Size

SCIENCE

by WARREN WEAVER

THE very big, the exceedingly tiny: there is a queer and almost universal appeal in each. Largeness is nearly always impressive, sometimes involves importance, often results in the beauty of grandeur, and in some instances evokes awe or even tear. Smallness can be mean and puny, but much more generally it is attractive. Kittens, puppies, rolls, miniature models — all these are pleasant and interesting.

To be precise about size, one must measure, and then state the numerical result. And to describe the flagrantly outsized, one has to deal with wry large and very small numbers.

To write such numbers easily, and to be able easily to multiply and divide them, we have to make use of one of the many convenient tricks of elementary algebra. We all remember (or should.) from our high school mathematics that 10² means the product of two factors, each equal to ten; that is, 10² is a short way of writing 10 times 10, or 100. Similarly 10³ means the product of three Lens. Every time you multiply by ten you add one zero to the product. Thus 10³ — 1000, and 106 = 1,000,000.

Once you get the basic idea that the exponent in the shorthand form stands for the number of zeros in the written-out form, the rest is easy. When you multiply anything by 104, the factor 10 4 adds four more zeros. Thus the product of 107 by 10 4 must have four more zeros than does 107. The answer is 107 + 4 or 10 11.

We have discovered an important but simple rule: To obtain the product of two powers of ten, just, add the exponents; 102 times 105 equals 107, and 1013 times 10 4 equals 1017.

The rule is true for any exponents, even though they be negative. Take, for example, 1018 times 10-4; the product exponent 14 is obtained by adding the positive number 18 and the negative number — 4.

A number with a negative exponent merely means the reciprocal of that number; thus 10-3 means one divided by 10³. Just as a positive exponent means “multiply,” a negative exponent means “divide.” So 10-6 means one divided by ten six times, or one one-millionth, and 10-9 means one onebillionth. One little point more and we are through with the algebra lesson: dividing by the reciprocal of a number gives the same result as multiplying by the number. Thus, 500 divided by 1/5 is 2500, just as 500 multiplied by 5 is 2500.

We are now almost ready to display some of the numerical records for sizes. But first we must have a sort of gentlemen’s agreement on one point. For the number which measures anything depends upon the size of the unit used. A distance of one mile is measured by the number 5280 if we choose to use feet as our linear unit, or by 63,360 if we prefer inches. The smaller the unit used, in other words, the larger is the number that measures any given physical quantity. Thus any contest to produce a big number could apparently be won by the man who just chooses a small enough unit of measure.

We will avoid this difficulty by agreeing to measure everything in the smallest possible unit that has any direct physical meaning, but not in any still smaller artificial units. As we go along we shall find this convention reasonably workable, even though vague.

Although science deals with myriads of composite or derived quantities, it deals with very few basic quantities. Of these basic quantities we shall speak of three — length, time, and number of things.

What is the biggest length, or more specifically, what is the largest number that can be associated with a length? The greatest distance that seems to have any physical meaning is the estimate which relativity theory assigns to the diameter of the universe. This is something like 10 29 when measured in centimeters. But according to our gentlemen’s agreement, we shall use for our unit of measure not a centimeter, but the smallest distance which seems to have reasonably direct physical meaning. This smallest length is the diameter of ' the smallest known thing in the universe: a proton the name for the nucleus of a hydrogen atom. Its diameter in centimeters is 10-13 This our

largest distance number is 10 29 divided by 10-13 or, what is the same thing, 10 29 times 10 13, which equals, by the rule of adding exponents, 10 42. If written out in full, with all its forty-two zeros, in the body type used in this magazine, this number would be about three inches wide.

How about lime? The longest, time interval to have any meaning would seem to be the age of the universe. The estimates of this vague and romantic quantity are of course highly variable. The age of our earth seems rather firmly set at about 3350 million years, or about 10 17 seconds.

The relativists calculate a similar figure for that remote moment which they would refer to as the beginning of the expanding universe, by which they mean the time when the explosive expansion of matter ceased and a more sedate natural expansion began. As to the original start of the explosive expansion, Eddington, the star performer in this speculative field, once ventured to guess that the total age of the universe is probably less than 2000 million million years, or some 10²² seconds.

But in what smallest unit shall we measure the age of the universe? A second is surely not small enough, for even horse races are timed in smaller units than that.

Time intervals as short as one thirty-millionth of a second have been measured in the laboratory, but since we can afford to be generous we will take for our unit of time the interval of a single wave (corresponding to the tick lock of a clocks pendulum) of the highest frequency phenomenon known to science— namely, cosmic rays.

