When do you need a dollar symbol before an arithmetic statement in bash?

Question

I am trying to get my head around bash but am struggling with some core foundations.

I can see an arithmetic statement is defined by double parentheses

((5-3))

but if I try to echo that or us it in shift I get an error

echo ((5-3))
-bash: syntax error near unexpected token `('

In examples it shows that prefixing it with a dollar symbol will make it work

echo $((5-3))

Why is that? The syntax looks as though I'm applying the arithmetic to a blank variable and then onto the echo function. Why can't I just pass the result of the arithmetic into the echo function? How do I know when to prefix an arithmetic statement with a dollar symbol in other instances.

Please excuse my ignorance. Im so used to web development languages and Bash is just so different.

Answer 1

((...)) is an arithmetic expression.

$((...)) is an arithmetic expansion.

The difference is that an expression is a unit on its own but an expansion just gets replaced with a value.

This is the same thing as the difference between.

$ echo foo
foo

and

$ $(echo foo)
-bash: foo: command not found