Learn English – What do I call the ′ in mathematical formulae

The precise name of a symbol in mathematics sometimes depends on what you're using it for. For example, × is often referred to simply as the multiplication sign, but if you need to distinguish scalar from vector multiplication, you might refer to it more specifically as the cross multiplication sign, vector multiplication sign, or something similar.

In your second example, the ・ symbol is called dot. The product of two numbers multiplied using the dot operator is the dot product. In some contexts, you might call it the scalar multiplication sign.

Outside of programming, we usually only use * for multiplication when we can't type ×. The * symbol is usually called the asterisk or star, though if you're using it as a multiplication sign, you might call it that, instead.

The last example appears to be using a period. Like *, I assume this is simply because they couldn't type ・. I could call this a period, but more usefully, I could call it whatever it represented: in this case, I might call it dot, since it's standing in for ・.

• The absolute value of the difference between S and
• the sum from i equals one through i equals n of
• the function f evaluated at t sub i times the width of each i
• is less than epsilon.

If it is clear that i and n are one-indexed, then "the sum from i equals one through i equals n" can be replaced by "the sum of the first n terms". "The width of each i" is an interpretation of "delta i".

• The function is one divided by the quantity x plus one close quantity, all divided by the square root of x.
• The integral from zero to infinity of the function d x
• equals the limit as s goes to zero of the integral from s to 1 of the function d x
• plus the limit as t goes to infinity of the integral from 1 to t of the function d x.

"Goes to" can be replaced by "goes toward", or (as Damkeng suggests) "tends to", or (as J.R. suggests) "approaches".

I often use a notation like "integral from x equals a to x equals b" instead of "integral from a to b". I also often say "to positive infinity" instead of "to infinity".

I "factored out" the definition of the function in the first sentence. If the function were easy to say (such as "x squared"), I would not "factor out" the definition of the function. Instead, I would include the "x squared" in the statements of the integrals, a la Damkeng's answer.