Puzzles, Games, and Tricks: Understanding the Mystery and Magic of Numbers
184
Puzzles, Games, and Tricks: Understanding the Mystery and Magic of Numbers
184eBook
Available on Compatible NOOK devices, the free NOOK App and in My Digital Library.
Related collections and offers
Overview
Product Details
| ISBN-13: | 9781510727816 |
|---|---|
| Publisher: | Skyhorse |
| Publication date: | 10/17/2017 |
| Sold by: | SIMON & SCHUSTER |
| Format: | eBook |
| Pages: | 184 |
| File size: | 25 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Read an Excerpt
CHAPTER 1
THE WORLD OF NUMBERS
* * *
EXPLORING BOTH ENDS OF OUR NUMBER SYSTEM
WE ORDINARY mortals live in a world of numbers just about halfway between the inconceivably large and the incredibly small. The numbers in our lives range between several million in the upper limit to about 1/10000 in the lower limit, and in the daily personal lives of most of us numbers greater than a few thousand or smaller than a sixteenth of an inch seldom appear.
When numbers get into the hundreds of millions or billions they have no meaning for us. We read of the population of the United States being 150,000,000 but can't possibly conceive such a large number, and numbers like the money for defense or the total national debt — running into the hundreds of billions — -make us yawn and say "Well, well." Of course if you are a banker or the owner of a high-circulation magazine, numbers in the millions don't faze you; if you are a highly skilled mechanic working in an airplane factory a ten-thousandth of an inch reading on a micrometer is nothing unusual. But who ever hears of a trillion or quadrillion or sextillion or an octo-vigintillionth of a gram in ordinary conversation? These terms, outlandish and impractical as they seem to us, belong to the scientist who not only "eats them for breakfast" but uses them to produce the thousands of scientific miracles that make our lives comfortable and our living standards the highest in the world.
To the astronomer a number like 5,000,000,000,000,000,000 miles is all in a day's work. It represents the distance in miles of a certain nebula or star cluster from the earth. It is 5 quintillion miles or somewhat less than one million light years. A light year is the distance that light, traveling at 186,000 miles per second, will travel in a year. To the atomic physicist a single gram contains about 1,000,000,000,000,000,000,000,000,000 electrons. This is more electrons than there are drops of water in the Atlantic and Pacific Oceans combined! Yes, unlike you and me, these scientists live in a world of numbers far beyond our comprehension, and they are continually playing with the very great or the very small.
Of course they don't write these giants or pigmies out in ciphers: that would be a waste of time and space. They merely represent them in powers of 10. The exponent tells the number of ciphers, so when you see 10 you know that it means 1 with 24 zeroes after the 1. When you see 10 you know that it is a fraction with 1 in the numerator and 1 followed by 18 zeros in the denominator:
1/1,000,000,000,000,000,000
When the physicist speaks of the mass of the earth being 5.9 × [10.sup.27] grams he saves himself from writing 5,900,000,000,000,000,000,000,000,000 grams. When he tells us that the energy locked up in a pound of matter is 4 × [10.sup.23] ergs he can't be bothered to write 400,000,000,000,000,000,000,000. We shall meet numbers in this section that are of this amazingly large or small order, so we might as well get used to the mathematical way of writing them.
NUMBER GIANTS–HOW MUCH IS A BILLION?
Suppose you had a billion dollars and, in your desire to be entirely unique, you decided to invest it, without interest or dividends, in a very bad stock company. Now suppose the corporation was so poorly run that it succeeded in losing a thousand dollars of your money every day in the week and still managed to stay in business. It would take more than two thousand years for you to lose that billion dollars!
Again assuming that you are seventeen years old and started now to count up to a billion, one count every second, day and night without stopping to eat or sleep. Of course you couldn't do this without going on shifts, so we'll assume that two of your seventeen-year-old friends help you out. By the time you reached a billion you would be in your late forties.
If your fountain pen were enlarged one billion times it would be 95,000 miles long and 8,000 miles high. The cap of your pen would be big enough to enclose the earth, whose diameter is roughly 7,900 miles. But a billion is only 10 and that is a mere nothing compared to the real giants. Suppose it were possible to tear a sheet of paper of a certain size and of about the thickness of this page, in half; and then to tear the pieces again in half, and then again in half, and to keep this up for fifty tearings. Each time you would be doubling the number of sheets of paper. The question is: How high a pile would fifty tearings make, and how big a sheet would you need to begin with, to get final sheets of about the size of this page? The answer is incredible but it can be easily verified. It is 2 raised to the 50th power and turns out to be 1,125,899,906,842,624 sheets of paper. Since each sheet would measure about one third of a square foot, this comes to 375,299,968,947,541 square feet or over 13,000,000 square miles, or a little less than one quarter the land area of the entire earth. Figuring 400 of these sheets to the inch, we would have a pile about 2,860,000,000,000 inches high. Divide this by 12 and we get about 230,000,000,000 feet. Now one million feet come to approximately 190 miles, so the pile would be more than 38,000,000 miles high!
