Take a Number: Mathematics for the Two Billion
240
Take a Number: Mathematics for the Two Billion
240eBook
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Overview
Author Lillian R. Lieber was head of the Department of Mathematics at Long Island University. In the 1940s, she wrote a series of good-humored, well-respected books about mathematics, including The Education of T. C. MITS (The Celebrated Man in the Street), a special edition of which was circulated among American GIs during World War II. Whimsical drawings by her husband, Hugh Gray Lieber, illustrate Take a Number and many of her other books.
Product Details
| ISBN-13: | 9780486822334 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 05/24/2017 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 240 |
| File size: | 32 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Read an Excerpt
Take A Number
Mathematics for the Two Billion
By Lillian R. Lieber
Dover Publications, Inc.
Copyright © 2017 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-82233-4
CHAPTER 1
WHY SHOULD YOU STUDY MATHEMATICS?
Most people probably grant nowadays that since we live in a scientific age, SOMEBODY has to study Mathematics, but is this subject necessary for EVERYONE? For all the two billion (2 X 109) people on this earth? Why not let those who like it struggle with it, and let the rest of us study something else, or maybe just take it easy and enjoy life? We shall try to tell you WHY everyone, including you, must study some Mathematics — and the more the merrier!
And, with this little book, perhaps you will even begin to like it — we hope!
First of all, you will admit that everyone must know a little Arithmetic, so that he can at least count his pennies or his sheep or what have you, and carry on whatever little business he has to do. "True," you will say, "but that is not much. How about Algebra, though, and Geometry and all that stuff? No doubt the scientists need it, but I am going to be a businessman, or a stenographer, or maybe a housewife, and I surely don't expect to use Math. What I need is good looks, money, common sense — but not Math." Now you know perfectly well that many times people who are not so good-looking know how to use what looks they have to better advantage than some very handsome ones. And that people without money can go out and make a million if they have the ability and if that is what they want. In short, what we need most is a knowledge of the world we live in and the "tools" needed to get along in it. Now the most important "tool" is certainly your own brain, so that if you learn to use that "instrument" you will really have something! Furthermore, it is easy to see that EVERYONE must learn to use his "bean," or else what one intelligent man or group builds up can be destroyed by the ape-men. So you see unless we ALL GET WISE TOGETHER, the world cannot be safe for anyone.
Now, since man's brain has been most successful in Science (including Physics et al), Art (including Music, Poetry, et al), and Mathematics, let us see if we can learn from THEM how to really use our brains And, to make it easier, let us combine them all into a single character called SAM — so that our problem is to help to bring SAM to the TWO BILLION, and show HOW he can help us. You will see that he can be a better LEADER than any individual human being!
Now it must be very clearly understood that SAM, who is of course very scientific, is nevertheless NOT just a gadgeteer who furnishes us with radios and automobiles and other gadgets to play with, though he HAS his pockets full of toys for us — but he will give them to us ONLY if the human race is GOOD — otherwise he will give us ATOMIC BOMBS instead! For in his heart SAM is really in agreement with the great RELIGIONS and will FIGHT for them and not allow us to ignore them any longer!
CHAPTER 2MATHEMATICS MADE EASY
One of the main difficulties with the study of Mathematics by the average person has always been that he was never let in on the basic rules of the game, but was given THOUSANDS of little details to remember, which did not seem to have much connection with each other, so he could never figure things out for himself. Can you imagine playing any game without knowing the EQUIPMENT of the game or the RULES? But just being pushed around without ever finding out what you were doing, or the difference between a goal and a foul?!
Now, in this little book, we shall try to show you that Mathematics is really like a game. And if you know the EQUIPMENT and the RULES (and you will be surprised to find how VERY FEW basic rules there are!), you can easily learn the game and even figure out your own "plays." This not only makes it easy to play, but you will be surprised to find what FUN it is! Try it and see.
CHAPTER 3THE EQUIPMENT
First, take Arithmetic. The equipment of Arithmetic consists of various kinds of NUMBERS, which we shall recall to your mind in this chapter. And you can then easily go from Arithmetic to Algebra by merely ENLARGING this equipment to include a kind of numbers called NEGATIVE numbers, which are very easy to understand, as you will see.
