For Math Nerds, On 3.14 Day: Why I Love Perfect Squares
I’m getting old now, and my best math days are behind me. Yet not to sound like Uncle Rico, back in my day, I was a huge math brain. I did all the trig and calculus classes in high school. Was on Math Team and Academic Challenge. Took the AP Calculus exam and “CLEP-ed” out of a bunch of college math classes. I was the youngest student in my college freshman math class.
I give all glory to God for that, because as John the Baptist said, “A man can receive only what is given to him from Heaven”. And I also give credit to Ms. Nancy McFaddin, and Mr. Wendell Robinson, my math teachers from 6th to 12th grade. Because before them I was a struggling student with low self-confidence. But as a teenager, there was no doubt what my reputation was: the math nerd.
I started college intending to be an actuary or some type of engineer. But God had other plans for me and after running from Him a couple of years, I ended up in Bible College studying to be a pastor. So that’s what I’ve done for 20 years. As such, my ability to do the highest levels of math has atrophied. I couldn’t solve a function with a limit if my life depended on it. As proof, I’m not even sure if that is the correct terminology (though I do remember talking about limits approaching infinity). 1997 Gowdy would be ashamed of 2022 Gowdy!
However, there is one aspect of math that I still adore to this day, that I use every single day of my life: arithmetic. In high school, I could do complex calculations in my head. Now, I can no longer do that, but numbers still fascinate me.
Statistics especially. I don’t want to hear that a basketball player averages 8 assists per game; I want to know if it’s 8.1 or 8.2 or 7.9. I know off the top of my head that 12/17 is 70.6% because I’ve figured it up as a quarterback completion % many times. I track statistics for Rambling Ever On articles regularly. I always want to know which articles have been viewed 1,000 times, if articles in a series trend up or down in terms of views, and all manner of other numbers-related information. I enjoy using my phone calculator. Or any calculator! It’s just how God created me.
And today I want to talk about a different yet equally enthralling subtopic under arithmetic: Perfect Squares. I remember memorizing them from 1×1 (1) to 25×25 (625) all those years ago, and I have continued to be able to cite them. Recently, they came back to me recently in a meaningful way.
While watching the show Fringe, genius Walter Bishop comments at one point that to battle insomnia he cited the Fibonacci Sequence in his head. Which is when the next number is the sum of the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34….and so on. This sequence gets into very big numbers very quickly so I decided when I had trouble sleeping, I would try to cite something more doable: the perfect squares.
It actually has worked quite frequently. Typically when I get to 81, or 256 or 361, I start nodding off. Yet there have been a few times it has not worked. But in those moments I have learned things about perfect squares I’ve never thought of before.
For example, as I thought of 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 a few times, I realized there is a symmetrical up and down or “mirror” pattern with the last individual digit, that continues on forever. The last digit will always go 0-1-4-9-6, peak at 5, then come back down just as it came up, 6-9-4-1-0. You can see it clearly in the first sentence of this paragraph.
But if that were all that I learned I doubt I’d be writing. There were more patterns, and they blew my mind. A second I realized as I got to numbers bigger than 25 was that the last two digits have a symmetrical “mirror” pattern that peaks at 25. I would say in my head “24 squared is 576, 25 squared is 625 and then 26 squared is 676”. And I reflected: 24 and 26 squared have the same last two digits–76.
Since I normally stopped at 25 in my youth, I don’t think I’d ever considered this. Yet, one occurrence doesn’t make a pattern so I kept going. And it kept going. 23 squared is 529 and 27 squared is 729. Same last two digits. 22 squared is 484 and 28 squared is 784. Again, same. And it keeps going all the way to 0 on one side and 50 on the other.
In fact, remember the first ten perfect squares: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81. Now look at the perfect squares from 41 to 50: 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500. Look at the last two digits of each one, and bask in the glory of that symmetry! It happens with all numbers going down and up from 25. I just showed the first few and last few for the sake of space.
I confess this dazzles my brain. I’m pretty sure when I realized this the first time while trying to fall asleep, it had the opposite effect: It kept me awake longer because my brain was so entertained. You may as well have turned every light on and blown an air horn in my ear I was so awake.
Another way to say what I’ve said is that the numbers equidistant of either side of 50 are increasing linearly multiples of 100 away from each other, which is why their last two digits are the same (since + or – 100 doesn’t change the last two digits of any number, except when crossing zero). 576 and 676 are 100 apart. 529 and 729 are 200 apart. 484 and 784 are 300. And on it goes till you see that 0 squared and 50 squared are exactly 2500 apart.
This happens (and let us understand why it happens! That was an invaluable lesson from Ms. Nancy and Mr. Wendall) because of one of the most obvious patterns of perfect squares you can see even if you are not a math person: They are the next odd number apart as they get bigger. Meaning, 1 to 4 is a difference of 3. 4 to 9 is a difference of 5. Then, 7, 9, 11, and so on. When you get from 24 squared to 25 squared it’s a difference of 49. When you get from 25 squared to 26, it’s 51. That’s 100 when put together. And that pattern keeps going with 47-53, 45-55, etc.
Finally, I’ll add that after you get to 50 squared, the pattern starts over. 51 squared is 2601. 52 squared is 2704. 53 squared is 2809. 54 is 2916. 55 squared is 3025. Look at those last two digits each time!
Even if you are not a math person, I hope you can appreciate the aesthetic beauty of this. Math really is beautiful sometimes. We do not typically associate art, or beauty, with math. Hard science is the opposite of art, right? Not exactly. Because God created numbers for us to use, even to worship him (a whole Bible book bears the name!), they are beautiful.
So today as we celebrate math with the day that reminds us of Pi to two places, and while some use it as an excuse to eat pie, I advocate for a celebration of numbers in general. Feel free to help us celebrate in the comments section below!
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Speaking of statistics, 63.9% of all statistics are made up on the spot.