# Talk:Difference of two squares

WikiProject Mathematics (Rated Stub-class, Low-importance)
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Field:  Basics

## Markup

The math markup is currently in "ba" (''b'' − ''a'') format, instead of "${\displaystyle b-a}$" ([itex]b-a[/itex]) format, because the latter fails to render properly. See bug 8373. grendel|khan 23:04, 24 December 2006 (UTC)

## DOTS disambig?

Could someone create a disambiguation page that also links DOTS here. I've seen DOTS as an abbreviation for this. [1] and [2]. However, there's another page that has DOTS as a redirect. I was just unsure how to do this. S♦s♦e♦b♦a♦l♦l♦o♦s (Talk to Me) 23:21, 5 December 2007 (UTC)

Oh, I'm not sure if the sum of two squares should also be added to this page. S♦s♦e♦b♦a♦l♦l♦o♦s (Talk to Me) 01:40, 6 December 2007 (UTC)

## Clarifying the illustration

Many students would find it helpful to be told more of the relationship between the equations and the illustration.

The geometric illustration does not point out that the two rectangles represent (a-b)b. A better account of the picture would vastly help some people understand the whole article.

(a-b)b + (a-b)(a-b) + (a-b)b =

ab-b^2 + (a^2 - 2ab + b^2) + ab - b^2 =

(a^2 - b^2)

A more explicit connection between the text and illustrations could help readers of wikipedia.(After all, mathematical prodigies don't need it in the first place) —Preceding unsigned comment added by 198.53.140.155 (talk) 06:55, 12 April 2008 (UTC)

There is also a formula derivable from a difference of two squares to receive the difference's value. when a is a perfect square, x^2 - a = (x - a)^2 - a(a + 2x -1) —Preceding unsigned comment added by 24.44.52.119 (talk) 01:14, 22 September 2010 (UTC)

## Moved from the page

This addition moved here for consideration. The identity is valid but didn't really fit well as placed. Charles Matthews (talk) 12:55, 23 September 2011 (UTC)

An alternative method, Momaney's Theorem:

(x^2-y^2)= 2dx-d^2 (Where "d" is the difference between the numbers)

## Section split

I think the contents of the section Difference of adjacent squares would make more sense in the article square number. Isheden (talk) 11:14, 14 November 2013 (UTC)

It actually makes sense to mention it both articles, here it is an example for another application of the difference of squares.--Kmhkmh (talk) 12:52, 14 November 2013 (UTC)

The statement about the difference of two consecutive perfect squares makes sense here, but why would statements about real numbers a and b with a certain distance between them be relevant? Yes, if ${\displaystyle a=b+1}$ then ${\displaystyle a^{2}-b^{2}=2b+1}$ and if ${\displaystyle a+1=b-1=x}$ then ${\displaystyle a^{2}-b^{2}=4x}$ and of course any real number can be expressed as 4x, but why would all this be worth mentioning? Describing two real numbers as adjacent is meaningless since they are uncountable and calling the square root "base" is misleading. Isheden (talk) 19:04, 14 November 2013 (UTC)

Sorry for my somewhat premature reverts.
I agree that current terminology is not helpful if it is understood as a statement about real numbers (and I couldn't find any book or paper using such terminology in a superficial search). The reason why I reverted this was that if restricted to natural numbers it somewhat redundant to the theorem about odd numbers following directly afterwards.
I'm not sure what's the best option to proceed here. One option might be to make (leave?) this a statement about real numbers of distance one and change the terminology accordingly. Another option might be to make make this statement about sequential integers or naturals (I suppose you intended that) and explain the terminology in that context (or modify it). In that case however we should merge that with theorem directly after to avoided having the same proof twice.
As far as "adjacent squares" in terms of real numbers are concerned, there is a way to make some sense of geometrically. That if the side length of 2 squares differ by one, then the difference in area in identical to the sum of their base sides (ignoring units). You can visualize that nicely by placing the squares on top of each and we could actually include such drawing into the article. Maybe that's even what the original author had in mind, but that's pure speculation on my side now.--Kmhkmh (talk) 00:35, 15 November 2013 (UTC)
I think what the "original" wording [3] of this section was referring to was that the difference of two consecutive square numbers, say the nth and (n-1)th, is equal to 2n-1, i.e. the sum of the square roots. I agree that the statement that an odd number can be expressed as a difference of (consecutive perfect) squares is redundant, so that sentence could simply be deleted.
The geometrical interpretation that you propose is already included in the square number article. I'm not sure what the additional benefit is in allowing for real numbers here. The concept of "adjacent" squares loses its meaning and actually any real number can be expressed as a difference of squares in infinitely many ways.
My view is that it would be more interesting to discuss how the difference of consecutive perfect squares leads to a recursive formula for calculating the nth perfect square from the previous one and mention the concept of squarity testing as described in square number. In any case, there should also be a link to that article. Isheden (talk) 11:47, 15 November 2013 (UTC)
Well allowing (positive) real numbers instead of natural numbers turns the whole thing into (trivial) geometric theorem (as stated above and the notion of bases makes sense as base side) rather than being just an geometric illustration of number theoretic theorem. Whether such a "geometric theorem" is worthwhile to mention here is imho a matter of taste/editorial discretion, it certainly doesn't need to be in the article, but I see no harm in mentioning it either.--Kmhkmh (talk) 22:06, 15 November 2013 (UTC)