'How to convert a fraction to binary?

I don't know how to convert from a fraction to binary. When I search it, there has a solution shows that:

    1              1
-- (dec)  =   ---- (bin)
10            1010


       0.000110011...
      -------------
1010 | 1.0000000000
         1010
       ------
         01100
          1010
         -----
          0010000
             1010
            -----
             01100
              1010
             -----
              0010

I don't know how and why to do it.

algorithm


Solution 1:[1]

Let's take a look at converting the decimal value of 0.625 to binary.

Step 1: Begin with the decimal fraction and multiply by 2. The whole number part of the result is the first binary digit to the right of the point.

Because .625 x 2 = 1.25, the first binary digit to the right of the point is a 1.
So far, we have .625 = .1??? . . . (base 2) .

Step 2: Next we disregard the whole number part of the previous result (the 1 in this case) and multiply by 2 once again. The whole number part of this new result is the second binary digit to the right of the point. We will continue this process until we get a zero as our decimal part or until we recognize an infinite repeating pattern.

Because .25 x 2 = 0.50, the second binary digit to the right of the point is a 0.
So far, we have .625 = .10?? . . . (base 2) .

Step 3: Disregarding the whole number part of the previous result (this result was .50 so there actually is no whole number part to disregard in this case), we multiply by 2 once again. The whole number part of the result is now the next binary digit to the right of the point.

Because .50 x 2 = 1.00, the third binary digit to the right of the point is a 1.
So now we have .625 = .101?? . . . (base 2) .

Step 4: In fact, we do not need a Step 4. We are finished in Step 3, because we had 0 as the fractional part of our result there.

Hence the representation of .625 = .101 (base 2) .

Decimal 1/10 converts to an infinite binary fraction.
In your question you said that 1/10 in decimal equals 1/1010 in binary. .1 (1/10) in decimal actually equals 0.00011001100110011... in binary.

Solution 2:[2]

Fractional value to Binary number conversion

  1. The fraction value is multiplied by 2 and result has a decimal (1 or 0) and a fraction value.

  2. take the faction value for step 1 operation.

  3. The repeat process until the fraction value reached to 0.

  4. collects a decimal value from bottom to up

    fraction = .125

          = .125 x 2    
      = 0.250 x 2
      = 0.50  x 2
      = 1.0

    fraction = 0.125 = 100

    Results
    given fraction value (base 10)= 0.125
    into binary bits (base 2) = 0.100

    Real number to binary conversion

Solution 3:[3]

In a binary weighted fraction each digit to the right of the decimal point is a power of (1/2) (or the negative of power of 2) smaller than the one to the left. The first rightward digit has a weight of 1/2, the second is 1/4, the third 1/8, and so on.

So a 0.111 (base-2) is:

1*(0.5) + 1*(0.25) + 1*(0.125) = 0.875

And a 0.0101 (base-2) is:

0*(0.5) + 1*(0.25) + 0*(0.125) + 1*(0.0625) = 0.3125

It's no different from binary integers, except we're just extending it to negative powers of 2 as we move right of the decimal point.

I hope that addresses at least part of your question.

Sources

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Source: Stack Overflow

Solution Source
Solution 1
Solution 2 clcoder
Solution 3 jongo45

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