Interlocking numbers
If you enjoy crossword puzzles, you might like this post.
We all learned in school that, for all integers, if A/B = C/D, then AD = BC. For example: 2/3 = 4/6 then 12 = 12. No problem.
But what happens when AD <> BC (the <> means not equal)? What if they are very close. Does that mean A/B and C/D are close? Maybe. Let’s play with the numbers.
- Pick AD and BC so that they are close (i.e. n,n+1) Let’s pick a pair out of the air. AD=49 and BC=50.
- Then find a pair of factors. In out example: A=7, D=7, and B=10, C=5.
- Now find A/B and C/D. For us, we have A/B=7/10 and C/D=5/7.
For the example above, 7/10 = 0.70 and 5/7 = 0.71, which are indeed very close! The two fractions 7/10 and 5/7 are related in this way.
Let’s enumerate from the very beginning.
| AD | A | D | BC | B | C | A/B | C/D |
| 1 | 1 | 1 | 2 | 2 | 1 | 1/2=0.50 | 1/1=1.00 |
| 2 | 1 | 2 | 3 | 3 | 1 | 1/3=0.33 | 1/2=0.50 |
| 3 | 1 | 3 | 4 | 4 | 1 | 1/4=0.25 | 1/3=0.33 |
| 3 | 1 | 3 | 4 | 2 | 2 | 1/2=0.50 | 2/3=0.67 |
| 4 | 1 | 4 | 5 | 5 | 1 | 1/5=0.20 | 1/4=0.25 |
| 4 | 2 | 2 | 5 | 5 | 1 | 2/5=0.40 | 1/2=0.50 |
| 5 | 1 | 5 | 6 | 6 | 1 | 1/6=0.17 | 1/5=0.20 |
| 5 | 1 | 5 | 6 | 3 | 2 | 1/3=0.33 | 2/5=0.40 |
Do you see the pattern? Any rational number (i.e. fraction) is related to just a few other rational numbers thru this particular operation. Wonder if it builds up an interesting structure.
Neato!
jtalics 
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