# Binary Fractions and Floating Point

~ on ~ 0 commentsExpressing binary fractions in the floating-point format may also be done using the same method we have developed above. We shall illustrate the technique by using a numerical example.

**Example 34:**Express decimal fraction ¼ in the binary floating-point format.

**Solution:**

**As the first step, we convert the given decimal fraction into regular binary fraction. The conversion yields**

1/4 = 0.25 =

**0. 010**

Now, since the given number is a fraction, we employ the

*reverse*of our previous technique, i.e., first multiply and then divide (instead of first dividing and then multiplying) for its floating-point representation. Also, since there are only two bits in the given fraction

**0.01**(the

**0**after the

**1**is not counted), it seems quite natural that we use 2

^{2}for multiplying and dividing it. However, we have imposed a restriction on the mantissa

*M*that for binary floating-point format it should lie in between ½ and 1. Imposition of this restriction means that we have to express the mantissa as

**0.1**, and

*not*as

**0.01**. This further means that we have to multiply and divide the number by 2

^{1}, and not by 2

^{2}. Performing this operation yields

X = (0.01 x 2

^{1})(1/2^{1}) = 0.1 x 2^{-1}
Now, the exponent of

*X,*which is a negative integer, must also be expressed in binary,as stated above. Carrying out this exercise, our floating-point representation of decimal fraction ¼ becomes
X = (0.01 x 2

^{1})(1/2^{1}) = 0.1 x 2^{1001}
where the first

**1**in the exponent represents the negative sign, and the next three bits, viz.**001**, represent decimal 1.
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