Another change in going from Arithmetic to Algebra will be in the use of LETTERS to stand for numbers. You will see how this simple device will make it possible to solve problems more EASILY and more GENERALLY. Although you may not realize it, you already know, from Arithmetic, several different kinds of numbers: for example, as a very small child, you learned about the WHOLE NUMBERS or INTEGERS, like 1, 7, 11, 79, etc. Later on, you also learned about FRACTIONS, like 1/2, 7/8, 11/9 and decimal fractions, like 1.25, .06, 7.09, and mixed numbers, like 1 1/2, 3 1/4, etc. Now, fortunately, it is not necessary to have so many different names, for all the numbers mentioned above are called
RATIONAL NUMBERS. Let us give you a clear definition of a RATIONAL NUMBER: This is ANY NUMBER WHICH MAY BE EXPRESSED AS A RATIO OF TWO INTEGERS: Thus obviously the fraction 7/8 is a rational number since it is the ratio of the integer 7 to the integer 8. Similarly the mixed number 1 ½ is a rational number since it can be written 3/2, which is the ratio of the integer 3 to the integer 2. Also, the decimal .06 is a rational number, since it can be written 6/100 or 3/50. And even an integer, like 7, can be written 7/1, which, again, is the ratio of the integer 7 to the integer 1, showing that the integers themselves are also rational numbers.
Now we want to introduce you to a kind of number which you have never had in Arithmetic. These are the NEGATIVE NUMBERS mentioned on page 9. They are very easy to understand, especially with the help of the following line:
[ILLUSTRATION OMITTED]
On this line you see that all the numbers which you had in Arithmetic are the zero and the numbers to the RIGHT of the zero, including integers, fractions, decimal fractions, and mixed numbers, each having its definite place on the line. But you also see that we can put numbers to the LEFT of the zero, putting a MINUS sign in front of each to distinguish it from the numbers on the right, called the POSITIVE numbers. A positive number may be written with a plus sign, thus: or more briefly, without the plus sign:
+2, +5.
Or more briefly, without the plus sign:
2, 5.
But of course you must NEVER omit the MINUS sign in front of a negative number, like -1 or -7.
The NEGATIVE NUMBERS are very USEFUL, for they enable us to write BRIEFLY:
a temperature of 5° BELOW ZERO simply as -5°; or a DEBT of 5 dollars simply as -$5. Etc. And as you go on, you will see other advantages of the NEGATIVE NUMBERS.
So you see that the first important thing about Algebra is that it has already opened up to you a new set of numbers which do not exist in Arithmetic, and which are very useful. And the second new idea in Algebra is the use of letters for numbers:
for instance, in Arithmetic, if you wish to find the area of a rectangle whose length is 4 feet and whose width is 7 feet, you say
area = 4X7 = 28.
In Algebra, you may use A for area, l for length, and w for width, and say
A = lw
which gives you the general FORMULA for finding the area of ANY rectangle, instead of a particular one as you do in Arithmetic. You will find some more formulas on page 211.
Another advantage of the use of letters is that, in solving a problem, you can do it more easily and more directly with the use of letters, for you may use a letter to stand for a quantity which is unknown, and still go ahead and work with it, and pretty soon the problem almost solves itself mechanically, and you find out the value of the unknown quantity! It is almost like magic!
CHAPTER 4THE RULES OF THE GAME
Now, in playing any game, you naturally want to know (1) what THINGS you play with — that is, the EQUIPMENT, and (2) what the RULES of the game are. Think of baseball or football or any other game and you will see that this is true.
It is the same way in Science or Mathematics, except that here we say "ELEMENTS" instead of "EQUIPMENT," and the RULES are called "POSTULATES." Thus, in ARITHMETIC, the ELEMENTS are the POSITIVE RATIONAL NUMBERS AND ZERO; and the POSTULATES are very familiar to you, although perhaps you never called them "postulates" before. Here are some of the postulates which you know well:
(1) The Commutative Law for Addition: that is, a + b = b + a, which means that when you ADD two numbers, it does not matter which one you write first: for instance,
3 + 4 and 4 + 3
both give the same answer, 7, and therefore
3 + 4 = 4 + 3.
By the way, notice how convenient it is to say
a + b = b + a
instead of merely
3 + 4 = 4 + 3,
because, when using letters, you are really saying that this Commutative Law for Addition applies to ALL the elements in Arithmetic, since the letters may stand for ANY numbers, and not merely for 3 and 4.
