Some numbers are so huge that it's hard to imagine what they would look like. But there is information about these numbers you can work out using simple maths and logic. What could you any of the following?
2 to the power of a trillion
100 ! (1 x 2 x 3 ....... x 99 x 100)
1234567834 x 1234567966
infinity
You can comment about any of these numbers or find your own. Your posting needs to include arguments to justify your claims.
HINTS
INFINITY
The digits going round the conveyor belt on my perpetual motion machine are all the same - the digit would be ____ ?
2 TO THE POWER OF A TRILLION
What digit would be at the end of 2 to the power of a trillion ?
ie., 6 is at the end of 16 and 256.
i'll have a go
2 to the power of a trillion would 1,000,000,000,000,000,000,000,000 because a trillion has 12 zeros..... even if i'm wrong about the zeros i know it'll end with a zero=)
2 to a power of a trillion
The discussion below falls into two parts - when I lose you in the first part move onto the second part.
2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384
32768 65536 131072 262144 524288 1048576 2097152
4194304 8388608 16,777,216 33,554,432 67,108,864 134,217,728
268,435,456 536,870,912 1,073,741,824 = 2 to the power of 30
2,147,483,648 4,294,967,296 = 2 to the power of 32
Obviously 2 to the power of a trillion is going to be an astronomically large number. However, there are at least two things we can say about the particular number.
First, what would the last digit be ?
If you look at the numbers above you'll see the recurring pattern 2 4 8 6 repeated for the last digit. Therefore if the power is divisible by 4 with no remainer then the last digit should be a 6. So is a trillion divided by 4 have a remainer - if not then the last digit should be 6. If it does, the last digit will be
2, 4 or 8.
[What would interest me is working out what the last two digits, last three digits, last four digits etc. would be. Note at 2 to the power of 22 the first two digits start a second cycle, ie., 04 08 16 - but that 02 is missed out. So can we expect several numbers to be missed out when the cycle for 3-digits starts, ie., 002 004 008?]
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Second, how many digits will be in the number ?.
This question is harder but can be estimated. if you look at the squares above you'll see the following pattern in the numbers
3 have 1 digit
3 have 2 digits
3 have 3 digits
4 have 4 digits
3 have 5 digits
3 have 6 digits
up to 10 digits the pattern goes 3-3-3-4-3-3-4-3-3-4.
If we assume that the 3-3-4 pattern continues then all you have to do
is subtract [3+3+3+4] from a trillion [ie., trillion - 13]. Then divide the
resultant number by 10 [which equals 3+3+4]. The answer will give you the number of times [3+3+4] is repeated as you move from 2 to the power of 14
[16,384] to 2 to the power of a trillion.
You then mutliply the answer to your division sum by 3 [because you have 3 digits in 3+3+4 so that each cycle represents an increase of 3 digits in your number] and add 4 [for the 3+3+3+4]. The resultant number is the estimate of the number of digits in two to the power of a trillion.
SO for 2 to the power of 33.
33 - 13 = 20 , 20/ 10 = 2, 2 x 3 + 4 = 10
and 2 to the power of 33 = 8,589,934,592
which has 10 digits.
[SEE THE NEXT POST FOR MY ESTIMATE FOR NU. DIGITS IN
2 to the power of a trillion]
Note that in the absence of mathematical proofs it is only an assumption
that the above procedures will give you accurate information about
2 to the power of a trillion.
how many digits in 2 to the power of a trillion ?
My Oxford Concise Oz Dictionary has a trillion as 10 to the power of 18 -
a million million million
1,000,000,000,000,000,000
which I read once was an estimate of the number of neurons in the human brain !
IN USA a trillion is a million million [10 to the power of 12].
Lets stick to 10 to the power of 18. To estimate the number of digits in 2 to the power of a trillion
[A] subtract 13 [in fact don't bother 'cos it's not going to affect the estimate]
[B] divide a trillion by 10, which equals 10 to the power of 17
100,000,000,000,000,000
[C] multiply [B] by 3, which equals 300,000,000,000,000,000
[D] add 4 to [C] which equals 300,000,000,000,000,004 my estimate of the number of digits in two 2 to the power of a million
Or there are [3 X 10 to the power of 17] + 4 digits in the target number.
conveyor belt
So we've cleared up that infinity is not a number, of sorts (No digits and all of that)
But if we were to take a trip back to the 1970s, when everyone thought the highest number was 45 billion, presumably, the digits on your conveyor belt would presumably be a row of nines
"Aha!" You might say. "but if you add 1 to that it becomes 100000000000000...!"
well then I'll say "Aha! but by then I've multipleid my number by 10 and added 9!"
This cycle would continue as long as that motor kept going. However, when the motor stops, you should probably get out of the room fast, because you have just trounced me in an illogical argument, and I won't be very happy about it.
how do you know ? & what is infinity ?
what people thought in the 1970s ?
I, on the other hand, was there but I must admit I never wondered which number was the largest & if I had, I doubt I'd chosen 45 billion - it doesn't seem particularly special
& of course, my perpetual motor machine producing an endless cycle of 9s is not infinity any more or less than any other such cycle [ie., 1010101 or even 000000000] - hence does it even make sense to think about infinity having a size - to me it's just an implication of counting, ie., you always have one more, you can always keep going [unless of course, you have a closed set]
perhaps this should be in the philosophy section ?
infinite sequence
would not need to be cyclical - or even have a consistent generating rule [or rules] - it only has to be never ending [but what about never beginning - surely in that sense it has an ending ?
& I don't think I'm mixing up eternal & infinite
100 ! [1 x 2 x 3 ......... ]
We now know something about what infinity might 'look like' and what 2 to the power of a trillion does look like so lets explore the above.
To get things going -
1 x 2 x 3 x 4 x 5 = 120
120 x 10 = 1200
gives a hint about one piece of information we could find out about 100 !