Is mental math really related to intelligence?

>>8742316
Mental math is about two things: How much information can you retain, and how good you are at optimizing your thoughts. I call this optimization "IQ Jumping".

Here is a good mental math problem for you to practice. Don't write down ANYTHING. Try to reason entirely in your mind. With this example it will be clear that retention and IQ Jumping are the two main characteristics of intelligence.

Let f be a function defined on all real numbers such that f(x+y) = f(x) + f(y) for all real numbers, x and y.

Level 1: Prove that f(0) must be equal to 0
Level 2: Prove that f(-x) = -f(x), for all x
Level 3: Prove that if f(1) = k, then f(n) = kn, for all natural numbers n
Level 4: Prove that if f(1) = k, then f(n) = kn for all Integers n.
Level 5: Prove that if f(1) = k, then f(n) = kn for all Rationals n.
Level 6: Assume that f(x) is a continuous function and that f(1) = k. Prove that f(n) = kn for all real numbers
Level 7: Prove that f(x) must always be a continuous function

For Level 1 all you have to do is retain information. You have to keep in mind that f(x+y) = f(x) + f(y) and use that to reach the conclusion. Simple.

For Level 2 there are two ways. You could retain all the information. Or you could first retain only the hypothesis, prove a result from it, then forget about the hypothesis and retain only your result and then prove the conclusion.

I call this "IQ Jumping". It is when you optimize your brain processes by retaining only the newest information and forgetting about everything else. It is as if you are jumping from point to point.

Level 3 I believe can also be done without IQ Jumping, but IQ Jumping is very important.

Level 4 the same.

Level 5 is, I believe, when it gets interesting. I do not believe that it is possible for a human to retain that much information. So from Level 5 and onwards you HAVE to do IQ Jumping.

>>8742316
>Level 7: Prove that f(x) must always be a continuous function

Oh wait. Correction:

Assume that f(x) is continuous at x=0. Then prove that f(x) must be a continuous function.

>>8742316
No

>>8742341
that was an interesting post. Doing problems like this is odd and sometimes I found myself in loops of thought for a few minutes. But I am not sure if my results are rigorous.
for 6 I said that the real number is a finite or infinite sum of rational numbers (for example by splitting up the decimal representation into a sum) and then doing the procedure of breaking up the fuction, applying what was proven in 4 and then getting the result of 6.
also is this function necessarily linear?

>>8742490
For 6 you can work better with series.

You can show that there is a sequence of rational numbers that converges to any irrational number. Then you use the sequential criterion of continuity to show why this is enough to know that it holds for irrational values as well.

>is this function necessarily linear

Actually yes. If you prove all of this then you can show that f has to be linear.

>>8742341
Was all this just an excuse to show off what you learned in RA this week?