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I know this may be a straight forward question, but I looked around on google for about 20 minutes to find nothing. I am currently a year 8 student and came across a problem where I have to find the highest number using the four following numbers: $9, 5, -8$ and $-2$. I could use any mathematical I wanted to use, so I just used factorials. I came up with $(9! \times 5!)\times(-8x-2)!$, then, I came up with $5!^{9!}$ and then $9!^{(-8 \times-2)!}$. It may come up in a test and I didn't demonstrate this in class. Does this equation work and is there a higher number I could get with mathematical notation I don't know about? (Note - I can't use infinity.) Thanks.

M47145
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Syzcai
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    If you are using factorials, why stop at just one? E.g., $(5!)!$? – Clayton May 21 '16 at 02:33
  • I was thinking about that. I just didn't want to make it over excessive and keep it clean :). It's not based around the highest number I don't think, I think it's based around creativity. But I will try it, thanks! – Syzcai May 21 '16 at 02:35
  • You can factorial a natural number and can then raise the result to a power as you have done. Example: $(3!)^2= 6^2=36$. – jdods May 21 '16 at 02:35
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    I see. My guess is that your teacher would want the mathematical symbols restricted to $+,-,\times,\div$ otherwise you have a never-ending sequence of steps as I mentioned above. You would still be allowed to use exponents as it doesn't introduce any new notation (just how I imagine the problem, though... I might very well be wrong). – Clayton May 21 '16 at 02:37
  • Thanks! Does a factorial of multiple powers work too? As shown in the second equation I made? – Syzcai May 21 '16 at 02:38
  • Certainly, you can raise one factorial to another; they are numbers after all. – J. M. ain't a mathematician May 21 '16 at 02:39
  • @Clayton , Its about demonstrating your knowledge of mathematical notation and coming up with the best idea. My first equation (9! x 5!)x(-8x-2)! was the best answer in the class because other members of the class only used +,−,×,÷. – Syzcai May 21 '16 at 02:40
  • @RecklessReckoner They did not specifty, I think they just assumed everyone was going to use basic arithmetic and did not know about other approaches. – Syzcai May 21 '16 at 02:43
  • I withdrew my earlier remark as precipitous. If you're going to "step up" to exponentiation, though, I think $$ 5!^{8!^{[(-2) \cdot (-9)]!}} \ $$ is even bigger... – colormegone May 21 '16 at 02:47
  • @RecklessReckoner Thats exactly what I tried to write man :). – Syzcai May 21 '16 at 02:50
  • As many of the responders show, this is a puzzle-game where some definite "boundaries" on permissible operations need to be set. – colormegone May 21 '16 at 02:51
  • With the change you made, you can't change its whole number value, as you've done to 9 in (−2)⋅(−9)]!. – Syzcai May 21 '16 at 02:51
  • @RecklessReckoner As I mentioned in previous comments, I don't think they expected anyone to go beyond basic arithmetic. – Syzcai May 21 '16 at 02:52
  • Oops, I forgot where the minus sign was "attached": so $$5!^{9!^{[(-2) \cdot \ (-8)]!}} \ \ , $$ which I wrote before carelessly "rethinking" it. – colormegone May 21 '16 at 02:55
  • @RecklessReckoner Its all good man, thanks heaps, that's what I'll write in an exam if I need to use it again! – Syzcai May 21 '16 at 02:56

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yes you can include factorial in the exponent. but simpler would be $9!!!!!!! $ or $9^9^9^9^9^9^9$ or even further forms to make numbers bigger see https://en.m.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

shai horowitz
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  • Thanks! This would work but if you do 9^9^9^9^9^9^9 you're just using the same number over and over again, which you can't really do, you can only use one of each number :) – Syzcai May 21 '16 at 02:46
  • ah I didn't see that part try 9 uparrow uparrow uparrow uparrow uparrow 5. it's far larger then anything computable. i.e. even if we wrote a single digit on every atom in the universe we wouldn't have written down the number. – shai horowitz May 21 '16 at 02:49
  • probably can't even say for sure how many digits there are even though it's a finite number – shai horowitz May 21 '16 at 02:52
  • Thanks man, I think that might be a more simple way of writing it. – Syzcai May 21 '16 at 02:55
  • (-8)! is a weird thing that could debatebly be larger but then it would be infinite rather it's any of a specific infinite set of natural numbers – shai horowitz May 21 '16 at 03:17