Then there is the story of the ancient king who, being under obligation to one of his subjects, offered to reward him in any way he desired. The subject, a man of mathematical mind and modest tastes, simply asked for a chess board with one grain of wheat on the first square, two on the second, four on the third and so on, doubling the grains each time until all the squares on the board were accounted for. The old king was delighted and relieved with this simple request but was soon sorry that he granted it. To account for every square on the board in this fashion the king had to supply [2.sub.63] + 1 grains of wheat which, in round numbers, is 9,460,000,000,000,000,000. Now, assuming that there are 250 grains in a cubic inch and 2,150 cubic inches in a bushel, one bushel will contain about 540,000 grains of wheat. This number, then, would amount to 17,382,000,000,000 bushels. Since the yearly output of wheat of the entire United States is a little more than 1 billion bushels it would take this nation 16,000 years to satisfy this "modest" subject of the king. The story goes no further but the chances are the king lost his temper and the subject lost his head long before the 64th square of the chess board was reached.
Now let us consider a real giant. Let us roughly try to determine the number of molecules of water on the surface of the earth. Of course such a problem is highly impractical and ridiculous but it will serve to show what a real giant looks like. Taking the area of the water on the earth as 140,000,000 square miles and the average depth of the oceans as 2 miles we have a total volume of water of about 280,000,000 cubic miles or, in other words, 2.8 × [10.sup.8]. Now there are 2.6 × [10.sup.10] square centimeters in a square mile and 1.6 × [10.sup.5] centimeters to a mile. Therefore there are 4.1 × [10.sup.15] cubic centimeters in a cubic mile. Multiplying this by 2.8 × [10.sup.8] we get 11.5 × [10.sup.23] or 1.15 × [10.sup.24] cubic centimeters of water on the face of the earth. It has been quite accurately calculated that there are 27 × [10.sup.18] molecules of water to the cubic centimeter so we conclude that the water molecules on this earth amount to approximately (1.15 × [10.sup.24]) × (2.7 × [10.sup.19]) = 3.1 ×[10.sup.43]. This is otherwise known as 31 million, billion, billion, billion, billion molecules. Written out it looks like this: 31,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
A staggering number is the mass of the sun in grams, which turns out to be 19 with 32 ciphers after it, or 19 ×[10.sup.32]. Of course this is completely incomprehensible and inconceivable, but it still is a dwarf when multiplied by the number of molecules of hydrogen in a gram which is 27 ×[10.sup.18]. We can then say that the approximate number of molecules of hydrogen which make up the mass of the sun is 510 × [10.sup.50]. If this number seems large to you, consider Eddington's estimate of the number of electrons in the universe. This turns out to be about 157 × [10.sup.77] which according to Eddington, who figured it out, is: 15,747,724,136,275,002,577, 605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,185,631,0 31,296.
Now we come to a giant of giants, a number that is so ridiculously large that it never could apply to anything. Oddly enough it can be written with three nines and is represented by 9[??]. This is 9 multiplied by itself 387,420,489 times. Written in numerals the size we are using, the number would stretch for a thousand miles. To read it normally would take more than a week. What it is nobody knows although it does begin with 428,124,773 ... and ends with 89. That number of bacteria would overflow the Milky Way. The number of snowflakes that have fallen since the earth began does not even approach this giant. It is more than 4 million times as large as the number of electrons in the universe, which we just discussed.
INFINITY
Another extremely important idea in mathematics is infinity. Any quantity that increases without ever stopping will ultimately become and remain greater than any definite number we can name. It will then be infinite. Of course infinity and the infinite are vital terms and concepts in higher mathematics. We have briefly touched on this subject in the section called More and More of Less and Less. The theory of limits and infinite series gives us among other things the trigonometric tables and tables of logarithms.
An excellent example of one practical application of infinity is an elementary principle in navigation. Astronomers and navigators consider that the earth is in the center of a celestial sphere commonly called the heavens. The radius of this sphere — the distance from the earth to the imaginary sphere itself — is infinity. The captain of a ship in midocean takes his sextant and sights on the north star (Polaris).
