Without going into why I was doing so, today I had the "opportunity" to answer the question, "What is the square root of 523,457?" And I learned some mathematical truths by trying to answer this question.
I began be trying to factor it down to prime numbers -- but that left me with 11 x 23 x 2069. Then I tried factoring down an approximation of the original number, in this case, 523,456 = 8 x 8 x 8179. 8,179 is a prime number.
I finally came at it from the answer instead of the question, homing in on the answer I needed. The quick version is as follows, with the square root of 523,457 represented as x:
700 < x < 800
720 < x < 730
723 < x < 724
Anyway, the point of going on about this isn't to figure out the square root of this odd number, but to share something I discovered (for myself, not for the mathematical world at large, which surely has already figured this out). The original number, 523,457, is obviously not a perfect square because perfect squares never end in 7. I'd never realized the ban on sevens before, but it makes sense when you consider how you multiply multi-digit numbers.
Looking into this further, I realized that perfect squares always have to end in 0, 1, 4, 5, 6, or 9. I discovered this by squaring each of the first 10 numbers.
But I also discovered a pattern that I hadn't noticed before. Looking just at the last digit, if you start to list the perfect squares in order, those final digits follow the pattern 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, ... I find not only the existence of the pattern intriguing, but also the fact that it's a symmetrical pattern with an intervening zero.
There's also a pattern to the rest of the perfect squares. If you drop the final digit of each perfect square, there is a stepwise addition pattern that shifts up whenever you reach a perfect square that ends with a 9. You'll have to look at it yourelf, but basically, you start at 0. When you hit 9, you start adding 1 to get each of the next perfect squares. When you hit the next perfect square that ends in 9 (49), then you start adding 2 to get the first digit of each number.
I know, most people will just think "so what?" But for me, I get a great feeling from discovering a truth for myself. I also love logic puzzles, and this is something of that sort. Now that I've figured this out, I'll have to try to find the underlying logic of the decimal system that makes this so. And I'll have to see if there is a similar pattern for perfect cubes.
5 comments:
This post couldn't have come at a better time! My professor gave us a bonus quiz regarding a very large number, and told us it was not a perfect square. We had to answer "Why?". Thanks for the insight!
If you want some brown-nosing extra credit, ask your professor if (s)he listens to Car Talk on NPR. The question your professor asked was the weekly puzzler on Car Talk this same week. Not only did you solve her/his puzzle, you solved her/his source.
You were talking about the first couple digits stepping up. I have found this also but If you look carefully it does it every 6 and 4 numbers, alternating back and forth. But the first six numbers start on the negative end of the number line. Not including zero of course.
First of all, great job! Whenever someone discovers something by him/herelf, they always remember it for a long time.
If you are willing to take say 10 minutes out of today, you will learn WHY it is symmetrical.
It starts with something called modular arithmetic (read up on number theory, you will be intrigued!).
When you read up on it, you will find that x = x - 10 (mod 10)implies x^2 = x^2 - 20x + 100 = x^2 (mod 10).
If you do not yet understand number theory (elementary N. T., then you probably will not understand the proof above).
As for perfect cubes...
The pattern you observed will NOT work in general (again N.T. will verify this as easily as before). It will turn out to be congruent to the negative of the number. Actually, this generalizes to base anything (except for the trivial "bases", 0 and 1). To be more mathematically precise, it's mod anything (modular anything if you prefer).
Another interesting result is that through number theory (specifically modular work), you can show (easily) that the period with respect to a given mod and a given power is at most the mod n (n is the designated mod). I.e. Note how 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0 has cycles of ten before repeating. Hmm... I've seen the word ten before in this problem. AHA! we're dealing with base ten and powers repeats in cycles of ten! Could this be a coincidence?!
It turns out that this result (again easily provable with elementary number theory) generalizes to any base.
Just some food for thought. But great job again for figuring out these things by yourself!
You couldn't have said it better
And plus i had to figure out a pattern for homework today so thanks!!!!
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