Convert Numbers to Rational Bases: An Example

In summary, the conversation discusses playing with numbers in different bases, specifically fractional bases. The speaker has found a way to convert numbers to fractional bases and has searched the internet for similar methods. An example is given of how this can be done. The conversation then moves on to discussing the use of rational and irrational bases, and even complex bases. The speaker recommends reading "The Art of Computer Programming" for further exploration and suggests trying negative bases and different examples of bases. Overall, the conversation highlights the endless possibilities for exploring and playing with different bases in mathematics.
  • #1
pseudogenius
7
0
I have been playing around with numbers in different bases and then I thought, what if they were in fractional bases. I found a way to convert numbers to fractional bases and have been searching on the internet and not found a similar way to do this. Anyway, here is an example of how I would do this:

The problem is: Write 5 in base 3/2

5*(2/3)= 3 remainder 1
6*(2/3)= 2 remainder 0
2*(2/3)= 1 remainder 1
1*(2/3)= 0 remainder 2

Take 2101 and divide each digit by 2.

5 in base 3/2 = 2/2 1/2 0/2 1/2

Check:

(2/2)*(3/2)^3+(1/2)*(3/2)^2+(0/2)*(3/2)^1+(1/2)*(3/2)^0=
27/8+9/8+0+1/2= 5

It worked.

Is it okay to extend digits to rational digits?
Is this a valid way to put integers in rational bases?

I have worked out a way to put rational numbers into rational bases but the process is complicated.
 
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  • #2
pseudogenius said:
Is it okay to extend digits to rational digits?
Is this a valid way to put integers in rational bases?

Yes to both. You can even use irrational bases; phi is a popular example.
 
  • #3
Thank you CRGreathouse,

Onward to irrational and complex bases!
 
  • #4
pseudogenius said:
Thank you CRGreathouse,

Onward to irrational and complex bases!

For complex bases, Google "quarter-imaginary base".
 
  • #5
You can read "The Art of Computer Programming" by Donald Knuth, vol I. A great (I don't mean sort of nice, I mean *great*!) book for anyone who has enough brain to ask a question like that. It is readable and humorous as well as seriously mind-stretching. Knuth doesn't need to be obscure to be impressive.
Have a go at negative bases. They are really entertaining.
Here is another, more important one: consider compare examples like base ten, hundred, and thousand. Also 2, 4, 8, 16 etc.
Then, for humorous effect, try root ten, root two, etc.
Try interconversions and arithmetic in each base.
It is a tremendous field for entertaining exploration.

Cheers,

Jon
 

1. What is a rational base?

A rational base is a number that is used as the base or the main number in a number system. It is usually expressed as a ratio of two integers, where the numerator represents the value of the base and the denominator represents the number of digits in that base.

2. How do you convert numbers to rational bases?

To convert a number to a rational base, you can follow these steps:

  1. Divide the number by the base value to get the remainder.
  2. The remainder becomes the rightmost digit in the rational base representation.
  3. Divide the quotient from the first step by the base value again.
  4. The new remainder becomes the second digit in the rational base representation.
  5. Repeat this process until the quotient becomes 0.
  6. The final remainder will be the leftmost digit in the rational base representation.

3. Why would you want to convert numbers to rational bases?

Converting numbers to rational bases can be useful in various areas, such as cryptography, computer science, and data compression. It can also provide a different perspective on numbers and help understand their relationships and patterns.

4. What are some common rational bases?

Some of the most commonly used rational bases include binary (base 2), octal (base 8), and hexadecimal (base 16). However, rational bases can be any number, as long as it is a ratio of two integers.

5. Can you convert any number to a rational base?

Yes, any number can be converted to a rational base. However, the resulting rational base representation may not always be unique. For example, the number 5 can be represented as 5/1, 10/2, or 20/4 in rational base 10.

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