Dizzy Arithmetic: When Numbers Talk


MANY people find large numbers difficult to understand and hard to appreciate. Hence they do not follow with ease many of the interesting results which have, during the last quarter of a century, flowed in so large and increasing a stream: results which refer to the size of the universe, to the distance of the stars, or to the minuteness of the atoms and of their constituent electrons.

It is worth an effort to simplify this difficulty or remove this barrier by a few simple examples and explanations, and thus to convey an impression of the relative sizes of the objects which surround us in our world.

Men when they walk take about one hundred and twenty paces in a minute, so that a foot is placed on the ground every half-second. Hence in a workingday of eight hours they would make 120 X 60 X 8 steps or paces and this number is 57,600. Continuing this for twenty days, they will have walked a million paces.

It is easily seen that they must continue for fifty-five years in order to have marched a thousand million paces. This great number is called in America a billion, a term which in England is reserved for a million times a million.

It would thus take, always at the same rate, 55,000 years to achieve the number of paces indicated by an English billion, or by a French trillion.1

There is, however, a simpler and more direct method of appreciation of the magnitude of large numbers. Trees breathe the air by means of little openings, mouths, or stomata on the underside of their leaves, and the number of these small mouths on a single leaf may exceed a million. On the other hand, the leaves of a single large tree may and often do exceed one million in number, so that their joint area would cover a city block measuring 400 feet each way. Thus by thinking of a tree with its million leaves, each with a million mouths, we have a quick road to appreciating a million times a million. This number would be represented by a 1 followed by twelve zeros, ciphers, or noughts (sometimes wrongly called o, the letter preceding p in the alphabet — an ill use due to telephone practice). It is very convenient to denote a million million by 1012, meaning twelve tens all multiplied together, or 1 with twelve noughts following it.

This French nomenclature (billion, trillion) originated in the sixteenth century, and was conserved by the English, while the French changed from the original, followed later by the United States.

But we can readily continue further with the trees, the leaves, and their small mouths, toward the idea of still larger numbers. The population of the world has been estimated at 1500 million souls. There are certainly more than a thousand times as many trees as men. Hence we can arrive at the idea of a million million trees, each with a million leaves, and every leaf with a million mouths! Hence there are at least a million million million million stomata or mouths breathing the same air that we do, extracting the carbon from the carbon dioxide in the atmosphere and emitting the oxygen, whereas men inhale the oxygen and burn it with their food, emitting carbon dioxide.

This beautiful balance of Nature depends upon sunlight, upon chlorophyll, — many chemical miracles taking place in the living plants, — and upon those 1024 stomata, more or less, as the lawyers cautiously insert in their deeds.

In dealing with large figures it is often wise to use what are called round numbers: thus 3,560,481,724 can just as well be written 35 × 108, for it is only the two figures on the left that really interest us, and the other digits or figures are not generally known with sufficient exactness to be quoted. Also 3.5 × 109 is equally good. Thus a man of 65 is more than 2 × 109 seconds old, a terrible number of seconds to be responsible for; and his heart has made about 3 × 108 beats.


Some examples of the quaintness of large numbers and of the unexpectedness of the results obtained may next be quoted.

Lake Ontario is a large lake, about 140 miles long and 40 miles broad, at the widest. Let us take its area at about 3600 square miles, or 1011 square feet. Imagine the whole population of the world placed in the lake, and they would have plenty of room to float or even to swim, for each person would have no less than 60 square feet to himself.

On the other hand, submerge them all beneath the waters of the lake and its surface would rise less than half an inch: a result easily obtained as follows. A cubic foot of water weighs 1000 ounces or 62½ pounds, so a man of 125 pounds occupies about 2 cubic feet; for a man’s body is about as dense as water, otherwise he could not just float with lungs inflated and sink when they are full of water. If then a man occupies 2 cubic feet and has 60 square feet to swim in, on submerging him the water must rise 2/60 of a foot, or ⅖ of an inch.

