By George Gamow
The writing of large numbers may seem too trivial a matter to which to devote several pages, but in the time of Archimedes, the finding of a way to write big numbers was a great discovery and an important step forward in the science of mathematics.
Very large numbers often pop up in what may seem at first sight a very simple problem, in which you would never expect to find any number larger than a few thousands.
One victim of overwhelming numbers was King Shirham of India,who,according to an old legend,wanted to reward his grand vizier Sissa Ben Dahir for inventing and presenting to him the game of chess. The desire of the clever vizier seemed very modest. ‘Majesty,' he said kneeling in front of the king, ‘ give me a grain of wheat to put on the first square of this chessboard, and two grains to put on the second square, and four grains to put on the third, and eight grains to put on the fourth-And so, oh king, doubling the number for each succeeding square, give me enough grains to cover all 64 squares of the board.’
‘You do not ask for much, oh my faithful servant, 'exclaimed the king, silently enjoying the thought that his liberal proposal of a gift to the inventor of the miraculous game would not cost him much of his treasure.‘ Your wish will certainly be granted.' And he ordered a bag of wheat to be brought to the throne.
But when the counting began, with l grain for the first square,2 for the second, 4 for the third and so forth, the bag was emptied before the twentieth square was accounted for. More bags of wheat were brought before the king but the number of grains needed for each succeeding square increased so rapidly that it soon became clear that with all the crop of India the king could not fulfil his promise to Sissa Ben. To do so would have required l8,446,744,073,709,551,6l5 grains !
Assuming that a bushel of wheat contains about 5 million grains, one would need some 4 * 1012 bushels to satisfy the demand of Sissa Ben. Since the world production of wheat averages about 2409 bushels a year, the amount requested by the grand vizier was that of the world's wheat production for the period of some two thousand years!
Thus King Shirham found himself deep in debt to his vizier and had either to face the incessant flow of the latter's demands, or to cut his head off. We suspect that he chose the latter alternative.
But although such numerical giants as the number of grains of wheat demanded by Sissa Ben are almost unbelievably large, they are still finite and,given enough time, one could write them down to the last decimal.
But there are some really infinite numbers, which are larger than any number we can possibly write,no matter how long we work. Thus the number of all numbers 'is clearly infinite, and so is ‘ the number of all geometrical points on a line’. Is there anything to be said about such numbers except that they are infinite, or is it possible, for example, to compare two different infinities and to see which one is ‘ larger '?
Is there any sense in asking: ‘Is the number of all numbers larger or smaller than the number of all points on a line?' Such questions as this, which at first sight seem fantastic, ere first considered by the famous mathematician George Cantor, who can be truly named the founder of the ‘arithmetics of infinity'.
If we want to speak about larger and smaller infinities we face a problem comparing the numbers that we can neither name nor write down, and are more or less in the position of a Hottentot inspecting his treasure chest and wanting to know whether he has more glass beads or more copper coins in his possession. But, as you will remember, the Hottentot is unable to count beyond three. Then shall he give up all attempts to compare the number of beads and the number of coins because he cannot count them ? Not at all. If he is clever enough he will get his answer by comparing the beads and the coins piece by piece .He will place one bead near one coin , another near another coin, and so on , and so on …If he runs out of beads while there are still some coins, If he runs out of coins with some beads left he knows that he has more beads than coins;and if he comes out even he knows that he has the same number of beads as coins.
Exactly the same method was proposed by Cantor for comparing two infinities: if we can pair the objects of two infinite groups so that each object of one infinite collection pairs with each object of another infinite collection, and no objects in either group are left alone, the two infinities are equal. If, however, such arrangement is impossible and in one of the collections some unpaired objects are left,we say that the infinity of subjects in this collection is larger, or we can say stronger, than the infinity of objects in the other collection.
This is evidently the most reasonable, and as a matter of fact the only possible, rule that one can use to compare infinite quantities, but we must be prepared for some surprises when we actually begin to apply it. Take , for example, the infinity of all even and the infinity of all odd numbers. You feel, of course, intuitively that there are as many even numbers as there are odd, and this is in complete agreement with the above rule, since a one-to-one correspondence of these numbers can be arranged:
2 4 6 8 10 12 14 16 18 20 etc
There is an even number to correspond with each odd number in this table, and vice versa; hence the infinity of even numbers is equal to the infinity of odd numbers. Seems quite simple and natural indeed。
But wait a moment. Which do you think is larger: the number of all numbers, both even and odd, or the number of even numbers only? Of course you would say the number of all numbers is larger because it contains in itself and all even numbers and,in addition,all odd ones.But that is just your impression,and in order to get the exact answer you must use the above rule for comparing two infinities. And if you use it you will find to your surprise that your impression was wrong. In fact here is the table of one-to-one correspondence of all numbers on one side, and even numbers only on the other:
1 2 3 4 5 6 7 8 etc
2 4 6 8 10 12 14 16 etc
According to our rule of comparing infinitis we must say that the infinity of even numbers is exactly as large as the infinity of all numbers. This sounds, of course paradoxical, since even numbers represent only a part of all numbers, but we must remember that we operate here with infinite numbers, and must be prepared to encounter different properties.
In fact,in the world of infinity a part may be equal to the whole!
-----摘自《英语精读文选》安徽科技出版社
