[Author: Bryan Dye. Originally published in MicroMath magazine, Autumn 1994]
In some world cultures, the concept of large is very different to our own. Some people have the rudimentary number system 1, 2, 3, many. There is the story of an African tribesman trying to understand the number of people killed in the Second World War. He was stunned when told it was more than 10! Surely, when it comes to truly immense numbers, the number of electrons, atoms and neutrons in the universe is pretty big; perhaps the biggest quantity you could think of? Not at all! That number is triflingly small! Dwarfed by the latent power of the digits 1, 2, 3 and 4.
Not long ago, at my school, a large city comprehensive, I ran a mathematical competition that ran:
This problem was adapted from one originally published in Scientific American a few years ago and discussed in detail in a fascinating book by Clifford A. Pickover ("Computers and the Imagination", Alan Sutton Publishers, 1991). There the conditions were tighter, in that +, * and ÷ were not permitted. Pickover discusses the largeness of large numbers, comparing for example, the number of trials necessary for a monkey to type Shakespeare's Hamlet by random selection of keys with the number of possible chess games and the number of electrons, atoms and neutrons in the universe. In what order of size would you put these numbers? In fact, the order, smallest first, is: the universe number 1079, then the Hamlet number, approximately 1040000, then the Chess number, 10a, where a = 1070.5.
An intriguing point to consider here is that both the Hamlet and the chess numbers are far larger than we can possibly imagine - if, that is, we agree that it is surely impossible for us, at our current stage of evolution, to imagine an entity that is larger than all known matter, let alone the particles that make up our own brains! Remember that the word for one million was coined only in the 13th century and billion in the 17th.
As always, I publicised the puzzle on the Maths notice board, and for the first time in the monthly newsletter that goes home to parents too. About 40 entries were received. I set the problem to my own middle ability Year 9 class. Most ideas were disappointingly unimaginative, though three pupils came up independently with the idea 432 ÷ .1 = 4320. Once the others had caught wind of the fact that someone had got over 4000 - a huge number! - there was a scramble to copy the idea and by the end of the lesson most had assumed they had caught me out and would each now win the prize. No one mentioned that you could beat this just by putting the digits in the order 4321, and I didn't have the heart to disillusion them. Actually, there were many entries from other pupils in the school that were smaller. The smallest was 3 x 421 = 1263. There were also a few of the kind 4 ÷ (1+2-3) = infinity, "the largest number there is", as one entrant put it. These were expressly forbidden in Pickover's competition, since he stressed that the answer must be finite. I decided also to disallow these entries, even though the rules did not bar them. My argument was that the result of dividing by zero is not infinity at all but simply undefined, although I knew that many pupils would not buy that! The top 10 entries in my competition are listed below, along with their values, as generated by the spreadsheet Excel.
The entrants seemed to fall into three categories. First there were those who struggled to succeed using addition and multiplication as their only means of getting at largeness. 4321 was the largest number produced by this group. Then there were those who discovered powers and assumed after their first attempt that they'd won the competition. A factor here may have been that calculators cannot display Entry 7 or higher, and some pupils may have interpreted the E displayed as meaning simply error, rather than calculator overload. The largest number my graphic calculator can cope with is 9.999999999 x 1099. Finally there were those who thought of powers early on but were imaginative in their use, particularly with the decimal point, and who also had some facility with logarithms. There were only two pupils in this group both of whom were highly able students, one, in Year 13, shortly off to Cambridge to study mathematics, and possibly the brightest student I have taught. It was interesting that these entries fitted the same restricted conditions as Pickover's original problem. According to Excel, these numbers also fall into three categories: those small enough to calculate; those indicated by #NUM! which lie outside the range that Excel can display; and those indicated by #DIV/0!, that are larger again but which Excel fails to cope with due to its mistaken estimation that part way through the calculation it is attempting to divide by zero.
I wondered what the largest number was that Excel could display, and after a few minutes' experimentation came up with 1.79769313486231 x 10308 (much larger than the Universe number). This was achieved by repeatedly increasing the current largest number by a factor of 1.1 until Excel returned a #NUM!, and then using the factor 1.01, then 1.001, and so on. What is the significance of this Excel number (so called)?
Pinky, a contributor to MathsNet, tried 3^(41^2) on a UCALC.EXE PC calculator and got:
We are clear now about the sizes of entries 5 to 10 and we know the others are bigger. At this point I turned from Excel to Derive to see what it could make of all this. It actually makes a better job of coping with the larger ones, since it either comes up with an answer or simply returns its best effort. It evaluated Entry 2 in my competition directly but the best it could do with Entry 1 was 3.333335.01237 x 106989. I decided to convert this and all the other top numbers into powers of 10, which at least gives you the number of digits in the answer. Quite what the largest number which Derive can cope with is I have failed to discover. Eventually, I obtained the combined listing below (the positions in Pickover's competition, which were from an entry of about fifty culled from throughout the USA, are shown in brackets):
|Number||Value||Number of digits|
|1 (1)||.3-(.2-(.1-4))||10a, a=2.62086 x 106989||106990|
|3||.1-(.3-(.2-4))||10a, a=6.29819 x 10326||10326|
|4||Chess number||10a, a=1070.5||1070.5|
|5 (2)||.1-(342)||10a, a=1.09418 x 1020||1020|
|6 (3)||2341||10a, a=1.09793 x 1019||1019|
|7 (4)||3421||10a, a=2.09840 x 1012||1012|
|10||Excel number||1.79769 x 10308||309|
|11 (6)||3421||7.37986 x 10200||201|
|12||4321||1.82498 x 10193||194|
|13||2431||5.54534 x 10129||130|
|14||Graphic calculator number||9.999999999 x 1099||100|
|16 (7)||24(3+1)||1.1 x 1077||78|
|17 (8)||3142||4.33702 x 1062||63|
|18||3241||5.14220 x 1061||62|
|19||2143||7.16852 x 1056||57|
Thus the number of "elemental bits" in the universe is a relatively minute fragment in comparison to the Excel or Chess numbers, and all but disappears entirely when set against what 1, 2, 3 and 4 can do.