idk if this already exists and is called something but...

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one day in grade school i noticed a pattern in square numbers as related to their roots that can be expressed like this:

x+(x-1)+(x-1)²= x²

1²=1
2²=4
3²=9
etc.

1+(1-1)+(1-1)²=1
2+(2-1)+(2-1)²=4
3+(3-1)+(3-1)²=9
etc.

when written down in succession a person can write all number squared from the smallest fraction on into infinity using only addition and no multiplication whatsoever. only having to look at preceding numbers and their squares. despite that a number squared is in essence supposed to be solved using multiplication.

is this a new way of looking at it or no?

I'm not sure it's new but at least you are correct.

n * n
= n + (n-1) * n
= n + (n-1) + (n-1) * (n-1)

It's not new. We did a proof on that in my abstract math class as an example.

Rather old way of looking at it actually. The ancient Greeks were aware of it, as were classical Indian mathematicians. It stems from geometry.

Draw a square 1 x 1. Then add an "L" shaped side to it to create a square that is 2 x 2. Repeat the process as many times as you like to generate as many integer roots and squares as you want.

The pattern is useful if you want to square larger numbers in your head quickly. As long as you know a nearby square, you can use the equations above to do some quick mental math.

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

Where I do it wrong? (It's been a long time since I don't do much math)
And how are you writing exponents by the way?

To write exponents I use the Windows character map that is in the accessories menu. The only exponents
available are x°,x¹,x²,x³, Linux character maps do offer more super and subscripts but not all browsers can display them properly.

You claimed that this calculates the square using only addition, and no multiplication whatsoever, but note that this is factually incorrect. Multiplication IS required to square the number before it. Since the previous square is lower than the next square by 2x-1, you have to add 2x-1 to make it even out.

What you stumbled upon is actually formally known as the difference of squares, which is a well-known relationship in mathematics. Please see: http://en.wikipedia.org/wiki/Difference_of_two_squares

Thanks for the answers. I guess it was a pretty stupid error from my part.