A more interesting example is, I think, due to Doctor Aston of Trinity College, Cambridge, a Nobel prizeman, famous throughout the world for his work on isotopes. The illustration is intended to convey the idea of the enormous number of molecules contained in a moderate quantity of water, each molecule consisting of one or possibly more groups of H2O. Take a glass of water and empty it in the sea. Wait until winds, tides, evaporation, clouds, rain, and snow have thoroughly remixed that water, originally in the glass, with all the water on the surface of the earth. Now dip your glass into the sea, and you will probably recapture, on the average, about 2000 of the very molecules which were originally in the glass. And the reason for this is clear enough. Scientific men can number the molecules in a tumbler of water, and they can calculate the amount of water in the seas. Using these numbers, it is found that the one is 2000 times the other. The present writer has extended this idea to a quaint ease not without interest.

There is a book written by Sir Arthur Shipley of Cambridge, called Life. He was asked to write this book by the Macmillan Company, publishers, with a view to ‘making undergraduates think.’ With so praiseworthy an object the book needs every encouragement, and yet so many people are busy trying to make the undergraduate think, that he often has to resist the pressure in a spirit of rebellion and self-preservation. The book helps quite ordinary people to think, too. Incidentally it points out that our two cubic feet of body consists more of water than of all other substances put together. Punch, in its review of this book, quoted a quaint passage: that ‘even the Archbishop of Canterbury consists of 56 per cent water.’

The cubic foot of water present in any ancient historic person, at any moment of his adult life, has by this time dispersed. Thus, for example, the 56 per cent water of Julius Caæsar at the moment of his death has long since spread and mixed with the waters upon the earth quite generally, and you, the reader at this minute, have probably within you about a million water molecules which belonged to Cæsar at the moment of his death; for, if a cubic foot equals 25 times the volume of a half-pint tumbler, then the probability increases, not as 25, but as the square of 25, or 625. Hence the 2000 may be raised to 1,250,000. And you, the reader, have now molecules of water which were once the property of Abraham, Napoleon, Alexander, William the Conqueror, George Washington, Judas Iscariot, or any other ancient person whom you please to name. Moreover, the air molecules which you now inhale and exhale were some of them breathed by any one of them, and in past time oxygenated their blood.

I do not know whether it is venturing too far to point out that at any time and at all times we have actually within us a part of the very body and very blood of that sublime and divine Master who nineteen centuries ago brought so dazzling and glorious a light to the rising race of man.

This conception of the use and reuse of materials was, like most things, familiar enough to Shakespeare, for we find in Hamlet:—

HAMLET. To what base uses we may return, Horatio! May not imagination trace the noble dust of Alexander, till he find it stopping a bung-hole?

HORATIO. ‘T were to consider too curiously, to consider so.

HAMLET. NO, faith, not a jot; but to follow him thither with modesty enough and likelihood to lead it; as thus: Alexander died, Alexander was buried, Alexander turneth into dust, the dust is earth, of earth we make loam, and why of that loam whereto he was converted might they not stop a beer-barrel?

In this passage we find no suggestion of transportation or movement of the material from the East to Denmark. But in the case of air and of water we do find, by winds, rain, tides, and currents, that great circulation and motion which ensure thorough and perpetual mixing. The pack is always being shuffled and reshuffled.

The molecules of water in our blood have thus visited — many of them often — all the oceans and continents, ascended as vapor, drifted with the clouds, fallen as tropical rain or arctic snow; they have formed an intimate part of countless living things, animal and vegetable, for a period which may be as great as a thousand million years.2

As a final instance of the consideration of large numbers we may cite a remarkable calculation due to one of the great astronomers — either William or John Herschel, but whether due to father or to son I have not been able to ascertain. The theme is discussed in what might be termed a Lucretian poem of the present Poet Laureate, Doctor Bridges.

