Question

https://www.math.hmc.edu/funfacts/ffiles/10001.5-8.shtml

Answer 1

@Marki

Answer 2

So it looks like we are making two lists, one with all the powers of 10 in base 2 and another list of all the powers of 10 in base 5. Then if you combine the two lists, there will be only one number with 2 digits, one number with 3 digits, one number with 4 digits, etc...

Answer 3

hehe ok lets try

Answer 4

for a start (just making an assumption )
i would think its related to
10=2*5

Answer 5

My intuition tells me we could do this and look at powers of 6 in base 2 and base 3 and have the whole counting digits thing work too.

Answer 6

Yes!! We are on the same page I agree with you! =D

Answer 7

Powers of 10 in base 10 do this on its own already. Maybe we can map them from this to that some how or use the fact that powers of 2 in base 2 and powers of 5 in base 5 have this property to themselves?

Answer 8

hmm i dont understand the pattern >.<

Answer 9

Which? The one I said or the one from the website?

Answer 10

web

Answer 11

so
\((10)_{10}=(2*5)_{10}=(10)_2*(101)_2\)

Answer 12

i'll see if this leads to something , i'll go for lunch nw, brb

Answer 13

Oh I'll sort of rephrase it, so first off we can count like this: \[\Large 20_5, 400_5,1010_2,13000_5, 310000_5, 1100100_2, etc...\] So we have 2,3,4,5,6,7 etc... if we count the digits. I put the little 5s and 2s to show what base they are in, but if I rewrite these numbers in base 10 we have: \[\Large 10, 100, 10,1000,10000,100, etc...\] since we are representing all the powers of 10 in base 2 and base 5, they show up multiple times.

Answer 14

ok I'll see what I can figure out while you're gone.

Answer 15

\(\begin{matrix}
\text{base 10} & \text{base 2} & \text{base 5}\\
10 & 1010 &20 \\
10^2 &1100100 & 400\\
10^3 &1111101000 & 13000\\
10^4 &10011100010000 & 310000\\
10^5& 11000011010100000 & 11200000\\
\vdots & \vdots & \vdots
\end{matrix}\)

Answer 16

ok so
10=1010
10^2=(1010*1010)---> to multiply we put all zeros first we have 2 of them
10^3=(1010*1010*1010)--->3 zeros before multiply
ect xD
done

Answer 17

thinking while eating is good xD
same thing with base 5
10=20
10^2=(20*20) --> 2 zeros
....ect

Answer 18

Woah awesome! That's really cool! XD

Answer 19

:O
thats it ?