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I'm aware of other questions regarding "increasing" mathematical intuition, but I wanted to explain my own context, since that way any advice posted on this question feed will be specific to myself.

I'm 16. I'm a (new) Sixth Former (roughly college level) student and has just today had my fifth Further Pure Mathematics lesson. When I entered the classroom there were two questions on the board, one of them was the following:

If $s$ and $r$ are integers, when will

$$\frac{6^{r+s}\cdot12^{r-s}}{8^r\cdot9^{r+2s}}$$

definitely be an integer?

There were then $4$ options to pick as an answer which I cannot all recall, but I do remember how the answer was attained:

$$\frac{3^{r+s}\cdot2^{r+s}\cdot2^{2r-2s}\cdot3^{r-s}}{2^{3r}\cdot3^{2r+4s}}$$ $$\frac{3^r\cdot3^s\cdot3^r\cdot3^{-s}\cdot2^r\cdot2^s\cdot2^{2r}\cdot2^{-2s}}{2^{3r}\cdot3^{2r}\cdot3^{4s}}$$ then, after cancelling the like terms; $$\frac{3^{-s}\cdot2^{-s}}{3^{3s}}$$ $$3^{-4s}\cdot2^{-2s}$$

therefore, the answer was (and after all this, quite obviously) "when $s\leq0$". The issue is, when originally sitting down, I did not come to this conclusion myself. Mind you we did not have much time to solve this problem, but even so; someone studying further mathematics should be able to prove simply and rather easily why this answer is what it is. From my understanding of what the term intuition means to mathematicians, I can see why it is so important to those working on rigorous research tasks; intuition is an important key in utilising knowledge effectively. Hence, my question is: how can I improve my own intuition, if at all? I've heard before that intuition comes simply from experience, but I don't like to believe that I'm destined to lack the desired intuition until I've come of age, mathematically.

Any responses are appreciated, thank you.

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  • $\begingroup$ It would improve intuition to consider a special case first, say, $s=r$. Then the fraction is $\frac{6^{2r}}{8^r9^{3r}}$. Then we see immediately the answer. $\endgroup$ – Dietrich Burde Sep 12 '17 at 13:06
  • $\begingroup$ You should have learned that $a^{-n}$, for n positive, is the same as $\frac{1}{a^n}$. For any integers $3^{-4s}= \frac{1}{3^{4s}}$ if s is positive. If s is negative, then we can write s= -m for positive me and $3^{-4s}$ becomes $3^{-4(-m)}= 3^{4m}$ with positive power so it is an integer. $\endgroup$ – user247327 Sep 12 '17 at 13:09
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    $\begingroup$ Learn about problems and ways to solve them. The more problems you explore how they behave and solutions you learn your brain will start poking you when you see something that is "similar enough" to something else you have seen earlier in life. $\endgroup$ – mathreadler Sep 12 '17 at 13:09
  • $\begingroup$ @user247327 I'm aware of the index property you listed, but that's exactly the question, if I myself wasn't intuitive enough to be reminded of the link between the question and that property, how can I train myself to be intuitive enough next time I encounter a similar question. $\endgroup$ – joshuaheckroodt Sep 12 '17 at 13:12
  • $\begingroup$ The following assumes your main intention is to increase your math proficiency (or closely related attribute) and you see increasing intuition as means to that end. Focusing on intuition is maybe not the most productive way to become more proficient in math. Instead focus on becoming proficient. It may be the case that you never become as intuitive as you hope, but you still become highly proficient. On the other hand you may notice your intuition increases alongside your proficiency - almost as a side effect. But focusing on intuition may yield great intuition, but not the proficiency u want. $\endgroup$ – Χpẘ Sep 12 '17 at 14:22
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I don't think there's an answer to this question that doesn't involve doing lots of mathematics.

What I find helpful is, in addition to learning new math, to go back and re-learn or somehow review math that I already know. This can be done by teaching, or by reading over texts a little bit below my own level. That might sound boring, but if it's an area of math that you find interesting, then seeing how some author covers it isn't so bad, and you can move a lot more quickly than when you are reading to initially learn. When I do this, I very commonly see something in a new way, just from having a slightly elevated perspective.

On the flipside, attempting to read something beyond your level can be of value, even if you only make it a page or two into a book. It sometimes seems that simply gazing at the page in bemused wonder can generate a spark, maybe in the moment, and maybe later. I wouldn't spend too much time doing this, though.

It really would be hard to overstate the value of teaching and tutoring, as far as deepening intuition. If you have friends taking classes you've already passed, for example, make yourself available to help them. You might end up benefiting from the interaction as much as, or more than, they do.

Additionally, reading recreational mathematics is useful. Doing logic puzzles is useful. Drawing or doodling geometric figures very carefully is helpful. Even something as trivial as doing mental arithmetic that arises in daily life can end up improving your intuition about algebra.

Ultimately, your mathematical intuition is likely to be related to the number of hours you spend simply sitting with mathematical thoughts in your mind. Ten hours is worth something; a hundred hours is worth more; ten thousand hours is getting somewhere!

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  • $\begingroup$ Just to add one detail - I think I have a decent intuition when it comes to working with matrices. The reason I have whatever intuition I do have is largely to do with the fact that, somewhere in my past, there are multiple notebooks completely filled with row operations. Doing it until I was bored with it, discovering my own shortcuts, coming back a year later and doing it again, but from a different book..... that's where I built that intuition. $\endgroup$ – G Tony Jacobs Sep 12 '17 at 16:19
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The reason piano students do scales (hours and hours of scales and Hanon exercises....you have to SuperGlue your butt to the bench and go until your fingers bleed) is to develop "familiarity with one's instrument." If you watch a trained pianist, you'll see him do amazing things. His head can be looking right and his left pinkie finger will shoot far down the keyboard and hit the right note and then shoot back toward the middle and again hit the right note 1/10 of a second later. Thanks to the hours of tedious finger exercises, he knows exactly where each key is. Also, the movement between every pair of keys is subtly different, and he knows instinctively each movement.

When we switch topics to math, then "scales" become "rote" and we roll our eyes and act like "rote" is the dumbest thing on the planet. But the plain fact is, as you quote, "intuition comes with experience." Which is a different way of saying "rote." By working lots and lots of "just plain" problems, you build familiarity with your tools. Intuition doesn't come all at once, but will slowly increase.

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I think that particular question isn't a good one to trigger the need for developing intuition, but yours is a good question.

I strongly recommend George Polya's Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy in Mathematics . I first encountered it when I was at about your age.

http://press.princeton.edu/titles/1735.html

http://uberty.org/wp-content/uploads/2016/01/Polya_Mathematics_and_Plausible_Reasoning1.pdf

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