The Difference Between Squares
Insights from the number line
*******************************************************************I did ok in maths when I took it in highschool. From what I can tell though, new stuff keeps getting figured out. And I can’t find where to download this years update. This article is the beginning of a series, Papers & Pictures, where I track my learning progress on things like particles, black holes, abstract algebra and a bunch of other stuff.
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Papers & Pictures Season : 1
Episode : 1 : The Difference Between Squares
Episode : 2 : The Summing of Squares
Episode : 3 : Find π For Yourself : Pi Part-1
Episode : 4 : …Another Slice of π : Pi Part-2
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😴 TL;DR 🥱
The difference of two squares (n+m)² - (n)² ; can be simplified as m(2n+m).
You’ve probably seen it as (a-b)(a+b) or just a² - b²
😪 TL;DR 🙄
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I was playing 🎮 around with an equation and wanted to do some quick calculations in my head 🧠. Sometimes pulling out a calculator and punching in numbers can take longer than just solving the problem in your head real quick and moving on.
So I listed the numbers from 1–10 in order.
Then squared ◼️ each number and wrote the result under.
I couldn’t help noticing a pattern though 👀.
When I looked at the list of squared numbers they looked like they were incrementing regularly. So I went in order and subtracted each square by the square that came before it.
😳 And out popped the odd numbers!?!? What?
To generalize the process in an equation I wrote out :
(big #)² - (small #)² = Answer
(3)² - (2)² = 5 and (4)² - (3)²=7 and (5)² - (4)²=9 and etc…
Noticing that the (big #) is going to just be (small #)+1 we can write out the equation above as :
(small # + 1)² - (small #)²= Answer
And to make it easier on ourselves while we’re moving stuff around let’s make (small #) = n
So our equation looks like this now :
(n+1)² - (n)²=Answer 👍🏾
This might not look any simpler than where we started but sometimes things need to get a little messy before we can clean up.
To simplify this we solve for (n+1)² :
(n+1)² = (n+1)(n+1) = n² +2n+1
Plugging this back in we see that we’re starting to get somewhere!
Any number subtracted from itself is always 0.
Our original equation simplifies to from n² + 2n + 1 - n² = Answer
🎉 2n+1=Answer 🎉
Now we can say that :
(n+1)² - (n)² = 2n+1
So if you have to find 81² - 80² you don’t even have to solve for the squares! You can double 80 and add 1! or even bigger! 223² - 222²
😎 Some math you can do in your head! 🤓 Now go impress your friends!
If you’re still following we’re about to go deeper.
Cut can we generalize even further? 🧐 Say if we wanted to take the difference of squares where one number was actually 3 away from another, or 5 away or 7?
We’ll have to bring back our equation from earlier to build on our understanding. (big #)² - (small #)²= Answer
So now our (big #) is going to be (small # + some #) and we’re going to write our formula like this :
(small # + some #)² - (small #)² = Answer
We’ve already decided that we can write (small #) = (n). Let’s do the same thing with (some #) and make that equal to (m). So (some #) = (m) and :
(n + m)² - (n)² = Answer
And solving for (n+m)² :
(n+m)² = n²+2nm + m²
Plugging this back into our formula and simplifying :
n² + 2nm + m² - n² = 2nm+m² = m(2n+m)
While a little more complicated, with a little practice, you’ll surprise yourself by how quickly you will be able to compute these values in your head. 🌞
🤖 m(2n+m) or really | m(2n+m) |
You may be thinking to yourself “Who Cares?!?!” I’ll still get the computer in my pocket 📱to calculate this for me! Lol, most of the time I will too 😂.
What I think we’ve found here is not just an answer to a² - b²=c² ,
oops 🥸 (big#)² - (small#)² = Answer , but a way to visualize all the answers at once. ⚛️
If you’ve made it this far THANKS! I’m new to this so I’d love the feedback!
You probably noticed that we never talked about the odd numbers pattern or what happens if instead of squaring we cube or go higher. Something for next time!
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Next Article
The Summing of Squares👉🏾
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