Imagine that 6000 years ago there were a man and a woman who had four children only, two boys, two girls; let those also in due course have four children, each pair. Let there be no deaths. Then, allowing 30 years for a generation, and supposing that the above process continues for 200 generations, there would be 2+4+8+16+ —

peoples all alive, where the series must be continued for 200 terms or so. This series is in geometric progression, each term being twice its predecessor, so that the sum is 2 (2200— 1) or about 201 twos all multiplied together. A large number! Would there be standing-room for these people? No—they would be piled in solid heaps upon each other. They would occupy 2202 cubic feet or 1060 cubic feet.3 Hence the mass of people would rise to a height of 1020 feet, far beyond the moon and indeed beyond the sun, for the distance from earth to sun is less than 100 million miles or 5 × 1011 feet!

This example of rapid and impossible increase shows that if there is to be youth and renewal of life, then death is inevitable; and if death is to threaten life, then there must needs be pain.


The question of a quick grasp and clear representation of small and minute numbers and sizes next demands attention. There is a very convenient notation used by scientific men which might with great benefit be adopted by the public. The fraction one millionth (1/1,000,000) is represented by 10-6, where the dash or minus sign before the six indicates that a fraction is contemplated. Thus an American billion is denoted by 109, while one billionth part is represented by 10-9.

For example, the diameter of an atom is of the order of 10-8 centimetres, where a centimetre is a legalized measure of length, and it is convenient to remember that an inch is a little more than 2½ centimetres (one inch = 2.54 centimetres). Now it is believed on most potent evidence that atoms are built up of protons and electrons, their sole known constituents. Thus, according to Rutherford and Bohr, an atom of hydrogen consists of a proton or positively charged electron, around which rotates rapidly (1014 times a second) a single electron or negatively charged particle. The proton is the more massive of the two, indeed 1800 times as heavy; and if mass is due to concentrated electrical energy, then the more massive must be smaller and more condensed. Theoretically it may be deduced that the electron has an effective radius of 10-13 centimetres, while the proton may perhaps have something of the order of 10-16 centimetres as a radius. Here we have arrived at the smallest known entity; information as to the ultimate constitution of electrons and protons is not likely to be known to this generation.

The Caspian Sea, latitude 38° to 48°, is landlocked, with fresh water in the north, very salt water in the south, fed by four rivers, especially by the great River Volga. Russian engineers have computed that these rivers discharge sufficient water to raise the level of the Caspian five and one-half feet in a year, and the rainfall in it would further raise the level at least another foot and a half, or about seven feet altogether. But the level remains almost constant, because the rise is balanced by loss from evaporation due to the sun’s heat.

We may perhaps pause to express amazement at the progress of physics during the present century, during which many of the mysteries of atoms have been unraveled through skillful experiment and acute reasoning by a small handful of men of outstanding genius. A similar tribute is due to biology, where the mystery of the Mendelian law is receiving some explanation in the marvelous properties of the subdivisions of chromosomes.

A word of caution is needed when the size of any object is stated. Take for example the sun, a spherical object of 400,000 miles radius. Is the sun wholly confined to that region? Does it not fling to us corpuscles which cause the aurora? Does it not produce magnetic effects which reach the earth and cause magnetic storms? Does not radiant energy, heat and light, pour forth and give us life upon this planet? But it may be argued that these energy manifestations have left the sun and no longer form a part of its fiery globe. That is perfectly true; nevertheless, when we consider the gravitational field of the sun, that certainly extends to us, and beyond us to Jupiter and Neptune, holding those great planets in their elliptical orbits around the sun by some means which, in spite of Newton and Einstein, still remains in blackest obscurity.

So also when the radius of an electron is quoted as 10-13 centimetres we must bear in mind that a moving electron carries with it an electric field and a magnetic field extending outside the radius, decreasing indeed in intensity as we go farther and farther from the electron, but not terminable anywhere in space. It seems to be true that everything is both where it is and where it is not. Or, if that sentence is repudiated as illogical and unintelligible, then it may be said that the total physical energy of any object, part of which we localize in our imagination and call matter, extends and permeates throughout space. Distant stars attract our sun and are attracted by it. The whole universe is an interwoven meshwork, not a conglomerate of separate and independent units.