Here again the situation is not very clear. There are surely cosmic rays whose energies correspond to frequencies of 10 25 per second; and there is recent evidence that the energies are so great as to correspond to frequencies of about 10 30 per second. Adopting, for good measure, this latter figure, the biggest sensible time number is found by measuring the age of the universe in units which are only 10-30 seconds long. Thus this greatest time number comes out about 10 47 if we take the ,-mc of the earth or the expanding-universe figure of the relativists, or 10 52 if we choose Eddington’s maximum figure. In either case the time number is, we see. substantially bigger than the greatest distance number we have chosen to consider.

How about numbers of things? There are over two billion people — or 2 t imes 10 9 — on earth, but that is a very mild fact indeed; for there arc roughly a million times as many cells in the body of a single man as there are people on earth. There are billions, of billions of stars (of which, incidentally, the average human eye can individually see fewer than 3000 at any one time). In fact, there are about 1011 stars in each galaxy, such as our own sun-planet-Milky Way system, and probably about an equal number, 10¹¹, of galaxies in the universe, making something like 10²² stars in all. But these stars are in turn composed of the elementary particles of the physicist, such as protons and electrons. The biggest number of things must be the total number of elementary particles in the universe.

It is an almost incredible feat of science that this number has been estimated. Again it was Eddington who was responsible for the reasoning. The details of ihe argument are by no means clear, nor are they universally accepted by physicists and cosmologists; but his conclusion was that the total number of protons and electrons in the universe is about 1080.

For the three entities so far examined — lengths, times, and numbers of things — we have estimated the maximum numbers 1042, 1052, and 1080. Are these three the largest numbers to which any sensible or direct meaning can be assigned?

The answer is a resounding negative. For example, modern technology has produced a much larger meaningful number. In telephoning from San Francisco to London the original energy of the human voice must be amplified time after time in the process of being transmitted to the listening ear in London. The over-all amplification factor of this process is about 10256. This number, if written out here with all its 256 zeros, would take about six lines. If there were as many universes as there are elementary particles in the one universe, these many universes would still contain a total number of elementary particles fantastically small compared with this number 10256.

And this is by no means the present record. For the logical processes of the mind have found definite use for numbers larger than this by far. It is perhaps not surprising that it. is the mathematicians who have made the extremest excursions. (Mathematicians make use, of course, of a symbol for “infinity”; but this is really not a number, but rather the name of the process of getting indefinitely larger. They have also invented transfinite numbers. We are concerned here only with numbers which can be directly associated with some describable finite process. Also, of course, we rule out the artificiality of breaking the record for the largest meaningful number by just adding, say, a million to produce a new number whose meaning then is that it is a million larger than the previous tit leholder.)

We do not stop to pay our respects to the number 10400, which a mathematician has very recently found to be involved in the proof of a theorem concerning prime numbers, but pass on directly to what appears at the moment to be the all-time record holder. In a way too complicated to concern us here, an English mathematician named Skewes, again in connection with a theorem about prime numbers, found specific use for what we will call the Skewes number, namely: ‘—

34

10

10

10

This symbol stands for the utterly fantastic number which, if written out in full, would have

34 10 1 0,000,000,000,000,000,000,(XX),000,OCX),000,000 10 = 10

zeros. Mind you, this is not the Skewes number; it only expresses the number of integers it would take to write out the Skewes number! What would be the task of writing out, in full, the Skewes number? Let us think of using tiny type in which we could print, say, a million microscopic integers per inch. Suppose we could, in some unimaginable way, print lines of integers which would be so long that they would stretch across the diameter of the whole universe. Suppose that, from the very beginning of the universe, 2000 million million years ago, a staff of two billion persons — all there noware on earth — had spent their entire time, night and day, printing out such lines of integers across the universe, and just to add fantasy to fantasy, suppose every person had printed at the rate of a million such lines every second of all this time. Through all that totally unimaginable effort only about 1072 integers could be printed. But 1072 integers, even though it is admittedly a few, is a wholly negligible number compared with the handsome number which stretches across the page a few lines above. Our crazy process would, in fact, not succeed in making any effective start whatsoever in printing out the Skewes number in full.

Thus the human mind, in the exploration of size, has penetrated far beyond the numerical limits, as we now can envisage them, of space, or time, or matter. Have we reached an ultimate meaningful number? Surely we have not. At the moment the Skewes number seems to be out in front, and by so comfortable a margin that it can afford to relax.