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Math Games for Middle School

Challenges and Skill-Builders for Students at Every Level

Chicago Review Press Incorporated

The Invention of Numbers

WHAT HAPPENED 20,000 YEARS AGO

Astronomers, the scientists who study the sky, tell us that our universe, with all its billions and billions of stars, was born in a tremendous explosion that they call the Big Bang, about 15 billion years ago (that's 15 followed by nine zeros!). They also tell us that the sun and our earth were born about 5 billion years ago. Recently, anthropologists — the scientists who study humankind — discovered evidence that we human beings appeared for the first time on the surface of the earth about 4 million years ago.

As you can see, it all started a long, long time ago. Yet math was invented only about 20,000 years ago, the day some genius among our ancestors realized that three stars, three stones, three trees, and three children had something in common: the property of "three-ness"! To record this great discovery, this brightest of people held up three fingers or cut three notches on a tree branch with a sharp stone.

Although we were not there to see it, I am quite sure that our bright ancestor got so excited at this discovery that he or she began counting everything, just as you did when you first discovered that you had one, then two, then three, four, and five fingers on each hand.

Actually, our language gives us fairly good evidence that our ancestors also counted on their fingers. The numbers from 0 to 9 are sometimes called "digits" in English, and the word "digit" comes from "digitus," the Latin word for finger. Whether numbers were "discovered" or "invented" is harder to say. I believe they were invented by people, as was the rest of mathematics, starting about 20,000 years ago.

ABSTRACTION

The idea of numbers is an abstract idea, meaning a thought in our minds about something that may not even exist. For example, we can think of a green cow with five legs, although real cows have only four legs and are not green. As people kept inventing more and more mathematical ideas, math kept becoming more and more abstract and, at the same time, more and more useful. The trouble is that many of us are not very good at abstract thinking. We find the abstractions of math difficult if not impossible to understand, and we end up hating mathematics.

If any of my readers are among the "abstract math haters," I have good news for them. The idea of numbers may be abstract, but numbers are extremely useful because they represent very real things. We use mathematics so often, every day of our lives, that we could not live without it. Just think of this: could you buy anything at a store if you could not use numbers to count?

Of course, as you already know, math goes beyond mere counting. You are living at a time when science influences every aspect of our lives. We could not fly without the mathematical science of aerodynamics needed to build airplanes; your doctor could not treat you when you are sick without the mathematics of chemistry needed to make pills; nor could you talk with your friends over the phone or look at television without the mathematics of electronics. In short, whether we like it or not, you and I must use math all the time. And we live better than our early ancestors because more and more math is being invented every day to make our lives healthier, easier, and more pleasant.

Let us now find out how mathematics was invented, beginning with counting. I bet you that by the time you finish this book, you will agree with me about what mathematics really is — a fascinating game that anybody can enjoy playing as well as a needed game that we all must play for our own good.

COUNTING

Our early ancestors may have held up the fingers of their hands or cut notches like / / / on tree branches to indicate how many cows they had in the corral or how many apples they had picked that day. But they must have soon realized that no tree branch could be long enough to count a very large number of cows or apples. They eventually invented names for groups of notches and, since we don't know what language they spoke, we might as well imagine that they spoke English and said "one" for /, "two" for / /, "three" for / / /, or "nine" for /////////, and so on. Thus did special words become useful substitutes for notches.

INTERNATIONAL COUNTING SYMBOLS

Our ancestors had an easy time using their special words to communicate with the members of their own tribe, but they soon realized that these words were not understood by members of tribes with which they wished to exchange goods, because they spoke a different language. We would have the same problem today. How could we sell 10 cars to a Spanish car dealer by using the word "ten" when 10 in Spanish is "diez"?

This problem was solved in due time by substituting numerals, more easily recognized shorthand symbols for the special words.

The word "shorthand" comes from a kind of special writing where symbols substitute for letters, words, and phrases, allowing people to write so fast that they can put a speech on paper while it is being delivered. For example, all the speeches of our congressmen and congresswomen in the House of Representatives are taken down in shorthand. To make the writing easier, typewriters are made that have shorthand symbols instead of letters.

Math is another kind of shorthand language that has grown easier and easier to use by becoming shorter and shorter. In math shorthand, at least in Europe, "one" became 1, "two" became 2, "three" became 3, "nine" became 9, and so on, in imitation of the numerals that were brought to Europe from India by the Arabs, many centuries ago. Nowadays the numerals 1, 2, 3, 4, and so on are called the Hindu-Arabic numerals (or simply the Arabic numerals), as a way of recognizing the importance of this contribution to mathematics.

Unfortunately the problem of communicating with numerals among all the people of the world has not been solved yet, because not all countries have adopted the same symbols. Yet more and more countries are doing so today. For example, although the Japanese language has its own numerals, the Japanese have adopted the Hindu-Arabic numerals for their telephone numbers. Hopefully, these numerals will soon be used internationally.