In the diagram, Figure 1, the circle represents the earth. The line OB makes an angle ? with the equator and this angle is of course the latitude of the point B. A tangent to the circle at B represents the horizon at latitude B. Now our captain sights on the north star P, which is almost directly above the north pole of the earth. If the radius of the celestial sphere were not considered as infinite the line of sight from B would meet the North Star P in the diagram. But because the radius is infinite, BA becomes parallel to OP. It follows then that the angle AB makes with the line OA is equal to the angle phi or the latitude. Our captain can then tell his latitude merely by the angle that Polaris makes with the horizon.
MUCH ADO ABOUT NOTHING
In conversation "nothing" is a word like any other word and as such can easily be confused with "something." The old riddle, "Which would you rather have, complete happiness in life or a ham sandwich?" is a good illustration. The answer is, "A ham sandwich. Nothing is better than complete happiness in life, and a ham sandwich is better than nothing." Here we use nothing as a definite something and, of course, the comparison is ridiculous. In the same way we prove that a cow has nine legs by saying that no cow has five legs and a cow has four legs more than no cow.
The mathematician does not stand for this sort of nonsense. While a layman like Willie's father may be puzzled and annoyed by the following question, the mathematician has a logical and complete answer for it:
"If I add nothing to 5 I don't add anything so I don't change its value. The same holds true when I take nothing away from 5. Now if I multiply 5 by nothing I don't multiply it by anything and, of course, I leave it alone and its value is not changed. Now I know that 5 times nothing is nothing but I'm sure that it should be 5. Why isn't any number times nothing equal to that number for the reason I have just given?"
Willie's father could not answer him satisfactorily. If he had been up in his mathematics he would explain the difference in meaning between the English word "nothing" and the mathematical term "zero."
The simplest way to answer Willie's dilemma is to show that multiplication is really repeated addition. Multiplying 5 by 3 is the same as adding two 5's to the original 5, thus: 5 + 5 + 5. In the same way 5 × 2 is the same as adding one 5 to itself, thus: 5 + 5; 5 times 1 is the same as adding no 5's to the original 5. The number of times we add the number to itself is always one less than the number we multiply by; so, in general, multiplying a number A by another number B is just the same as adding A to itself B 1 times. In multiplying 5 by zero we add 5 to itself 0 - 1 times or, in other words, we add -5 to the original 5 and the result is zero.
Another and more complete mathematical way to explain Willie's problem is this: Instead of calling zero "nothing," let us call it a fraction with some numerator, say 1, whose denominator is larger than any number we can name. The fraction 1 divided by 100 quintillion is so small that for all practical purposes it could be called zero. If we make the denominator still larger such as 100 octillion, the fraction is well on its way toward the vanishing point or zero. Now you can readily see that 5 times this inconceivably small fraction will make no difference at all. It will still be, for all practical purposes, equal to zero. Reasoning in this way you can see that when the denominator of a fraction becomes and remains larger than any number that can be named — becomes infinite, in other words — the value of the fraction will become equal to zero and any number in the numerator, no matter what, cannot keep that fraction from equaling zero, hence zero times any number is always zero.
Of course you must have observed from this that as the denominator of a fraction approaches infinity (written 8) the value of the fraction approaches zero. We can then say that zero equals 1/∞.
Now we can see why any number to the zero power equals 1. Let the number be A. Raising A to the zero power, from what we have just said, is the same as taking the infinite root of A or
[A.sup.0] = [A.sup.1/∞] = [??]
We can show that the infinite root of any number is equal to 1 regardless of the size of the number. Take 64 for example:
The square root = [square root of 64] = 8.00000
The cube root = [cube root of 64] = 4.00000
The 4th root = [??] = 2.8284 ...
The 10th root = [??] = 1.5156 ...
The 100th root = [??] = 1.0425 ...
The 1000th root = [??] = 1.0069 ...
The 10,000th root = [??] = 1.0004 ...
* * *
The infinite root = [??] = 1.00000
That is, [64.sup.0] = 1
You can see that the infinite root or zero power of any positive number will behave in the same way and ultimately be equal to 1.
Zero and infinity are two of the most important concepts in all mathematics. The calculus, or the study of the ratio of two infinitesimals, gives us the instantaneous rate of change of one variable with respect to the other. In doing so it becomes one of the most vital tools in science.
