How Do Symmetrical Base n and Base m Decimals Relate?

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Discussion Overview

The discussion revolves around the relationship between symmetrical base n and base m decimals represented by two fractions, F1 and F2. Participants explore methods to solve equations derived from these decimal representations without resorting to trial and error. The focus is on mathematical reasoning and problem-solving techniques.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses the relationship between F1 and F2 in base n and base m, providing equations for both fractions based on their decimal representations.
  • Another participant questions the introduction of a variable 'y' in the equations and reports difficulties in solving the equations without it.
  • A different participant highlights the symmetry in the numbers involved and suggests that this symmetry may provide insights into solving the problem, while also noting the lack of progress in finding a solution.
  • There is a call for assistance in solving the simultaneous equations presented, indicating a need for further clarification or guidance on the method.

Areas of Agreement / Disagreement

Participants do not seem to reach a consensus on how to solve the equations simultaneously, and there are differing opinions on the approach to take, particularly regarding the significance of the symmetry in the numbers.

Contextual Notes

Some participants express uncertainty about the method of solving the equations and the implications of the symmetry, indicating that assumptions about the values of n and m may affect the outcomes.

jmich79
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When fractions F1 and F2 are written as base n decimals, F1=(.7373)base n and F2=(.3737)base n. In base m, F1=(.5252)base m and F2=(.2525) base m. Find (m+n). No trial and error please.
 
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Saying "F1= (.7373) base n" means F1= 7/n+ 3/n2+ 7/n3+ 3/n4. Similarly, saying "F1= (.5252) base m" means that F1= 5/m+ 2/m2+ 5/m3+ 2/m4. Putting those together you know that 7/n+ 3/n2+ 7/n3+ 3/n4= 5/m+ 2/m2+ 5/m3+ 2/m4. Likewise, F2= (.3737) base n and F2= (.2525) base m gives 3/n+ 7/n2+ 3/n3+ 7/n4= 2/m+ 5/m2+ 2/m3+ 5/m4. That gives you two equations to solve for the two unknown numbers m and n.

Edited to replace "y/n3" by "7/n3". My finger slipped!
 
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where are you getting y/n^3 from
I tried doing this method without the y variable and was not able to solve for it. I came up with some crazy nubers. Can you please explain further?
 
dOES ANYONE KNOW HOW TO SOLVE THIS EQUATION SIMULTANEOUSLY?
 
It seems to me that you're supposed to spot the weird symmetry in the numbers here. '7373', '3737', '5252', '2525'. HallsofIvy outlines a general method of solving this problem when the given numbers have no special pattern, but fk only knows how you'd solve those simultaneous equations!

My thoughts. Try adding F1 and F2 in base m and in base n. Assume, just to make it easier, that n>10 and m>7 (if that's not the case I'm sure it'll turn up in some contradiction or other)


(eta: well, I'm not making much progress... but I still think that it's important to note the symmetry here. It means something, I'm sure of it.)
 
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