So I've picked up on a pattern involving square numbers. I've been running numbers all morning since I spotted the pattern and I'm convinced that this is actually a thing. Okay, here goes:

If AxA=B

And (A-1)x(A+1)=C

Then C=B-1

Or

4x4=16

(4-1=3)x(4+1=5)=15

15=16-1

5x5=25

4x6=24

6x6=36

5x7=35

And so on. I haven't yet found a set of numbers that this fails on, provided that you start with a number timesed by itself. Is this like a known thing? Does it have some kind of a name? Because it's pretty cool - useless, but cool.

I have never noticed that. o_O

ReplyDeleteHave passed your post to some mathematician buddies, to see if they can answer your questions - I'll be really interested to see what they say.

Ta :)

DeleteSo the new mob are throwing organs? Waste of good protein if you ask me.

DeleteYup, I did ask if any of them wanted to take some home to make stew. No one was interested. Ah well.

Delete"There's kids in Ethiopia", as my gran used to say. She never got round to explaining the significance of there being kids in a place called Ethiopia with regard to my own food consumption, but it was self-evidently a reason to eat food.

DeleteGood spot, Joey - it's the difference of two squares, and you're right, it IS a really neat pattern. Don't worry that it's not a new discovery, a lot of the joy of maths is in working things out even if they've been worked out before.

ReplyDeleteWhat about numbers four below a square? Or 49 below a square?

Can you prove it algebraically? Try multiplying out the (a-1)(a+1) bracket and seeing what happens. How about (a-b)(a+b)?

Play with it and see how far you can take it :)

aa=a^2

Delete+

(-1)(+1)=-1

+

(a)+(-a)=0

=

a^2-1

Okay, so the concept has an algebraic proof.

Subbing in a b for the 1 seems too easy.

aa = a^2

(-b)(+b)= -b^2?

Hmm...

Taking base 6^2 as an example:

6^2 = 36

4x8 = 32

Difference is -4, or -2^2. Again:

7^2 = 49

4x10 = 40

Difference is -9, or -3^2

Following this logic, the original (+1)(-1) should be -1^2, which still = -1 for all practical purposes. But then a negative times a negative is a positive, so for it to make mathematical sense it should read -(1^2), or in the general sense (a+b)(a-b) = a^2-(b^2).

Maybe I'll graph this...

Possibility of rendering the square as negative by making b greater than a?

Delete1) 7±9?

7^2 = 49

-2x16 = -32

2) 7±10.5

7^2 = 49

-3.5x17.5 = -61.25

3) 7±10

7^2 = 49

-3x17 = -51

4) 7±9.5

7^2 = 49

-2.5x16.5 = -41.25

Hmm...

I don't think this one'll have a bang-on proof, more like a near-enough job.

Same concept but where the square can also hit an integer at root 4?

Delete16

4x4 = 16

No.

A square of a cube? First is 64.

8x8 = 64

-2x18 = -36

-3x19 = -57

-4x20 = -80

Big leaps. No point going into decimals as a decimal at one end means a decimal at the other. It'll just mean going into smaller decimals.

What kensson said!

ReplyDeleteExcept to add that it's NOT useless! It's a useful tool for factorising certain types of quadratic equations (which have all sorts of uses- here's a list I found: math.tutorvista.com/algebra/applications-of-quadratic-equations.html )

Ok, so in real life the right type of quadratic is unlikely to crop up as there are so many more of the wrong types, but thinking about things and experimenting as you have can only serve to deepen your understanding of quadratics (in this case) and of mathematics (and therefore the universe) in general.

It could also be argued that there's a certain amount of fun to be had in playing with numbers, and fun is never useless.

I'm impressed, and I'm heartened to know there are folks out there getting stuck in and messing around with numbers. Keep at it, and share your findings!

Thank you all. It's cool to know that this is a thing. I didn't really get quadratics that well at GCSE, so maybe this'll add a new perspective. I'll play about with it and see what sticks.

ReplyDeleteThanks :)