Biggest decimal integer is odd ... or even ... or odd ...

[bensaei]

We can represent any integer by the sum of multiplication of its digits by 10 to the power of its place:
345 = 5 + 40 + 300.
Same goes to binary too:
1101 = 1 + 2*0 + 4*1 + 8*1 or 13
So lets try to produce the largest possible binary number:
1 + 2*1 + 4*1 + 8*1 + ...
essentially this would look like this in binary: 111111...
Note that the only odd number that is being added is the very first lonely 1, everything after that is a multiple of 2 and therefore even. so everything else is being added is even and then we add 1 to it. by definition it becomes odd! we can try the same in decimal or any other even base:
9 + 9*10 + 9*100 + 9*1000... = 9999999...
and it is an odd number.
Interestingly in all odd bases, the largest number is even :/
8 + 8*9 + 8*81 + 8*729 + .... = 88888888...

 

[HallsofIvy]

The difficulty is that there is no such number! There is no largest integer.

 

[Bob Brown MSEE]

We can represent any integer by the sum of multiplication of its digits by 10 to the power of its place:
345 = 5 + 40 + 300.
Same goes to binary too:
1101 = 1 + 2*0 + 4*1 + 8*1 or 13
So lets try to produce the largest possible binary number:
1 + 2*1 + 4*1 + 8*1 + ...
essentially this would look like this in binary: 111111...
Note that the only odd number that is being added is the very first lonely 1, everything after that is a multiple of 2 and therefore even. so everything else is being added is even and then we add 1 to it. by definition it becomes odd! we can try the same in decimal or any other even base:
9 + 9*10 + 9*100 + 9*1000... = 9999999...
and it is an odd number.
Interestingly in all odd bases, the largest number is even :/
8 + 8*9 + 8*81 + 8*729 + .... = 88888888...

• HallsofIvy was correct, there is no largest integer. However if you give up the definition of what "big" means (but keep the arithmetic) you get the base-adic numbers. Something really wonderful happens. Add one to the three integers that you defined and in each case you get 0.
  
  
  ...11111111 base 2 + 1 = ...000000000 base 2
  ...88888888 base 9 + 1 = ...000000000 base 9
  ...99999999 base 10 + 1 = ...000000000 base 10
  
  • • This "largest" number in each case must be minus 1 in that base of base-adic numbers. WOW -- you discovered an entirely new kind of number!
      
  • . PS: For technical reasons base-adic numbers are only defined for bases that are prime. So if you google for more information search "p-adic numbers"

 

[Bob Brown MSEE]

What is ...3333333334 in base 10?

What is ...3333333334 in base 10?

If you multiply by 3 you get ...000000002.
...3333333334 in base 10 must be 2/3 in 10-adic numbers.