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My first grader is very advanced in math. Rather than doing more and more math and making school math even more boring for him, I recently decided to start going "deeper" rather than "faster."

Some of the questions we've explored recently are:

  • Why do we use a numbering system? (Because otherwise we'd need an infinite number of names)
  • Why is multiplication commutative? (Because you could switch the number of items and the number of people and you'd still need the same number of items in the 'rectangle')
  • What might math look like without the number 0?
  • Why are 2, 5, and 10 the only numbers whose divisibility tests only need the last digit? (because they're the factors of 10)
  • Why is a number divisible by 9 if its digit sum is divisible by 9? (becuase every power of 10 has remainder 1 when divided by 9)

Are there any resources I can use to find more questions/explorations like this?

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    $\begingroup$ Maybe you would get better answers at matheducators.stackexchange.com $\endgroup$
    – GEdgar
    Commented Oct 26, 2023 at 9:38
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    $\begingroup$ Speaking from personal experience, the book called "The Number Devil" by Hans Magnus Enzensberger could be a fun read for a math loving first grader. I loved that back when I was in early elementary school. It isn't very deep, but it does explore some cool topics. $\endgroup$
    – Arthur
    Commented Oct 26, 2023 at 10:01
  • $\begingroup$ "What might math look like without the number 0?" Apart from exotic rings , every ring has a null-element , so I do not think that we can do reasonable math without $0$ , not even if we work in a different really useful number system. In base $b$-systems where $b>1$ is an integer we need $0$ and other systems have little to no merit. $\endgroup$
    – Peter
    Commented Oct 26, 2023 at 14:25
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    $\begingroup$ @Peter That's a very...modern perspective. I would argue that a lot of good math was done for thousands of years (e.g. Babylonian approximation of $\sqrt 2$) before zero gained full-fledged number status. $\endgroup$
    – Mark S.
    Commented Oct 26, 2023 at 15:24
  • $\begingroup$ The number $0$ , perhaps , but they needed the digit $0$. $\endgroup$
    – Peter
    Commented Oct 27, 2023 at 16:03

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