It is customary to change the unit of measurement with the size of the object measured. This common usage is not necessary or even desirable, for with a little practice powers of 10 are readily appreciated. Regardless of repetition, the writer again emphasizes the idea that 1024 is not hard to appreciate by one who comprehends a million. The thousand million people of the world might have, each in the world, on an average, a thousand trees, each tree with a million leaves, each leaf with a million mouths or stomata. Hence 1024 stomata, or — written in full — 1,000,000,000,000,000,000,000,000 stomata. Note the convenience of the short notation.

We measure the diameter of a fine wire in mils or thousandths of an inch; the height of a man in feet; the height of a horse in hands; the depth of the sea in fathoms; the wave-lengths of radio in metres or kilometres; the radius of the earth in miles; the distance of the stars in light-years or in parsecs; and the confines of space in astronomical units or the distance from sun to earth.

Now all these might quite appropriately be expressed in centimetres with the correct number of tens affixed. In fact, such usage is common in scientific work, and it is used because it is the quickest in calculation and — what is more important — the clearest for appreciation.


The now wearied reader, if he has followed this disquisition so far, may be relieved at a prospect of the final lap, wherein are set forth some distances, fair samples in ascending grade, from the radius of the proton (10-16 centimetres) to the supposed radius of the space assigned to each one of us, whereof each one of us is consummate lord, and at the centre of which space each one of us sits or stands triumphant, wherever we be in space and whatever time is for us now.

This radius is declared by Doctor Silberstein to be 7 × 1012 astronomical units; and an astronomical unit is 93 million miles or 15 × 1012 centimetres; so that the so-called radius of space is 1026 centimetres, and a line three times as long as that would perhaps go round our space, and, if so, is the longest straight line possible.4

All of which is noted with due reservations, awaiting interesting observations now in progress!


Radius of proton or positive electron (perhaps) l0-16
Radius of an electron 10-13
Radius of an atom (about) 10-8
Wave length of visible light 10-5
Average distance between moleio cules in air 10-5
Height of a man 10 2
Height of Mount Everest 10 6
Radius of earth 10 8
Distance to moon 10 10
Distance to sun 10 13
Distance to nearest star 10 18
Distance to Pole star 10 19
Ten thousand light-years 10 22
‘Radius’ of space-time 10 26

In front of each number, multiplying it, should stand a number between nought and ten, which does not interest us here because we are thinking of the order of sizes, not of the exact sizes.

This article has been written with a purpose, which is probably by this time clear enough to any reader. It is a plea for the general adoption in books, journals, and newspapers of a clear and useful notation which has stood the test of usage in scientific work.

The remarkable interest of a large part of the civilized world in the principle of relativity reveals a growing eagerness to grasp the conditions of our existence and the nature of our universe. There are other regions of modern science equally enthralling, but the language and symbols and units of science have drifted far afield from those used by the general public. The difficulty of bridging this gulf is great. A greater uniformity of numerical notation would remove one obstacle. A wise choice of units would be another aid; but this last great subject is too large and inflammatory for discussion at this time and place.

  1. American English French
  2. 106 million million million
  3. 109 billion milliard or billion
  4. 1012 trillion billion trillion
  5. 1015 quadrillion
  6. 1018 trillion
  7. 1024 quadrillion
  8. Two acute-minded friends who read this paper quickly put a critical question: How long does it take on the average for a molecule of water to circulate from the ocean to the clouds and back again? A precise answer would be difficult, but a suggestive guess can be given.
  9. Now the average depth of the ocean is 13,000 feet, so that in middle latitudes it would take nearly two thousand years to evaporate the sea away, if there were no rain or rivers to feed it. Near the equator much less time would be required, and in arctic regions much more. In the cool depths of the ocean there is little movement of water due to currents and change of temperature, and diffusion must be extremely slow. Such water takes little part in the circulation we are now considering.
  10. Over the land, where there is vegetation, a large proportion of the rainfall is promptly evaporated. Taking all these things into consideration, it might be estimated that a small particle or molecule of water circulates from ocean to cloud and from cloud back to the ocean in an average time of something like a thousand years.
  11. Let 10z = 2202
  12. then x = 202 log 2
  13. so x =60
  14. Curiously enough, if R is the radius of space, its circumference is stated to be πR and not 2πR,