Most people use the word infinitesimal in the wrong sense. You cannot have an infinitesimal quantity of a thing because the definition of the word is a quantity becoming smaller than any fixed number we can think of, no matter how small. Strangely enough, the limiting ratio of two infinitesimals is not always zero divided by zero but, instead, an exact value. If an automobile travels 60 miles in one hour would you say that that car traveled at the rate of 60 miles an hour? Of course not. It might have stopped somewhere along the line and made up time later on. We can only say that its average speed is 60 miles an hour. If now we cut down the distance and the time to a mile a minute (the same ratio) we give it less chance to average and the velocity of the car becomes more exact. We can keep doing this as much as we please. It travels 88 feet per second and here the velocity is much more exact. It follows from this that as the distance and time get smaller and smaller and approach zero as a limit, their ultimate ratio (distance divided by time) is the exact value or instantaneous rate of 60 miles an hour. This instantaneous rate at any given instant is of the utmost importance in higher mathematics and science.
One of the best ways to appreciate zero is to consider permutations and chance. The number of ways three objects can be arranged, for example, is 1 × 2 × 3 which in mathematical terms is called factorial 3 and written 3! This is easily shown for A, B and C as follows:
ABC BAC CAB
ACB BCA CBA
If now we take on the letter D, we see that it can be placed first, second, third or fourth in each of the above six combinations. We then have 4 × 6 or 24 combinations. You can verify this for yourself by setting up four columns of six combinations each, the first column beginning with A, the second with B, the third with C and the fourth with D. So factorial 4 (4!) is 1 × 2 × 3 × 4, or 24.
In the same way the number of ways eight different objects can be arranged is factorial 8 (8!), or 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8, or 40,320.
Now try to imagine the number of arrangements that can be made in a single pack of cards, using all the cards for each arrangement. This, of course, is 52! which turns out to be 1.06 × [10.sup.68]. This means that the chance of each bridge player getting 13 cards of the same suit in order of suits and rank from a shuffled pack in a single deal is 1 in 1.06 ×[10.sup.68]. We can safely say that, for all practical purposes this is no chance at all or, in other words, that the chance is zero. If everybody in the world, men, women and children, played bridge night and day, dealing hands at the rate of 1 a minute, this particular situation would come up once in [10.sup.48] years. Since the entire solar system is not yet 4 × [10.sup.9] years old, the chance of each bridge player getting 13 cards of the same suit in proper order and suits can be safely put as "never." But you must remember that the reasoning applies to any specific collection of hands whatever, provided each hand is specified in detail as to the order in which the cards are dealt.
(Continues…)
Excerpted from "Puzzles, Games, and Tricks"
by .
Copyright © 2019 Jerome Meyer.
Excerpted by permission of Skyhorse Publishing.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
Preface,
The World of Numbers,
Exploring both ends of our number system,
Number giants — how much is a billion?,
Infinity,
Much ado about nothing,
Mathematical midgets,
More and more of less and less,
NUMBER SYSTEMS OTHER THAN OURS,
How did the Romans multiply and divide?,
Babe Ruth hit 111100 home runs in one season,
The magic of numbers by Royal V. Heath,
OTHER NUMBER FACTS AND CURIOSITIES,
Casting out nines,
Obvious but interesting,
Interesting Identities,
Magic numbers,
Multiplying without repeating digits,
Reversed-digit playground,
Rare prime decades,
Magic multiplication square,
FIBONACCI NUMBERS,
POWERS,
The sum of the consecutive odd numbers is a square,
Square roots by the Pythagorean theorem,
Triangular numbers,
Odd squares also form a triangle,
Cubes, squares and triangular numbers,
π, i, e AND LOGARITHMS,
π,
i,
e and logarithms,
The sine and cosine tables,
THE WONDERS OF THE NOMOGRAPH,
MAKING A GOOD SLIDE RULE WITH AN ORDINARY RULER,
How to use your slide rule,
BIRD'S-EYE VIEW OF PLANE TRIGONOMETRY,
CURVES THAT CONTROL OUR LIVES,
The circle,
The ellipse,
The parabola,
The hyperbola,
MATHEMATICAL HOW-TO-DO'S,
Making a date indicator,
MATHEMATICAL FALLACIES AND PROBLEMS,
The fallacies explained,
THE CIRCLE IS SQUARED AND THE ANGLE TRISECTED — OR ARE THEY?,
INTERESTING PROBLEMS THAT REQUIRE SOME THOUGHT,
Answers,
Table of Logarithms,