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

Page 1 of 1 [ 11 posts ] 

Seanmw
Veteran
Veteran

User avatar

Joined: 25 Jul 2009
Age: 33
Gender: Male
Posts: 3,639
Location: Bremerton, WA

26 Jul 2009, 5:08 pm

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?


_________________
+Blog: http://itsdeeperthanyouknow.blogspot.com/
+"Beneath all chaos lies perfect order"


CloudWalker
Veteran
Veteran

User avatar

Joined: 26 Mar 2009
Age: 34
Gender: Male
Posts: 711

26 Jul 2009, 5:35 pm

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)



Orwell
Veteran
Veteran

User avatar

Joined: 8 Aug 2007
Age: 34
Gender: Male
Posts: 12,518
Location: Room 101

26 Jul 2009, 5:57 pm

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


_________________
WAR IS PEACE
FREEDOM IS SLAVERY
IGNORANCE IS STRENGTH


Aoi
Veteran
Veteran

User avatar

Joined: 16 Jul 2009
Age: 55
Gender: Male
Posts: 683

26 Jul 2009, 6:38 pm

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.



Seanmw
Veteran
Veteran

User avatar

Joined: 25 Jul 2009
Age: 33
Gender: Male
Posts: 3,639
Location: Bremerton, WA

26 Jul 2009, 11:24 pm

Aoi wrote:
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.
interesting, i didn't know that


_________________
+Blog: http://itsdeeperthanyouknow.blogspot.com/
+"Beneath all chaos lies perfect order"


richie
Supporting Member
Supporting Member

User avatar

Joined: 9 Jan 2007
Age: 65
Gender: Male
Posts: 30,142
Location: Lake Whoop-Dee-Doo, Pennsylvania

27 Jul 2009, 6:29 pm

Seanmw wrote:
Aoi wrote:
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.
interesting, i didn't know that


The difference between any two consecutive squares is always an odd number some of which
are also squares....That is how we are able to prove the existence of an infinite number of
Pythagorean Trios ie: 3²+4²=5², 5²+12²=13², 7²+24²=25²etc....


_________________
Life! Liberty!...and Perseveration!!.....
Weiner's Law of Libraries: There are no answers, only cross references.....
My Blog: http://richiesroom.wordpress.com/


Tollorin
Veteran
Veteran

User avatar

Joined: 14 Jun 2009
Age: 42
Gender: Male
Posts: 3,178
Location: Sherbrooke, Québec, Canada

03 Aug 2009, 9:37 pm

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?



richie
Supporting Member
Supporting Member

User avatar

Joined: 9 Jan 2007
Age: 65
Gender: Male
Posts: 30,142
Location: Lake Whoop-Dee-Doo, Pennsylvania

04 Aug 2009, 5:37 pm

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.


_________________
Life! Liberty!...and Perseveration!!.....
Weiner's Law of Libraries: There are no answers, only cross references.....
My Blog: http://richiesroom.wordpress.com/


lau
Veteran
Veteran

User avatar

Joined: 17 Jun 2006
Age: 75
Gender: Male
Posts: 9,618
Location: Somerset UK

05 Aug 2009, 9:36 am

Tollorin wrote:
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?

On your first line:
Code:
x+(x+1)+(x-1)^2=x^2
    ^


_________________
"Striking up conversations with strangers is an autistic person's version of extreme sports." Kamran Nazeer


immersive
Hummingbird
Hummingbird

User avatar

Joined: 7 Jun 2009
Age: 40
Gender: Male
Posts: 18

05 Aug 2009, 2:22 pm

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



Tollorin
Veteran
Veteran

User avatar

Joined: 14 Jun 2009
Age: 42
Gender: Male
Posts: 3,178
Location: Sherbrooke, Québec, Canada

05 Aug 2009, 9:35 pm

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