(2) The Commutative Law for Multiplication that is, ab = 6a, which means that when you MULTIPLY two numbers, it also does not matter which one you write first: for instance,
2 X 5 = 5 X 2.
Notice that with LETTERS it is not necessary to write the times sign (X) between them, as you do with numbers: thus, ab means a X b, but 2 X 5 cannot be written 25, since this means "twenty-five."
Perhaps you think that, since ADDITION and MULTIPLICATION are both COMMUTATIVE, then all operations must be commutative. But this is NOT so! For instance, DIVISION is NOT COMMUTATIVE, since 12 ÷ 6 = 2, but 6 ÷ 12 = 1/2; or, in general,
a ÷ b ≠ b ÷ a.
Of course, all this is really perfectly familiar to you, for you learned it as a small child, and now do it automatically. Let us mention, therefore, only a few more of these familiar postulates, which we shall need later.
(3) The Associative Law for Addition:
a + b + c = (a + b) + c = a + (b + c).
This means that when you have to ADD THREE NUMBERS together, it does not matter whether you add the sum of the first two to the third, or add the first number to the sum of the last two: for instance, to add 2 + 7 + 11, you may say either
2 + 7 = 9 and then 9 + 11 = 20,
or you may say
2 + (the sum of 7 and 11, which is 18)
and then 2 + 18 = 20; you see that you get the same answer. Notice that the parentheses " associate " the first two numbers in the one case, and the last two numbers in the other:
(2 + 7) + 11 = 9 + 11 = 20,
or
2 + (7 + 11) = 2 + 18 = 20
and therefore
(2 + 7) + 11 = 2 + (7+ 11).
Or, in general,
(a + b) + c = a + (b + c),
that is why this is called the ASSOCIATIVE LAW.
(4) The Associative Law for Multiplication:
abc = (ab)c = a(bc).
And
(5) The Distributive Law:
a(b + c) = ab + ac.
An illustration of this is:
if you wish to MULTIPLY a number, like 5, by the SUM of two other numbers, say, 2 + 7, you may say either
5 X (2 + 7) = 5 X 9 = 45
or you may multiply 5X2 and 5X7, and add these results together, obtaining
5 X (2 + 7) = 5 X 2 + 5 X 7 = 10 + 35
which gives the same answer, 45. Thus 5 (x + y) is the same as 5x + 5y or 7c + 7d is the same as 7(c + d).
When you say that 5(x + y) is equal to 5x + 5y, you are doing a MULTIPLICATION example. But when you say that 5x + 5y is equal to 5 (x + y) you are FACTORING; just as 3 X 2 = 6 is a MULTIPLICATION example but when you say
6 = 3X2
you have split 6 into its two FACTORS, 3 and 2. Note that when you split 6 into 5 + 1 you have NOT factored the 6. To FACTOR a quantity you must split it into a PRODUCT of other quantities. Thus, to factor 14 you get 7X2; and to factor 3x + 9y you get 3(x + 3y), etc.
To be sure that you understand this you should practise a little on page 191 #1. It is fun!
As we have already told you, in Algebra we have also NEGATIVE NUMBERS (see page 13), so that here the elements will be ALL THE RATIONAL NUMBERS, POSITIVE and NEGATIVE, and ZERO. And, although we have introduced new things to "play with" (the negative numbers), still, we do not change any of the above-mentioned RULES of the game (the POSTULATES), since, after all, we are still playing practically the same game, Algebra being only a sort of "glorified" Arithmetic.
CHAPTER 5HOW THE GAME IS PLAYED
And now let us see how the game is played. First, take Arithmetic: As soon as you learned the integers and the postulates (the rules), you were then taught how to perform the FOUR FUNDAMENTAL OPERATIONS (ADDITION, SUBTRACTION, MULTIPLICATION, DIVISION) with the integers. And later, when you had fractions, you again had to learn how to perform these FOUR FUNDAMENTAL OPERATIONS with fractions: that is, how to ADD FRACTIONS, how to SUBTRACT them, how to MULTIPLY them, and how to DIVIDE them. (And, similarly with DECIMAL fractions and mixed numbers.) So that, by now you should know how to perform the FOUR FUNDAMENTAL OPERATIONS with all the POSITIVE RATIONAL NUMBERS. But in case you have forgotten this, and since it is very important, you had better take a little time out to review this. Try your skill on page 191 #2.
