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zk/Signed_and_unsigned_numbers.md
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@ -4,15 +4,18 @@ tags: [binary, binary-encoding]
# Signed and unsigned numbers # Signed and unsigned numbers
## Summary
- To represent negative integers in binary we use signed numbers.
In order to represent negative integers in binary we use signed numbers. In order to represent negative integers in binary we use signed numbers.
**Signed binary** is basically binary where negative integers can be **Signed binary** is basically binary where negative integers can be
represented. **Unsigned binary** is standard binary without negative integers. represented. **Unsigned binary** is standard binary without negative integers.
In order to represent negative integers alonside positive integers a natural In order to represent negative integers alonside positive integers a natural
approach is to divide the available approach is to divide the available
[encoding space / word length](Binary_encoding.md) [encoding space / word length](Binary_encoding.md) into two subsets: one for
into two subsets: one for representing non-negative integers and one for representing non-negative integers and one for representing negative integers.
representing negative integers.
The primary method for doing this is to use _two's complement_. This method The primary method for doing this is to use _two's complement_. This method
makes signed numbers possible in a way that complicates the hardware makes signed numbers possible in a way that complicates the hardware
@ -83,9 +86,8 @@ inverting the bits.)
The chief advantage of the two's complement technique of signing numbers is that The chief advantage of the two's complement technique of signing numbers is that
its circuit implementation is no different from the adding of two unsigned its circuit implementation is no different from the adding of two unsigned
numbers. Once the signing algorithm is applied the addition can be passed numbers. Once the signing algorithm is applied the addition can be passed
through an through an [adder](Half_adder_and_full_adder.md) component without any special
[adder](Half_adder_and_full_adder.md) handling or additional hardware.
component without any special handling or additional hardware.
Let's demonstrate this with the following addition: Let's demonstrate this with the following addition:
@ -119,8 +121,8 @@ The ease by which we conduct signed arithmetic with standard hardware contrasts
with alternative approaches to signing numbers. An example of another approach with alternative approaches to signing numbers. An example of another approach
is **signed magnitude representation**. A basic implemetation of this would be is **signed magnitude representation**. A basic implemetation of this would be
to say that for a given bit-length (6, 16, 32...) if the to say that for a given bit-length (6, 16, 32...) if the
[most significant bit](Half_adder_and_full_adder.md#binary-arithmetic) [most significant bit](Half_adder_and_full_adder.md#binary-arithmetic) is a 0
is a 0 then the number is positive. If it is 1 then it is negative. then the number is positive. If it is 1 then it is negative.
This works but it requires extra complexity to in a system's design to account This works but it requires extra complexity to in a system's design to account
for the bit that has a special meaning. Adder components would need to be for the bit that has a special meaning. Adder components would need to be