POSITIONAL SHORTHAND

So far, so good. Our nine numerals from 1 to 9 work very well if we never have to count more than nine objects, but can you imagine how tough it would be to invent and memorize numerals for every number bigger than nine? Actually it would be impossible, because there is no limit to how high you can count. There is no largest number, nor is there a smallest number!

We must be grateful to the Hindus of India for the solution of this difficult counting problem and particularly to the Hindu mathematician Bhaskara, who published the complete solution to the problem in 1150 A.D. It was already known at the time that one more numeral was needed for counting all possible numbers, a numeral the Hindus called "sunya" (meaning "empty"), and we call "zero." (With zero we can make the mathematical statement: "I have zero apples," instead of the verbal statement: "I have no apples.") Bhaskara suggested that all numbers, large or small as they might be, could be represented by the sum of a sequence of numerals chosen among the ten symbols 0 to 9, called digits, by simply positioning them to the left or to the right of a decimal dot. Those to the left of the dot are understood to be multiplied by increasing powers of 10 and those to the right of the dot are understood to be divided by increasing powers of 10, depending on their distance from the decimal dot, according to the following rules: "The first digit to the left of the dot is to be multiplied by 1, the next digit to its left by 10, the next by 10 × 10, the next by 10 × 10 × 10, and so on. The first digit to the right of the decimal dot is to be divided by 10, the next to its right divided by 10 × 10, the next by 10 × 10 × 10, and so on."

For example, the Bhaskara number 234.567 represents:

2 × 10 × 10 + 3 × 10 + 4 × 1 + 5/10 + 6/(10 × 10) + 7/(10 × 10 × 10)

= 2 × 100 + 3 × 10 + 4 × 1 +5/10 + 6/100+ 7/1000

as shown in Table 1.

You may read this as "Two hundreds, three tens, four ones, five tenths, six hundredths, and seven thousandths."

This simple way of representing any number by means of the ten digits 0 to 9 is called "the Hindu-Arabic system of number representation" or simply the decimal system, because it is based on the powers of 10.

You may be interested to know that, while Bhaskara gave the first complete explanation of the Hindu-Arabic representation system in 1150 A.D., the Mayas of Central America had their own number system, which included the number zero, in the fourth century A.D., about 750 years before Bhaskara!

EXPONENTIAL SHORTHAND

As you will learn in Section 2 of Chapter Three, multiplication of the same number, say 10, by itself, say 6 times, may be represented by the exponential shorthand symbol 106 where 6 is called the exponent and 10 is called the base. Any number n (other than zero) with 0 as the exponent represents the digit 1, or n0 = 1. Similarly, any number, say 5, divided 3 times by any other number, say 7, may be represented by the negative exponential; thus 5/(7 × 7 × 7) = 5 × 7-3. By means of these exponential shorthand symbols, the decimal positional representation of the number 234.567 can be written:

2 × 102 + 3 × 101 + 4 × 100 + 5 × 10-1 + 6 × 10-2 + 7 × 10- 3.

Decimal notation is particularly easy to use because powers of 10 are easy to remember. For example:

104 = 10,000; 103 = 1,000; 102 = 100; 101 = 10;

100 = 1; 10-1 = 1/10; 10-2 = 1/100; 10-3 = 1/1,000; 10-4 = 1/10,000; and so on.

Math Camp 1

Since you are going to use numbers every single day of your life, you should become familiar with them and use them with the same facility with which you might hit a baseball or shoot a basketball. Practice with the following games in Math Camp 1. You might have fun playing the games with a friend and competing against one another to see who gets the correct solution first.

1 Write the following numbers in the decimal representation system:

Three hundred fifty;

Four thousand nine hundred twenty-five;

One million, two hundred thousand, three hundred forty-five.

2 Write the following numbers in English:

421.25;

42,375.374;

21,300,750,221.045.

3 Write the following numbers as the sum of their components in powers of 10:

321;

254,321.456;

500,401,302,002.

4 Write the following numbers in their decimal representations:

5 × 103 + 0 × 102 + 3 × 10 + 4;

6 × 109 + 5 × 107 + 2 × 105;

3 × 104 + 2 × 102 + 5.

5 Write the following numbers in their decimal representations:

2 × 102 + 3 × 10 + 4 × 100 + 3 × 10-1 + 5 × 10-2;

4 × 10-1 + 5 × 10-2 + 6 × 10-3;

1 × 104 + 2 × 103 + 4 × 10-1 + 6 × 10-2.

