Numbers - How Big is Big?

The Natural Numbers form the set of numbers beginning with 1 and sequentially adding 1 to go 1,2,3,... ad infinitum. Although infinite, the Natural Numbers do not display an infinity as "big" as that of the real numbers; however, they are certainly big enough to permit the counting of very large quantities: to wit, the distances in miles between stars, and the number of sand grains on a given beach. Yet as big as the previous quantities might be, we can think of still larger quantities. So how big is big?

I guess it all depends on how big an imagination you have. For example, let us take the speed of light. Light moves at 186,282 miles per second in a vacuum, which is outer space. That's a pretty big number. But how far does light travel in a year? If you take the number of days in a year times the number of hours in a day times the number of seconds in an hour, you get your answer, which is 5.8 trillion miles. This quantity is better known as a light year. A bigger quantity related to light, and lesser known by people, is the parsec, which is an abbreviation for parallax of one arc second.

The parsec is equal to about 3.3 light years or about 19 trillion miles. The distances between galaxies and to galaxies from earth are usually measured in parsecs. For example, the distance from the earth to the Triangulum Galaxy and the Bode's Galaxy are respectively, 900,000 parsecs and 3.6 million parsecs! Since each parsec is about 19 trillion miles, or 19 x 10^12 in exponential notation, we have 900,000 x 19 x 10^12 or 17 x 10^18 miles for the Triangulum Galaxy, and 3.6 x 10^6 x 19 x 10^12 or 68.4 x 10^18 miles for the Bode's Galaxy.

What kinds of numbers are those? Well, most of you know that after millions (10^6) comes billions (10^9), and most even know that after billions comes trillions (10^12). But what after that? The numbers do have names and we use the Latin prefixes to construct them. Thus after trillions we have quadrillions (10^15); then quintillions (10^18); sextillions (10^21); septillions (10^24); octillions (10^27) and even nonillions (10^30). Thus the Triangulum Galaxy is 17 quintillion miles from earth and the Bode's Galaxy is 68.4 quintillion miles from our mother planet.

The number quintillion was made famous by that curious puzzle the Rubik's Cube, which took the country by storm in the 1980's. You see the number of possible combinations of the 3x3 Rubik's cube is approximately 43 quintillion. That number lies somewhere between the distance from the earth to both the Triangulum and Bode's Galaxies. And yet as big as these numbers are, they are just a short walk along the elements in the set of Natural Numbers, as this set never ends.

To get another perspective, imagine the set of Natural Numbers laid out on a line, going out toward space forever. Since many claim that space is infinite, this seems like a reasonable possibility. Let a light beam travel along this so-called number line. If we space the numbers an inch apart, then, using the speed of light and the conversion from miles to inches, in one second the light beam would land exactly on the 11,802,827,520th natural number. After one hour, the light beam would land on the 42,490,179,072,000th natural number. After just one day, the light beam will have surpassed the one quadrillionth natural number. How far out in the set will the light beam be after a thousand years? A million years?

And yet as far out as the light beam will be after let us say one million years, in some far distant galaxy of natural numbers, that number will only be the beginning of another thousand or million year journey by the very same light beam. So how big is big? I guess it all depends on your perspective and on the imagination of your mind.