in the question mark after all? Why not leave the space blank and just fill in the answer as, in fact, is usually done? Well, the purpose of symbols is to make life simpler. The eye tends to skip over a blank space, and you have no way of reading a blank space. You want to fill the space with a mark of some sort just to show that something belongs there, even if you don't know exactly what for a moment. Suppose, for instance, you had a number of apples, but weren't sure exactly how many. How ever, a friend gave you five apples and, after that gift, you counted your apples and found you had eight altogether. How many did you have to begin with? What this boils down to is that some number plus five equals eight. You don't know what that "some number" is until you think about it a little. The "some number" is an unknown. So you can write ? + 5 = 8 and read that as, "What number plus five equals eight?" If you had tried to do away with symbols such as a question mark and just left a blank space, you would have had to write +5 = 8 and you will admit that that looks funny and is