floating point (ieee 754)
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Using the binary number system we know:
To express fractional numbers, we start by extending the scheme of the binary positional number system. Numbers to the right of the "binary point" simply have negative exponents.
1 is 2^0 = 1 10 is 2^1 = 2 100 is 2^2 = 4
To express fractional numbers, we start by extending the scheme of the binary positional number system. Numbers to the right of the "binary point" simply have negative exponents.
.1 is 2^-1 = 1/2 (.5) .01 is 2^-2 = 1/4 (.25) .001 is 2^-3 = 1/8 (.125) So, 101.11 = 5.75 [ 4 + 1 + .5 + .25]
| Binary | Decimal | Description |
| 0.1 | 0.5 | a half |
| 0.01 | 0.25 | a quarter |
| 0.001 | 0.125 | an eighth |
| 0.0001 | 0.0625 | a sixteenth |
| 0.00001 | 0.03125 | a thirtysecond |
| 0.000001 | 0.015625 | a sixtyfourth |
>> dec to fp converter >> hex to fp converter
IEEE 754
IEEE 754 uses this representation as a basis for its scheme to represent floating point numbers. (see wikipedia ieee 754-1985)
It assigns bit fields to 3 things:
1. sign (1 bit) 0 positive, 1 negative
2. exponent (8 bits) unsigned, bias 127 (stored as exponent + 127)
3. fraction (23 bits) normalized: leading one not represented
(image from wikipedia)
In the example shown above, the sign is zero so sign is positive, the exponent
is 124 so becomes -3 (124-127), and the fraction (significand or mantissa)
(.0100...) becomes 1.01 when the leading 1 is supplied
So we have +1.01 x 2^-3
Shifting the binary point left 3 places, 1.01 becomes .001012
In decimal, then, the represented number is: .125 + .03125,
which is +0.1562510
OR (another way to approach the conversion)
==
1.012 = 1.2510 x 2^-3 (.125)
1.25 x .125 = .1562510
The represented number is therefore: +1.25 x 2^-3,
which is +0.1562510
(image from wikipedia)
In the example shown above, the sign is zero so sign is positive, the exponent
is 124 so becomes -3 (124-127), and the fraction (significand or mantissa)
(.0100...) becomes 1.01 when the leading 1 is supplied
So we have +1.01 x 2^-3
Shifting the binary point left 3 places, 1.01 becomes .001012
In decimal, then, the represented number is: .125 + .03125,
which is +0.1562510
OR (another way to approach the conversion)
==
1.012 = 1.2510 x 2^-3 (.125)
1.25 x .125 = .1562510
The represented number is therefore: +1.25 x 2^-3,
which is +0.1562510