Large Numbers

It is said that our ancient ancestors would characterize the number in a group of objects as one, two, many…, with no real grasp of how different many can be. Infants and small children probably face the same limitation. Adults benefit from being able to count, but we also quickly reach the point where we only understand a very large number in an intellectual way. And the mind boggles at trying to understand fundamental physical constants, such as Avogadro's Number (the number of atoms in the gram molecular weight of an element), cited in the adjoining illustration. We never even knowingly experience a million of the same items, or to visualize what a trillion such items might look like.

So consider this. Consider a table top that measures exactly 1 meter by 1 meter, recalling that a meter, at 39.37 inches, is just slightly larger than a yard. Now suppose that a grid of columns and rows has been formed on this surface, with each inscribed line separated from its neighbor by 1 millimeter, or about 1/25th of an inch. This gives birth to a large number of little boxes, each measuring 1/25th by 1/25th of an inch, a little piece of area that seems crystal clear. Now, what does a million look or feel like? Just look at the table top, and grasp the reality that you see a million, or 106, little boxes. That's what a million looks like.

Now, a million is a large number, a billion, 109, much larger, and a trillion, 1012, much larger yet. While our mind can easily capture the difference between 1 and 10, it seems to lose sensitivity once the numbers get very large. What really is a trillion? Well, if you had a flat piece of land that measured 1 by 1 kilometers (about 5/8 by 5/8 mile), it would contain a trillion of those tiny squares referenced above. That's a lot of little squares, and this example puts in perspective just how large a trillion really is. Taking another step, a cubic kilometer would contain a million trillion, or 1018, of these cubic millimeter boxes, something hard to fathom. What about a billion trillion, or 1021? Well, it turns out that there are about a half billion square kilometers on the surface of our earth, so, from what has been said before, a billion trillion of our little millimeter squares would cover both Earth and Venus, which are of similar size. And a million trillion trillion, or 1030, turns out to be the number of little cubic millimeter boxes that would entirely fill our earth, since its volume is roughly a trillion cubic kilometers.

A very large unit of measure is the light-year, namely the distance that light would travel in one year at its speed of 186,000 miles per second. It turns out that this distance is about equal to 10 trillion trillion (1025 ) meters, or 1028 millimeters.

Let's consider another example. As a very rough estimate, suppose that California's coast, about 2000 km, consists entirely of sandy beaches whose average depth is 2 meters, and average width is 25 meters. The total volume of sand is then computed to be 0.10 cubic kilometer. Now suppose that a typical grain of sand is sized at about 1/5th of a millimeter, i.e. if we string 5 such grains together they comprise a millimeter in length. Thus a cubic millimeter contains, ideally, 125 grains of sand, but let's round this off to 100, or 102, grains, accounting for the fact that irregular shaped grains do not perfectly mesh, allowing for some tiny air pockets. Now how many grains of sand populate California's beaches? Stringing together the above thoughts, we get

N = (102)(0.1)(1018) = 1019 = 10 million trillion grains of sand

This is a huge number, but is easily digested by the universe in which we live. Astronomers tell us that there are roughly 200 billion stars in the typical galaxy, and that there are also an estimated 200 billion galaxies. That amounts to 4x(10)22 stars in our universe, of which our sun is just one rather average member. Thus

The number of stars in the universe exceeds the number of grains of sand on a beach that is 1000 times the size of that in Santa Monica, California.

Consider a somewhat less obvious gigantic number. We have all seen the long string of letters and numbers that make up the tracking code of a UPS package, and you may have wondered if all those digits were really necessary. Let's make up our own tracking code, a 15 symbol sequence made of the ten numerals and the 26 letters of the alphabet, both upper and lower case. Thus we have 62 symbols at our disposal, and a typical tracking number sequence would be D25rp8U9y7Ee48g. Until read for the first time here, or else copied from here, the likelihood is almost zero that this particular sequence was ever viewed by any human, past or present, or will be ever viewed again, except here. For, assuming symbol selections are uncorrelated from one another, the number of possible tracking codes is

Nt = (62)15 = 7.689 x 1026

This is an incredible number, much larger even than the grains of beach sand in California. The likelihood of generating any specific sequence is the incredibly small reciprocal of Nt, namely 1/Nt.

The earth's population is now well over six billion, but its average population since 0 AD is closer to 500 million. If everyone since that time had owned a laptop computer that continually generated a million of the above described 15 symbol random sequences each second, the number of such sequences generated since 0 AD would be

Nc = (500x106 people)(2000years)(365 days/yr)(24 hrs/day)(3600 sec/hr)(106 calc/sec) = 3.154x1025 sequence calculations

The likelihood that the specific sequence noted above would have been generated at some time by some computer is then given by Poisson statistics as

P = 1. - exp( -Nc/Nt ) = 0.042

Thus there is but a 4.2% chance that our desired sequence would have popped out anywhere, not a good bet. A lesson to be learned is that a very large number of samples does not necessarily guarantee a sought for success.

Regarding our own earth, a great many of its physical parameters had to meet very tight tolerances in order for it to support life as we know it. Now there are even more planets (and moons) than stars in our universe, but the likelihood that any arbitrary planet would be similar to our earth is vanishingly small. This likelihood is very difficult to quantify, so, at least for now

The existence, in the universe, of other intelligent life as we know it, while seemingly likely, is still an open question.

For an interesting perspective on natures's number scaling, visit Powers of Ten