Midpoint between two real numbers

Question

Well, it's quite clear that if I try to count mean average of any two numbers, I always get the midpoint. Is it really the whole intuition behind this concept? Or is there something hidden behind this that proves why this really works.

Answer 1

I think that it depends on your definition of "mean average."

If you intend the definition to be "Arithmetic mean," yes, the midpoint formula is clearly sufficient because it is also the definition

Alternatively, if you want to look at the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x)=x$, then the mean will be defined by: $$\frac{1}{b-a} \int_{a}^{b}f(x)dx$$

then the mean is: $$\frac{1}{b-a} \frac{1}{2} (b^2-a^2)=\frac{1}{b-a} \frac{1}{2}(b+a)(b-a)=\frac{b+a}{2}$$

As you might have expected, this is still correct.

I suppose it depends on which definition you want to start with. This idea is in some sense the "Continuous" version of arithmetic mean. There are many other senses of "mean" in mathematics.