Converting between bases without base 10 mid-step

In summary, the conversation discusses methods for converting between two non-decimal number systems, with the main challenge being the time constraint of 45 seconds. The suggested method is to use the division algorithm, but it is noted that this may be difficult due to the need to familiarize oneself with multiplication tables of different bases. An alternative method is also mentioned, which involves regrouping the bits.
  • #1
smize
78
1
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.
 
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  • #2
hi smize! :smile:

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

there's a general rule that logab = logxb/logxa …

does that help? :smile:
 
  • #3
tiny-tim said:
there's a general rule that logab = logxb/logxa

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?
 
  • #4
smize said:
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 [itex]a[/itex] be your number and let p be the base you want to convert it. Use the division algorithm to write

[tex]a=b_1p+r_1[/tex]

Use it again on [itex]b_1[/itex]:

[tex]b_1=b_1p+r_2[/tex]

Keep doing it until a [itex]b_n=0[/itex]. Then we have

[tex]b_{n-1}=0b_n+r_n[/tex]

Then [itex]r_n...r_2r_1[/itex] is the number you want.
 
  • #5
micromass said:
Let [itex]a[/itex] 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.
 
  • #7
Could you give an example of using that to convert between let's say, base 6 and base 4?
 
  • #8
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.
 
  • #9
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?
 
  • #10
smize said:
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...
 
  • #11
micromass said:
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.
 
  • #12
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
 
  • #13
Thank you coolul007. It is a very interesting method.
 

What is "converting between bases without base 10 mid-step"?

"Converting between bases without base 10 mid-step" refers to the process of converting a number from one numerical base to another without using base 10 as an intermediate step. This method involves directly converting the number from its original base to the desired base.

Why would someone want to convert between bases without using base 10 mid-step?

There are several reasons why someone may want to convert between bases without using base 10 as an intermediate step. One reason is that it can be a more efficient and accurate method, as using base 10 as an intermediate step can introduce errors. Additionally, some computer systems use different bases for their calculations, so converting without base 10 can be more compatible with these systems.

What are the most commonly used numerical bases?

The most commonly used numerical bases are base 2 (binary), base 8 (octal), base 10 (decimal), and base 16 (hexadecimal). Base 2 is commonly used in computer systems, while base 10 is the most commonly used base in everyday mathematics.

What is the conversion process for converting between bases without using base 10 mid-step?

The conversion process involves breaking down the number into its individual digits, multiplying each digit by its corresponding power of the original base, and then adding these products together to get the final converted number. This process can be repeated for each digit in the number until all digits have been converted.

Is it possible to convert between any two bases without using base 10 mid-step?

Yes, it is possible to convert between any two bases without using base 10 as an intermediate step. However, the conversion process may become more complex for bases that are not commonly used, such as base 3 or base 12.

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