Converting between bases without base 10 mid-step

[smize]

I am currently trying to go between any two bases which are between and including base 2 to base 10. (i.e, base 10 to base 3, base 4 to base 6, etc...). Is there an equation or set of formulas for a one-step transition between the bases (or one-way method, rather than converting to base-10 then to base-x.

 

[tiny-tim]

hi smize!

i'm not sure what you're trying to do

there's a general rule that log_ab = log_xb/log_xa …

does that help? ​

 

[smize]

there's a general rule that log_ab = log_xb/log_xa

Yes; I know the general log rules. I am part of the Math Academic team at my school and we are having to convert, for example, 2012₃ to Base 6. The issue at hand is we, at most, have 45 seconds to do the calculations and guarantee they are correct. We are wondering if there is a way to do this without have to convert it to Base 10 first. A.K.A. is there a direct way to convert between two non-decimal number systems?

 

[micromass]

Yes; I know the general log rules. I am part of the Math Academic team at my school and we are having to convert, for example, 2012₃ to Base 6. The issue at hand is we, at most, have 45 seconds to do the calculations and guarantee they are correct. We are wondering if there is a way to do this without have to convert it to Base 10 first. A.K.A. is there a direct way to convert between two non-decimal number systems?

Yes, there is a way to do this, but all depends on how fast you can calculate.

Let $a$ be your number and let p be the base you want to convert it. Use the division algorithm to write

$$a=b_1p+r_1$$

Use it again on $b_1$:

$$b_1=b_1p+r_2$$

Keep doing it until a $b_n=0$. Then we have

$$b_{n-1}=0b_n+r_n$$

Then $r_n...r_2r_1$ is the number you want.

 

[smize]

Let $a$ be your number and let p be the base you want to convert it. Use the division algorithm to write

you didn't define the division algorithm...And using this, a will have to equal a non base-10 number to another non base-10 number of a different number system.

 

[micromass]

http://en.wikipedia.org/wiki/Division_algorithm

 

[smize]

Could you give an example of using that to convert between let's say, base 6 and base 4?

 

[micromass]

Let's pick 2201 base 4 and let's convert it base 6. So a=2201 and p=12 (remember to express p also in base 4).

2201= 12*103 + 11
122= 12*10 +2
10 = 12*0 +10

Thus 2201 in base 6 is 10 2 11. If we put 10=4 and 11=5, then we get 425.

 

[smize]

So the division algorithm does work for all bases, it is just a matter of familiarizing yourself with the multiplication tables of the other bases, correct?

 

[micromass]

So the division algorithm does work for all bases, it is just a matter of familiarizing yourself with the multiplication tables of the other bases, correct?

Correct. And that might be a bit difficult.

Maybe there are other methods, but I doubt it...

 

[smize]

Correct. And that might be a bit difficult.

Maybe there are other methods, but I doubt it...

it is either that or converting to base 10 then the other bases. In some cases the multiple choice answers we have to choose from are also of 2 different bases -.- And we'll only have 45 seconds to answer.

 

[coolul007]

converting from power of 2 you just regroup the bits. base 2, 4, and 8 is easy same for base 3 and 9. 67 base 8 to base 2 110 111 then to base 4 11 01 11 or 313
for base 9 to base 3, 87 base 9 is 22 21 base 3

The others are a bit tougher

 

[smize]

Thank you coolul007. It is a very interesting method.