Differences Of Squares

[ishwar 0]

Hi guys , I just found out that the differences of squares form an arithmetic series with d = 2 , but I can't figure out why :

 

 

3 5 7 9 11 - notice the difference

1 4 9 16 25 36 - the squares

1^1, 2^2, 3^3 , 4^4 , 5^5, 6^6

Edited September 27, 2006 by ishwar (see edit history)

[-bLiNd- 0]

What a good combination, I see what you mean I don't understand why this is so........

[franz see 0]

It's actually quite simple

 

given d = y^2 - x^2, where y = x + 1,

 

d = (x+1)^2 - x^2

d = [ (x+1) * (x+1) ] - x^2

d = [ x^2 + 2x + 1 ] - x^2

d = 2x + 1

 

And if x = {1, 2, 3, 4, 5, ...}, we would have d = {3, 5, 7, 9, 11, ...}

 

Furthermore, you can do the reverse....

 

d = 2x + 1

d - 1 = 2x

(d - 1) / 2 = x

d/2 - 0.5 = x

 

THUS, combining with y = x + 1, we will have

 

d/2 - 0.5 = y - 1

d/2 + 0.5 = y

 

 

This is the basis of the trick i used to do back in elementary (yes, i was a bored little man back then hehehe ).

 

given any number, d, i can find its x and y such that y^2 - x^2 = d (it's just a matter of dividing d by 2 then adding and subtracting 0.5 to get y and x respectively)

 

it was a neat trick back then, but after being taught algebra, it got demystified

[elevenmil 0]

So you both are pretty young then eh? This is kind of easy to understand, but then I'm a Math Geek. I might have to contribute in this particular forum on a little bit of information on base number systems.

[ishwar 0]

Hi guys , thanks for all your help, I figured it by a simple diagram ,

 

I started pondering on the squares , and then I realized it was called square because of a square(shape)

 

its hard to explain this wihtout a diagram,

 

you can see this for yourself by drawing blocks of squares. You can then see the difference of squares increasing.

 

Ishwar

[pixieloo 0]

hey, that's pretty cool! i've never noticed that before. but there are a lot of patterns in math, so i'm not surprised.

[ghostrider 0]

That is amazing. I've always wondered about squares. Ever since grade school I wondered why adding and subtracting squares would never work right, only until 3 years later when I explained to my algebra class. Think about addition and subtraction. The only reason they really work is because the difference between each number is the same as all the other differences.2 - 1 = 13 - 2 = 14 - 3 = 1But with squares,4 - 1 = 39 - 4 = 516 - 9 = 7

[ishwar 0]

So did any of you guys try drawing little bunch of squares?

[me_boxer_dude 0]

Whoa. I have never been good at maths but decided to try to comprehend it in my head. I have reached the conclusion that understanding Mathematics stuff would be the hardest thing for me to try.

[inky-blue-white 0]

Squares are quadratic [state the obvious, I know] and these patterns will always have a second common difference.[Linear sequences have a first common difference that is. Cubic sequences have a third common difference etc.]A common difference by the way, is......-1,0,1,2,3,4,5,6...in this case, it's one. Because the numbers are all one more than the previous....-1,0,2,5,9,14...in this case, the differences go......1,2,3,4,5...So that's a second common difference of one.Maths is a wonderful subject! Sorry that my explaination was shody though!

[rsf 0]

(X+1)^2 - X^2 = 2x + 1That's why.It's also the same thing if you do it a bit backwards. (You don't have to start with 7)7*7 = 496*8 = 485*9 = 454*10 = 40...the differences are 1, 3, 5, ...This one is like:(x+n)(x-n) where n is the difference between the number and the starting number is:X^2 - n^2and that brings us back to the first example