Autosave: 2024-03-20 07:50:03

zk/Signed_and_unsigned_numbers.md

@@ -9,131 +9,10 @@ tags: [binary, binary-encoding]
 - To represent negative integers in binary we use signed numbers._Signed binary_
   includes negative integers, _unsigned binary_ does not.
 
-- The principal methnod for encoding signed binary numbers is called **two's
+- There are different methods of implementing the encoding of signed numbers.
-  complement**.
+  Two prominant approaches are **two's complement** and **signed magnitude
+  representation**.
 
 ## Related notes
 
-In order to represent negative integers alonside positive integers a natural
+[[gktb#detail|two's complement ❯ Detail]]
-approach is to divide the available
-[encoding space / word length](Binary_encoding.md) into two subsets: one for
-representing non-negative integers and one for representing negative integers.
-
-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
-implementation of the binary arithmetic operations as little as possible.
-
-## Two's complement
-
-Signed numbers can be implemented in binary in a number of ways. The differences
-come down to how you choose to encode the negative integers. A common method is
-to use "two's complement".
-
-> The two's complement of a given binary number is its negative equivalent
-
-For example the two's complement of $0101$ (binary 5) is $1011$. There is a
-simple algorithm at work to generate the complement for 4-bit number:
-
-1. Take the unsigned number, and flip the bits. In other words: invert the
-   values, so $0$ becomes $1$ and $1$ becomes $0$.
-2. Add one
-
-![](/img/unsigned-to-signed.png)
-
-To translate a signed number to an unsigned number you flip them back and still
-add one:
-
-![](/img/signed-to-unsigned.png)
-
-### Formal expresssion: $2^n - x$
-
-The heuristic account of deriving two's complement above can be formalised as
-follows:
-
-> in a binary system that uses a word size of $n$ bits, the two's complement
-> binary code that represents negative $x$ is taken to be the code that
-> represents $2^n - x$
-
-Let's demonstrate this, again using -5 as the value we wish to encode.
-
-Applying the formula to a 4-bit system, negative 5 can be derived as follows:
-
-$$
- 2^4 -5
-$$
-
-$$
- 16 -5 = 11
-$$
-
-$$
-11 = 1011
-$$
-
-So basically we carry out the subtraction against the word length and then
-convert the decimal result to binary.
-
-We can also confirm the output by noting that when the binary form of the number
-and its negative are added the sum will be `0000` if we ignore the overflow bit:
-
-$$
-  1011 + 0101 = 0000
-$$
-
-(This makes sense if we recall that earlier we derived the complement by
-inverting the bits.)
-
-### Advantages
-
-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
-numbers. Once the signing algorithm is applied the addition can be passed
-through an [adder](Half_adder_and_full_adder.md) component without any special
-handling or additional hardware.
-
-Let's demonstrate this with the following addition:
-
-$$
-    7 + -3 = 4
-$$
-
-First we convert $7$ to binary: $0111$.
-
-Then we convert $-3$ to unsigned binary, by converting $3$ to its two's
-complement
-
-$$
-0011 \rArr 1100 \rArr 1101
-$$
-
-Then we simply add the binary numbers regardless of whether they happen to be
-positive or negative integers in decimal:
-
-$$
-0111 \\
-1101 \\
-====\\
-0100
-$$
-
-Which is 4. This means the calculation above would be identical whether we were
-calculating $7 + -3$ or $7 + 13$.
-
-The ease by which we conduct signed arithmetic with standard hardware contrasts
-with alternative approaches to signing numbers. An example of another approach
-is **signed magnitude representation**. A basic implemetation of this would be
-to say that for a given bit-length (6, 16, 32...) if the
-[most significant bit](Half_adder_and_full_adder.md#binary-arithmetic) is a 0
-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
-for the bit that has a special meaning. Adder components would need to be
-modified to account for it.
-
-## Considerations
-
-A limitation of signed numbers via two's complement is that it reduces the total
-informational capacity of a 4-bit number. Instead 16 permutations of bits giving
-you sixteen integers you instead have 8 integers and 8 of their negative
-equivalents. So if you wanted to represent integers greater than decimal 8 you
-would need to increase the bit length.

zk/Twos_complement.md

@@ -1,7 +1,7 @@
 ---
 id: gktb
 title: Two's complement
-tags: []
+tags: [binary, binary-encoding]
 created: Tuesday, March 19, 2024
 ---
 
@@ -12,7 +12,13 @@ created: Tuesday, March 19, 2024
 - _Two's complement_ is a method for representing signed numbers (negative
   integers) in binary.
 
-- The two's complement of a given binary integer is its negative equivalent.
+- It is derived by inverting the values of an unsigned binary integer to create
+  its signed equivalent.
+
+- A benefit is that hardware implementation of the binary arithmetic of signed
+  and unsigned numbers can be handled in the same manner as unsigned numbers,
+  requiring no additional handling. A drawback is that it halves the
+  informational capacity of the given word length for the binary system.
 
 ## Detail
 
@@ -82,8 +88,12 @@ $$
 
 - In a 4-bit system instead of 16 total unique encodings of integers you have 8
   encodings for positive integers and 8 encodings for the their signed
-  equivalent.
+  equivalent. For integers larger than denary 8 you would need to increase the
+  bit length of the system
+
+- Consequently two's complement can necessitate larger overall word lengths.
 
 ## Related notes
 
 [[Signed_and_unsigned_numbers|signed_and_unsigned_numbers]]
+[[Binary_addition|binary addition]]