feat: add two's complement post

src/pages/posts/twos-complement.md

@@ -0,0 +1,42 @@
+---
+layout: ../../layouts/MarkdownPostLayout.astro
+title: "Two's Complement in 100 Seconds"
+pubDate: "2023-11-16"
+author: "MimmyJau"
+description: "Key concepts in two's complement."
+tags: ["architecture", "memory", "learning in public"]
+---
+
+If there's only one thing you need to understand about two's complement, it's the following:
+
+>  Negative numbers should have binary representations such that subtraction can be accomplished using the same algorithm as addition.
+
+In order for this to work, the following is **sufficient**:
+
+> For all $x$, let $\overline{x}$ be the binary value such that $x + \overline{x} = 0$ under binary addition. Then $\overline{x}$ is the binary representation of $-x$.
+
+For example, knowing that
+1) $0011_2$ has been defined as $3$, and [^binary]
+2) $0011_2 + 1101_2 = 0000_2$,
+
+it follows that $1101_2$ **must** be defined as $-3$. 
+
+Almost everything else about two's complement is a consequence of these ideas. For example, this explains why the common trick for multiply by $-1$ is to flip all the bits and add one.
+
+## Why this works (optional reading)
+
+To prove that this representation of negative numbers satisfies subtraction, we can prove the following two statements. 
+
+> Prove that the binary representation of each negative number is unique.
+
+*Exercise left to the reader.*
+
+> Let $\odot$ represent binary addition via two's complement. Prove that for any arbitrary $x$ and $y$, $x \odot (-y) = x - y$.
+
+If $x >= 0$ and $y <= 0$, then the statement is directly applicable since $\odot \equiv +$ when both operands are non-negative numbers. 
+
+If both $x$ and $y$ are non-negative and $x >= y$, then $(x-y) \odot y = x - y + y = x$, where the first equality is true since binary addition of two positive number is simply addition. Applying $\odot \, (-y)$ to both sides we get $(x-y) \odot y \odot (-y) = x \odot (-y)$, with the left-hand side simplifying to $x - y \odot 0 = x - y$, satisfying the equality.
+
+*The rest is left as an exercise to the reader.*
+
+[^binary]: The subscript $2$ in $0101_2$ means this is a binary number.