Fractions! UGH!
You hear that all the time. But let’s have a little fun instead. First, check out these (almost exact) decimals for the fractions from 1/7 to 6/7.
| 1/7 = | 0.142857 |
| 2/7 = | 0.285714 |
| 3/7 = | 0.428571 |
| 4/7 = | 0.571428 |
| 5/7 = | 0.714285 |
| 6/7 = | 0.857142 |
To be precise, we have to say that these aren’t quite exact, because these are all repeating fractions. Their decimal expansions keep going forever. (Try it on your trusty calculator or spreadsheet – especially if they show a lot of digits.)
But that’s
NEAT THING #1:
Each fraction above repeats in the exact pattern already started . . . forever!
(For example 1/7 = 0.142857142857142857142857 . . .)*
* The ‘math’ way to say that exactly in a short space is to put a bar across the top of the repeating part,but the formatting here won’t allow that.
And then there’s
NEAT THING #2:
Look at ‘repeating part’ patterns!! Each ‘repeating part’ has the SAME digits as all the others, but in a different rearranged order!
That’s so cool, that experience shows that even 5th and 6th grader math students are amazed!
These patterns are beautiful in/of themselves (if you ask me), but it’s only been very recently that I’ve learned that there’s a very neat ‘cherry on the top’! Which leads us to
(Incredible Bonus Super-)NEAT THING #3:
Let’s ignore the ‘0.’ for each fraction and put the repeating parts into this square (well, OK, the formatting made it a rectangle, but . . .
| 1 | 4 | 2 | 8 | 5 | 7 |
| 2 | 8 | 5 | 7 | 1 | 4 |
| 4 | 2 | 8 | 5 | 7 | 1 |
| 5 | 7 | 1 | 4 | 2 | 8 |
| 7 | 1 | 4 | 2 | 8 | 5 |
| 8 | 5 | 7 | 1 | 4 | 2 |
It turns out (drumroll, please!) that EVERY row and EVERY column of this ‘square’ add to the same number (namely 27).
That’s not quite an official magic square (which are fascinating also), because there are repeated digits, and the two diagonals don’t each add to 27, too.** (Also, it’s often true that magic squares use consecutive digits in their make-up, but that’s not required.)
** NEAT THING 3.5: The two diagonals together DO add to 54 (twice 27!), which is suspiciously neat (there ‘should’ be a reason?!)
SO . . . working and calculating with fractions themselves can be tedious, but – like humans – don’t judge a book by its cover! Fractions themselves can often have beautiful patterns and interesting facts!! (For another example, check out these fun facts.)