Conversion from INT to Binary

**chris1985** · 12-30-2004

Originally Posted by Sebastiani

>> A quick question: How do you convert an integer to a binary number?

just to clarify: an integer is a value and a base is a representation of a value. by representation I mean a string of characters where each character is referred to as a digit and each digit corresponds a value in the range of 0 through base-1. so technically, you don't convert an integer to a specific base - you convert one base representation of an integer to another base representation. here's a similar thread that you might find helpful.

While we're talking of base numbers, maybe you could help with this question:

How do you represent a number in base minus one?

Now my thoughts are that using the sequence below it is impossible to even have a base positive 1 yet alone minus 1.

Base 10 has numbers 0,1,2,3,4,5,6,7,8,9
Base 8 has 0,1,2,3,4,5,6,7
Base 2 has 0,1
Base 1 has 0

Since based 1 has only 1 value then it cannot represent information. So how does base minus 1 work?

The question in bold above, was from the book Hard Drive which is a history of the Microsoft company. Apparently this question used to be used to test employees in the early days.

**Sang-drax** · 12-30-2004

I always think of base one like this:
Position n is worth 1^n = 1, so every position is equally worth.
1=1
2=11
3=111
4=1111
...
This isn't 100% consistent with the other bases as you have to drop the zero.
Also, non-integers are hard to represent like this.

EDIT:
As for the conversion, I'm quite sure I've posted such code here before. Do a board search and I'm sure you'll encounter code by me or someone else.

Here's how it could look like:

Code:

string num2str(unsigned num, unsigned base)
{
  const char digits['0','1',...'9','A',...'Ö']; //the digits we will use to build the number
  string output;
  while (num > 0)
  {
    output = output + digits[num%base];
   num /= base;
  }
  return output;
}

Add some error checking and it should work.

**Tronic** · 12-30-2004

Only reason that wouldn't work in electronics, is because it wouldn't know the "start" or the "end" of things.

You would have to have base 2 if you were to have a start and an end. But representing it on paper, and not a duration with a stop/start value, works just fine. It just doesn't work well in the world of electronics.

**sigfriedmcwild** · 12-31-2004

Originally Posted by chris1985

How do you represent a number in base minus one?

After a quick look on google and mathworld I'd say it is impossible.
The representation of a number in base b is equivalent to a polynomial
x1 b^0 + x2 b^1 + x3 b^2 + ... + x4 b^n

where n is floor(log(number, base))

now since for any number (except 1 and -1) the logarithm of that number in base -1 is a complex number I fail to see how you could write the polynomial (most likely due to my limited knowledge of mathematics)

More over the digits are computed as:
xn = floor(numberi/basei)

where numberi (i is a subscript) is defined as

numbern = number

number(i-1) = numberi - (xi base^i)

again this formula fails for base = -1 (infinite recursion in the case of a complex exponent for base)

Now, I'm not a mathematician and if anyone who actually has studies this stuff wants to shed further light on the subject I'd be grateful

**Dave Evans** · 12-31-2004

Originally Posted by sigfriedmcwild

After a quick look on google and mathworld I'd say it is impossible.
The representation of a number in base b is equivalent to a polynomial
x1 b^0 + x2 b^1 + x3 b^2 + ... + x4 b^n

where n is floor(log(number, base))

now since for any number (except 1 and -1) the logarithm of that number in base -1 is a complex number I fail to see how you could write the polynomial (most likely due to my limited knowledge of mathematics)

More over the digits are computed as:
xn = floor(numberi/basei)

where numberi (i is a subscript) is defined as

numbern = number

number(i-1) = numberi - (xi base^i)

again this formula fails for base = -1 (infinite recursion in the case of a complex exponent for base)

Now, I'm not a mathematician and if anyone who actually has studies this stuff wants to shed further light on the subject I'd be grateful

This is true:

The representation of a number in base b is equivalent to a polynomial
x1 b^0 + x2 b^1 + x3 b^2 + ... + x4 b^n

The base, b, does not have to be positive.

If you are representing positive integers in a system with positive base b the expession floor(log(number, base)) gives a way of determining the number of base-b digits that would be required. (But it has nothing to do with the representation itself, or with the definition.)

Using base -2, here are a few numbers

Code:

 Base 10     Base -2

 
     -10        1010
      -9        1011
      -8        1000
      -7        1001
      -6        1110
      -5        1111
      -4        1100
      -3        1101
      -2        10
      -1        11
       0        0
       1        1
       2        110
       3        111
       4        100
       5        101
       6        11010
       7        11011
       8        11000
       9        11001
      10        11110

Regards,

Dave

**sigfriedmcwild** · 01-01-2005

@Dave Evans
With a base of -1 though you are still limited to a string of +1 and -1 that are multiplied by some coefficient (my guess is that you'd have a single coefficient: 0) making the results an infinite string (or just plain impossible).

Or maybe I'm looking at this completely wrong.

**Dave Evans** · 01-01-2005

Originally Posted by sigfriedmcwild

@Dave Evans
With a base of -1 though you are still limited to a string of +1 and -1 that are multiplied by some coefficient (my guess is that you'd have a single coefficient: 0) making the results an infinite string (or just plain impossible).

Or maybe I'm looking at this completely wrong.

You are, of course, correct. My example shows that it is possible to have negative number base systems. It just doesn't lead anywhere for base -1. (In other words, I committed the classic mistake of answering a question that wasn't asked --- I just didn't think it through.)

Regards,

Dave

**sigfriedmcwild** · 01-01-2005

Well according to mathworld there have been mathematicians using bases that weren't even rational numbers. Someone for examples used numbers written in base pi (not quite sure how that one worked).

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