can’t brain today … i has a tired -found on Facebook-

I before E except after C disproved by science. -Mariana Eye Institute, Saipan, Northern Mariana Islands-

She’s between a rock and a hard place. -WSHE 103.5 FM Ft. Lauderdale-

How Quickly Large Our Numbers Grow

On a recent test a student who either did not understand the question or the concept gave an answer that has set me to thinking. Some of these thoughts get philosophical pretty soon.

The question was “How many bits are needed to store 1,024 distinct values?” and the expected response was “Ten.” However, this student wrote “8192.” Clearly, 2⁸¹⁹² is a huge number. A very huge number. An astronomically large number, so much so that it is hard to get one’s head around it.

How can we conceive of numbers so large, or how can we relate to them? A quick Internet check tells me a single grain of sand weighs 3 micrograms. That is,

1 grain of sand = 3 × 10^(-6) grams

and if my calculator is correct, 2⁸¹⁹² = 1.0907 × 10²⁴⁶⁶; that is,

2^(8192) = 1.090748135619415929462984244733782862448264161… × 10^(2466)

so if we had 2⁸¹⁹² grains of sand, that would be around

3.272244406858247788388952734201348587344792483… × 10^2460 grams

and converting that to kilograms gives us it would be

3.272244406858247788388952734201348587344792483… × 10^410 kilograms

Hmm. 3.27 × 10⁴¹⁰ kg. I still can’t get that in my mind. Surely, it must be bigger than the sun? Looking up the mass of the sun and finding

mass of the sun = 1.98892 × 10^30 kilograms

we see that

2^8192 grains of sand = 1.6452368153863643527084813538007303397546369300… × 10^380 suns

Let’s state that again: 2⁸¹⁹² grains of sand would be about the size of 10³⁸⁰ suns! Not, “more than the mass of the entire solar system,” but much, much more than the entire solar system.

Even with such a comparison, the immense size of such a number is hard to comprehend. But wait a minute! 8,192 bits is 1,024 bytes or 1 kilobyte. How big is that, in today’s world of computers? The laptop computer I carry around has 3 GB of main memory, and a 380 GB internal hard drive.

If 2⁸¹⁹² is such a big number it’s hard to comprehend, what about 2^{3,000,000,000} or 2^{380,000,000,000}. How big would those numbers be?

And it’s not even a very powerful computer.

The scientific name for an animal that doesn’t either run from or fight its enemies is lunch. -Michael Friedman-

The best pilots fly more than the others; that’s why they’re the best. -Chuck Yeager-

Unfortunately, many people do not consider fun an important item on their daily agenda. For me, that was always a high priority in whatever I was doing. -Chuck Yeager-

The car was invented as a convenient place to sit out traffic jams. -Evan Esar-

If you can walk away from a landing, it’s a good landing. If you use the airplane the next day, it’s an outstanding landing. -Chuck Yeager-

Economics speaks to human greed, while politics speaks to fear. -George Friedman-