Tiny VoltBinary representations of integers
Four representation types are illustrated here: uint, two's complement, one's complement and biased representation.
| Shorthand | Meaning |
|---|---|
| uint | Unsigned integer |
| 2C | Two's complement |
| 1C | One's complement |
| bias | Biased binary representation |
A binary representation can be seen as a mapping from n bits to a decimal number:
| Encoding | Weights | Bias |
|---|---|---|
uint |
||
2C |
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1C |
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biased |
Representation:
Change the bit values and bias in the widget below to see how it changes the representation of numbers:
+ + - =
Note: bias is 0 for representation
Unsigned integer
This is the simplest representation.
- Range: , so for
- Every bit pattern maps to a different number
- The lexicographic ordering of bit strings is the same as numeric order:
Two's complement
Sign of the top weight is flipped to . Other weights stay the same.
- Range: , so for
This is the same ring as uint. This is because it changes half of the numbers by . And in the ring .
Bias
Read the bits as a plain uint, then subtract a fixed constant . Here .
- Range: , so for
Subtracting a constant does not reorder anything, so bigger bit pattern means bigger number. You can compare two biased numbers, signs and all, with an unsigned comparison of their bits. That is why float exponents are stored this way.
One's complement
The top weight is . Negating a number is just flipping every bit.
Range: , so for
The mapping is not injective:
000and111both map to .
Connection between two's complement and one's complement
Since in the ring, we have:
This equation explains the well known relationship: to get the two's complement representation of a number, you can take the one's complement and add 1.
Lexicographical ordering of bit strings vs their numeric values
| bits | uint |
2C |
1C |
bias () |
|---|---|---|---|---|
- Color coding: smallest to largest