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It is important to distinguish between axioms (things you assume), definitions and theorem (things you prove from axioms, stated in terms of definitions). So, for basic arithmetic, your assumptions might be the defining properties of numbers: associativity, distributivity, etc. In this setting, 1 + 1 = 2 is basically a definition, while (-1)* (-1) = 1 is something to be proved:

    0 = 0* (-1) (0* x = 0 for all x)
      = (1 + (-1))* (-1) (x + -x = 0)
      = 1* (-1) + (-1)* (-1) (distributivity)
      = (-1) + (-1)* (-1) (1* x = x for all x)

    1 = (-1)* (-1) (equality; add 1 to both sides)
Finally, slope = rise/run is a definition. It is not something you "see" to be true.

So assumption vs. definition vs. theorem can explain part of it. Another part is familiarity. 1 + 1 = 2 is ubiquitous in daily life. Depending on what you think about, (-1)* (-1) = 1 may not be. You probably have lots of interpretations of 1 + 1 = 2 (counting, number line, shopping, money, etc.) How many such interpretations of (-1)* (-1) = 1 do you have?

I highly recommend this book, written by two mathematicians:


It's now about mathematics per se, but about learning how to learn and think better. Good luck.

Awesome! Really liked the way you proved (-1)(-1)=1.

What I am doing is trying to relate every single new thing I am learning with real life examples, only to make sure that I have learned* that thing. Most of the times I fail at that. But I guess it's not always possible.

That may not always be possible (or even necessary) to relate to a real-life example, but as long as you develop some sort of intuition, of "feeling" about what you learned, you did not fail at learning that.

For example (taken from the article I mentioned on the other topic), when you think of a derivative, what is the first thing that comes to your mind?

The equation that defines it? (http://calnewport.com/blog/wp-content/uploads/2008/11/deriva...)

Or this intuitive image? http://calnewport.com/blog/wp-content/uploads/2008/11/tangen...

I am yet to tackle Calculus, so nothing comes to mind. But I am agree with you that, relating all of math concepts to real life won't be necessary as long as I truly understand it.

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