# Confused about XORing and addition modulo $2$

It's my understanding that when you XOR something, the result is the sum of the two numbers mod $$2$$.

Why then does $$4 \oplus 2 = 6$$ and not $$0$$? $$4+2=6$$, $$6%2$$ doesn't equal $$6$$. I must be missing something about what "addition modulo 2" means, but what?

100 // 4

010 // XOR against 2

110 = 6 // why not zero if xor = sum mod 2?

The confusion here stems from a missing word. A correct statement is "The result of XORing two bits is the same as that of adding those two bits mod 2."

For example, $(0+1)\bmod 2 = 1\bmod 2 = 1=(0\text{ XOR }1)$

and

$(1+1) \bmod 2= 2\bmod 2 = 0 =(1\text{ XOR }1)$

The xor of two one-bit numbers is their sum modulo 2. But the xor of two $n$-bit numbers can't possibly be their sum modulo 2: any value modulo 2 is either zero or one but the xor of two $n$-bit numbers could be anything between 0 and $2^n-1$, inclusive.

You are confusing operations on a single bit with operations on a byte,or word.(Multiple bits) A single bit represents either 0 or 1 depending on its value. If you add two bits, and ignore the carry, you are adding "mod2". 0+0 = 0 0+1 = 1 1+0 = 1 1+1 overflows, or carries, and you have 0

this is exactly the same as XOR

However it is NOT true for words. Consider a nibble (4 bits) 0110 + 0111 = 6 + 7 = 13 = 1101 0110 xor 0111 = 0001 = 1

I think you're misunderstanding the property.

I believe that property is more concerned with the boolean result of the XOR function, not the numeric value. What I mean is that each "number" you're adding is actually just a boolean value of True (1) or False (0) and adding those together, mod 2, is the same as XOR of those values.