And now all you have to learn in Algebra is how to perform the FOUR FUNDAMENTAL OPERATIONS with NEGATIVE NUMBERS and with LETTERS. Then you can apply it to the solution of practical problems
CHAPTER 6ADDITION
Now you know that in Arithmetic,
7 + 4 = 11;
and, of course, this is still true in Algebra. But in Algebra you might have an example like this:
7 + (-4).
What does this mean? Well, if you are talking about money, a positive number (like 7) represents money that you HAVE, or that is COMING IN, whereas a negative number (like -4) represents money that you OWE, or that is GOING OUT (like expenses). And so, if you have $7 and a debt of $4, your account is worth $3, so that
7 + (-4) = 3
In other words, you may think of ALGEBRAIC ADDITION as "balancing an account." See if you understand the answers to the following examples:
[FORMULA OMITTED]
These examples may also be written horizontally, thus:
(1) 4 + 7 = 11; (2) 4 - 7 = -3; (3) -4 + 7 = 3 (4) -4 - 7 = — 11; (5) -3 + 7 + 11 - 5 = 10; (6) 1 + 0 - 1 + 4 = 4; (7) -3 - 5 - 6 = -14
At first, in doing such examples, it might be easier for you to have a little "Account Book," and put on the RIGHT-HAND page the heading "Coming In," and on the LEFT-hand page the heading "Going Out"; then, in doing these "addition" examples, put all POSITIVE numbers on the RIGHT-hand page, and all NEGATIVE numbers on the LEFT-hand page, and then "balance" your account. Thus, in example {5) above, 7 and 11 would go on the RIGHT, -3 and -5 on the LEFT, then, in "balancing" you see that $18 are "coming in," $8 are "going out," so that the result is that you would have $10 left over. Whereas, in example (7), since all the items are "going out," the result is a debt of 14, or -14. Notice that the meaning of the word "add" is NOT exactly the same as in ARITHMETIC, but had to be modified so that we can apply it to NEGATIVE numbers; and yet, when the numbers happen to be all POSITIVE, (as in example (1) above), the answer IS exactly the same as in Arithmetic, so that the new definition of " addition " DOES NOT CONTRADICT the old one, but merely enlarges it to make it applicable to the new elements (the negative numbers).
You will see, as you go on in Mathematics, that, to MAKE PROGRESS POSSIBLE,
(1) Old definitions have to be modified,
(2) New elements have to be introduced, and
(3) Even new postulates have to be introduced.
Thus the old ideas are NOT ENTIRELY discarded, but are MODIFIED as the need arises. This is absolutely essential to MAKE PROGRESS POSSIBLE!
So you see that in Mathematics we do not say, "What was good enough for my grandfather is good enough for me," but modify our ideas to suit the times we live in.
Notice that the MINUS sign in front of the 7 in example (2) on page 32 does NOT mean "subtract," for this is an ADDITION example; it merely says that the 7 is A NEGATIVE number. And you will soon see that you can do all sorts of things with negative numbers, you can ADD THEM, SUBTRACT THEM, MULTIPLY THEM, and DIVIDE THEM; so that from now on you must NOT think that every minus sign means "subtract," as it did in Arithmetic.
(Continues...)
Excerpted from Take A Number by Lillian R. Lieber. Copyright © 2017 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Is mathematics a subject that everyone must study? You bet it is, and this playful guide shows why all of the people on this planet need to know something about math, and the more, the merrier! Mathematics is like a game and if you know the equipment and the rules, you can use it for both work and recreation. This illustrated book suggests practical and charming methods of studying and practicing math. From addition, subtraction, multiplication, and division to algebra and geometry, the brisk, lighthearted approach offers fun-to-solve problems and complete answers.
Author Lillian R. Lieber was head of the Department of Mathematics at Long Island University. In the 1940s, she wrote a series of good-humored, well-respected books about mathematics, including The Education of T. C. MITS (The Celebrated Man in the Street), a special edition of which was circulated among American GIs during World War II. Whimsical drawings by her husband, Hugh Gray Lieber, illustrate Take a Number and many of her other books.
Dover (2017) republication of the edition originally published by The Jaques Cattell Press, Lancaster, Pennsylvania, 1946.
www.doverpublications.com