just do the the conversion to base 7 by doing all your arithmetic in base 3.
You repeatedly divide the base three number you want to convert by 7, which is written 21 in base 3.
If you want to use the base 10 symbols 3, 4, 5, 6 then when remainders (in base 3) 10, 11, 12, 20 occur you will need to replace them by the symbols 3, 4, 5, 6.
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az_lender say: Yes. Note that
1_7 = 1_3
2_7 = 2_3
3_7 = 10_3
4_7 = 11_3
5_7 = 12_3
6_7 = 20_3
10_7 = 21_3
11_7 = 22_3
12_7 = 100_3
the next stuff we'll find useful is a base 7 equivalent of 1000_3; it's 36_7.
And a base 7 equivalent of 10000_3 is
144_7.
A base 7 equivalent of 100000_3 is
465_7, if I haven't made a mistake.
Once you have a bunch of powers of 3 translated to base 7, then take ANY number written in base 3, and add up its components...but you have to do the addition in base 7, which might take some practice!!!
Example: 20212 base 3
= 2*144_7 + 2*12_7 + 1*3_7 + 2*1_7
= 144_7 + 144_7 + 12_7 + 12_7 + 3_7 + 2_7.
When you add up the rightmost digits of all this, you get 23_7, so put down the 3 and carry the 2. When you add up the 2nd-to-right digits, you get 13_7, plus the 2 you carried is 15_7, so put down the 5 and carry the 1. Finally add 1 + 1 plus the 1 you carried, and put down the 3. The result is 353_7.
Check in decimal notation:
20212_3 = 162 + 18 + 3 + 2 = 185.
353_7 = 147 + 35 + 3 = 185.
So it worked.
But, since you are used to thinking in decimal numbers, that will usually be the easier way!!
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roger say: yep.
the method is the same regardless of base.
divide by the new base. Write down the remainder.
repeat.
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nick say: the only bases I know of are:
Base 2 (binary)
Base 8 (octal)
Base 16 (hexadecimal)
And of course base 10.
I've never heard of base 3 or base 7.
In case you don't know, here are the conversations for hexadecimal from base 10.
1 = 1
2 = 2
3 = 3
This continues up to 9.
9 = 9
Now things change.
10 = A
11 = B
12 = C
13 = D
14 = E
15 = F
Keep in mind that base 16 (hexadecimal) means digits from 0 to F (15) which is 16 digits in total. the same way that base 10 means digits 0 - 9.
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Ash say: yes you can go direct
log₃x = log₇x / log₇3
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llaffer say: If bases are always a base of a power of 2, there are shortcuts you can take, but from 3 to 7, you usually do have to go to some common ground since most people don't have the understanding of how base 3 and 7 works as well as we do with base 10.
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