THE BINARY NUMBER SYSTEM

You learned in Section 6 that it is efficient and convenient to express numbers by their location to the left and the right of the decimal dot, with the understanding that their position implies multiplication by the exponential powers of 10. For example, the number 312.54 is the positional decimal representation of:

3 × 102 + 1 × 101 + 2 × 100 + 5 × 10-1 + 4 × 10-2 = 300 + 10 + 2 + 0.5 + 0.04 = 312.54

On the other hand, there are situations where it is more convenient to represent a number with a base other than 10. The binary number system uses the powers of 2 and just two numerals, 1 and 0 (rather than all ten numerals, 0 to 9, of the decimal system). Binary numbers are essential for electronic computers because computers are actually just a bunch of switches that can only be on or off, and these two conditions are conveniently represented by two numerals, 1 (for on) and 0 (for off).

Table 2 gives the binary representations of the decimal numbers from 0 to 10 in the second row.

Let us see an example of binary representation by expressing powers of 2 exponentially and using 1 or 0 to indicate the number of times each power of 2 must be taken. Table 3 has all the information needed to do this.

We can now interpret the binary number 101001.11001 (bottom row of Table 3) as

1 × 25 + 0 × 24 × 1 × 23 + 0 × 22 + 0 × 21 × 1 × 20 + 1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5.

Remembering that any nonzero number to the power 0 means 1, this is

1 × 32 + 0 × 16 + 1 × 8 + 0 × 4 + 0 × 2 + 1 × 1 + 1 × 1/2 + 1 × 1/4 + 0 × 1/8 + 0 × 1/16 + 1 × 1/32,

or, in the decimal representation,

32 + 8 + 1 + 1/2 + 1/4 + 1/32 = 41.78125.

You realize, of course, that the representation of a number in the binary system is usually much longer than that in the decimal system. For example, in Table 2, the single decimal number 9 requires four binary digits. And in our example just above, the decimal number 41.78125 has seven decimal digits but its binary representation, 101001.11001, has eleven binary digits. Nonetheless, since a computer is really just a bunch of switches, each of which is either on or off, it is cheaper to have these switches represent binary numbers (off = 0, on = 1) than to have them represent decimal digits.

Let us now translate a decimal number into a binary number:

21.75 = 2 × 10 + 1 × 1 + 7 × 10-1 + 5 × 10-2

Math Camp 2

1 Translate the following decimals into binary numbers:

21;

248.625;

2,003,481;

0.3125.

2 Translate the following binary numbers into decimal notation and then do the indicated operations (if any):

1010;

11001;

1100 + 011;

1010 + 101 -10;

111 + 1010 -11;

1001 × 110.

Adding, Multiplying, Subtracting, and Dividing Numbers

ADDITION

Let us assume that you bought two pairs of blue socks and three pairs of red socks. To find out how many pairs of socks you bought, you could do what our ancestors did (and children still do): you could count by ones on the fingers of your hands and say:

"(One and one) and (one and one and one) pairs of socks is five pairs of socks."

But mathematicians have invented a shorthand operation for this long way of counting. It is represented by the symbol + (plus) and is called addition. So, we write the long count of our ancestors simply as:

2 + 3 = 5.

Of course, as we talked about in Chapter One, you will really appreciate the advantage of using the Hindu-Arabic or decimal number system when you have to perform the addition of large numbers, or the addition of a large number of numbers.

Notice that addition has the property of being commutative; that is, the sum of adding two or more numbers is independent of the order in which they are added:

7 + 1 + 4 = (7 + 1) + 4 = (1 + 4) + 7 = (4 + 1) + 7 = 12.

Suppose you bought two baskets of apples, one with 29 and the other with 37 apples. How many apples did you buy altogether?

When adding in the decimal representation, it is advisable to add the numbers vertically, spelling out the numbers multiplied by the increasing powers of 10. For example, in adding 29 to 37, write 29 as 20 + 9 and 37 as 30 + 7. Then add the unit numbers 9 + 7 = 16, and write 16 as 10 + 6. Continuing in this manner, the addition 29 + 37 becomes:

29 20 + 9 37 + 30 + 7 __ = _______________________ = ? 50 + 16 = 50 + 10 + 6 60 + 6 = 66

Once you have become familiar with the decimal number representation, addition can be shorthanded by putting the 1 from 10 + 6, which comes from 9 + 7 = 16, above the tens column and simplifying the writing to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following example illustrates this shorthand writing of addition with more and bigger numbers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Math Camp 3

1 Perform the following additions:

479 + 237; 1,245 + 2,377; 10,003,700 + 2,549;
$3.75 + $2.99 + $4.75 + $0.35 + $1.01;
25,432 miles + 32,567 miles + 298,000 miles + 12,354 miles.

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Excerpted from Math Games for Middle School by Mario George Salvadori, Joseph P. Wright. Copyright © 1998 Mario Salvadori and Joseph P. Wright. Excerpted by permission of Chicago Review Press Incorporated.
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