How Quickly Large Our Numbers Grow<\/b><\/p>\n
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.<\/p>\n
The question was \u201cHow many bits are needed to store 1,024 distinct values?\u201d and the expected response was \u201cTen.\u201d However, this student wrote \u201c8192.\u201d Clearly, 28192<\/sup> 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.<\/p>\n 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,<\/p>\n 1 grain of sand = 3 \u00d7 10^(-6) grams<\/strong><\/p>\n and if my calculator is correct, 28192<\/sup> = 1.0907 \u00d7 102466<\/sup>; that is,<\/p>\n 2^(8192) = 1.090748135619415929462984244733782862448264161… \u00d7 10^(2466)<\/strong><\/p>\n so if we had 28192<\/sup> grains of sand, that would be around<\/p>\n 3.272244406858247788388952734201348587344792483… \u00d7 10^2460 grams<\/strong><\/p>\n and converting that to kilograms gives us it would be<\/p>\n 3.272244406858247788388952734201348587344792483… \u00d7 10^410 kilograms<\/strong><\/p>\n Hmm. 3.27 \u00d7 10410<\/sup> 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<\/p>\n mass of the sun = 1.98892 \u00d7 10^30 kilograms<\/strong><\/p>\n we see that<\/p>\n 2^8192 grains of sand = 1.6452368153863643527084813538007303397546369300… \u00d7 10^380 suns<\/strong><\/p>\n Let’s state that again: 28192<\/sup> grains of sand would be about the size of 10380<\/sup> suns! Not, \u201cmore than the mass of the entire solar system,\u201d but much, much<\/em> more than the entire solar system.<\/p>\n <\/p>\n 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.<\/p>\n If 28192<\/sup> is such a big number it’s hard to comprehend, what about 23,000,000,000<\/sup> or 2380,000,000,000<\/sup>. How big would those numbers be?<\/p>\n And it’s not even a very powerful computer.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 \u201cHow many bits are needed to store 1,024 distinct values?\u201d and the expected … Continue reading From the Archives: How Quickly Large Our Numbers Grow<\/span>