# What is the difference between Hamming Distance and Manhattan Distance for non-binary data?

What is the difference between Hamming Distance and Manhattan Distance for non-binary data (specifically I am comparing points in $$\mathbb{R}^2$$)? I understand Manhattan sums the absolute difference in the and x and y directions but doesnt hammming distance do the same thing?

The Hamming distance between two length-$$n$$ vectors is the number of coordinates in which they differ. I've only ever seen it on finite alphabets, i.e. vectors in $$\Sigma^n$$ where $$|\Sigma|\in \mathbb{N}$$. In theory there is no problem with extending this to $$\mathbb{R}$$, but you may have to be careful with how you use equality of floats for instance.
Specifically over $$\mathbb{R}^2$$, if we consider $$x=(x_0,x_1)$$ and $$y=(y_0,y_1)$$: $$\begin{cases} dist_\textsf{Manhattan}(x,y)=|x_0-y_0|+|x_1-y_1|\\ dist_\textsf{Hamming}(x,y)=\delta_{x_0,y_0}+\delta_{x_1,y_1}\end{cases}$$
Where $$\delta_{a,b}=\begin{cases}1 \text{ if }a\neq b \\0 \text{ otherwise}\end{cases}.$$
The two coincide in $$\{0,1\}^n$$, because in that case we have $$\delta_{a,b}=|a-b|$$.