0 upvotes, 5 direct replies (showing 5)
View submission: Ask Anything Wednesday - Engineering, Mathematics, Computer Science
Hello, this is very weird, but I wanted to know how maths formulas work. I found a "formula", i found it from 5 * 6 - 2 = 4 * 7 ; 9 * 10 - 2 = 8 * 11. It would be a(a+1)-2 = (a-1)(a+2). I call it a -2 formula. I wanted to know how it's different from a^2 - b^2 = (a-b)(a+b) that we learn in school. Everybody tells me mine isn't really a formula. I know it cancels out fast, but every formula does, doesn't it? Why is it different?
Comment by UWwolfman at 26/06/2024 at 16:19 UTC
3 upvotes, 0 direct replies
The formula you discovered is fine. There's nothing wrong with it. There's ways to generalize it. For example what happens if you try -6 or -12.
At the same time, the expression a^2 - b^2 arises frequently in math and science. So learning the trick to factoring it can be very helpful. While your formula may have uses, and it's interesting, it doesn't arise nearly as frequently. There's nothing wrong with that.
Comment by Polaric_Spiral at 26/06/2024 at 16:35 UTC
3 upvotes, 0 direct replies
Your "formula" is, in fact, a formula.
More so than testing a couple of examples, you can quickly verify it with a bit of algebraic manipulation.
(a - 1)(a + 2)
a^2 + 2a - a - 2
a^2 + a - 2
a(a + 1) - 2
A formula is just a generally useful mathematical shortcut, and can cut out as many or as few steps as you like. If there's an application for it, you're free to use it, but something like this comes up infrequently.
Generally you'd just get from one side of the "formula" to the other via algebraic manipulation as above, since it's easier to jot down a few extra lines than to memorize and apply a gigantic set of generic algebraic formulas.
Comment by Letartean at 26/06/2024 at 16:27 UTC*
1 upvotes, 0 direct replies
a(a+1)-2 = a^2 +a-2=a^2 +2a-a-2=a(a+2)+(-1)(a+2)=(a+2)(a+(-1))=(a+2)(a-1)
So, yeah, the formula is right. You can get from one form to the other by using a factorization method (the trick is to add and remove something that helps in algebraically go from an addition to a multiplication by using the distributive nature of the multiplication on an addition; you have to create a form where something appears twice in multiplications separated by an addition (or a subtraction); a(b+c)=ab+ac; if you're unsure why it works try and start doing the algebraic moves from the other end). Now, is that formula very useful? Not really... Why do we learn a^2 -b^2 =(a+b)(a-b) and not your -2 formula? Most probably because in the grand scheme of things, you are more likely to face problems where getting from a^2 -b^2 to its factors is important. But, all in all, those two "formulas" are of the same ilk.
Note that you can do the same factorization trick with a^2 -b^2 .
a^2 -b^2 = a^2 -ab+ab-b^2 =a(a-b)+b(a-b)=(a-b)(a+b)
Comment by t4m4 at 26/06/2024 at 16:30 UTC
1 upvotes, 0 direct replies
(a-1)(a+2)
= a^2 + 2a - a - 2
= a^2 + a - 2
= a(a+1) - 2
--------------------------------------------------------------------------------
Generalization 1:
(a-1)(a+n)
= a^2 + n.a - a - n
= a^2 + (n-1)a - n
= a (a+(n-1)) - n
--------------------------------------------------------------------------------
Generalization 2:
(a-m)(a+n)
= a^2 + n.a -m.a -m.n
= a^2 +(n-m)a - m.n
= a (a+(n-m)) -m.n
--------------------------------------------------------------------------------
So, you see, in the end it's just algebra, really.
Comment by LanceWindmil at 26/06/2024 at 16:22 UTC*
0 upvotes, 0 direct replies
Yeah, this holds up, and is actually a specific example of a more general principle.
If we think of things a little differently
(a-x) * (a+x) = a^2 - x^2
So in your example
(5.5-.5) * (5.5+.5) = 30.25 - .25
(5.5-1.5) * (5.5-1.5) = 30.25 - 2.25
2.25 - .25 = 2
That's where your 2 is coming from!
You could take this farther
(5.5+2.5) * (5.5-2.5) = 30.25 - 6.25
And sure enough 8×3 is 24
I use this pretty frequently to estimate squares and square roots in my head.
For example if I want know what 6.666^2 is I can do
(6.666-.666) * (6.666+.666) =
6 * 7.333 = 44
44 = 6.666^2 + .444
6.666^2 = 44.444
Edit: oops subtracted at the end instead of added