IEEE 754 Notation
Example: Converting to Float
Convert the following single-precision IEEE 754 number
into a floating-point decimal value.
1 10000001 10110011001100110011010
- First, put the bits in three groups.
Bit 31 (the leftmost bit) show the
sign of the number.
Bits 23-30 (the next 8 bits) are the
Bits 0-22 (on the right) give the
- Now, look at the sign bit.
If this bit is a 1, the number is negative.
If it is 0, the number is positive.
This bit is 1, so the number is negative.
- Get the exponent and the correct bias.
The exponent is simply a positive binary number.
10000001bin = 129ten
Remember that we will have to subtract a bias from this exponent to find the
power of 2. Since this is a single-precision number, the bias is 127.
- Convert the fraction string into base ten.
This is the trickiest step. The binary string represents a fraction,
so conversion is a little different.
Binary fractions look like this:
0.1 = (1/2) = 2-1
0.01 = (1/4) = 2-2
0.001 = (1/8) = 2-3
So, for this example, we multiply each digit by the corresponding
power of 2:
1*2-1+ 0*2-2 + 1*2-3 +
1*2-4 + 0*2-5 + 0 * 2-6 + ...
1/2 + 1/8 + 1/16 + ...
Note that this number is just an approximation on some decimal number.
There will most likely be some error. In this case, the
fraction is about 0.7000000476837158.
- This is all the information we need. We can put these numbers in
(-1)sign bit *
2 exponent - bias
= (-1)1 *
The answer is approximately -6.8.
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