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{{Refimprove|date=December 2009}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
{{mergefrom|Conjugate (algebra)|date=January 2012}}
In [[mathematics]], the '''difference of two squares''', or the difference of perfect squares, is a [[Square (algebra)|squared]] (multiplied by itself) number subtracted from another squared number. It refers to the [[identity (mathematics)|identity]]


:<math>a^2-b^2 = (a+b)(a-b)\,\!</math>
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


in [[elementary algebra]].
Registered users will be able to choose between the following three rendering modes:


==Proof==
'''MathML'''
The [[mathematical proof|proof]] is straightforward. Starting from the [[Sides of an equation|right-hand side]], apply the [[distributive law]] to get
:<math forcemathmode="mathml">E=mc^2</math>
:<math>(a+b)(a-b) = a^2+ba-ab-b^2\,\!</math>,
and set
:<math>ba - ab = 0\,\!</math>
as an application of the [[commutative law]]. The resulting identity is one of the most commonly used in mathematics.


The proof just given indicates the scope of the identity in [[abstract algebra]]: it will hold in any [[commutative ring]] ''R''.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


Conversely, if this identity holds in a [[ring (mathematics)|ring]] ''R'' for all pairs of elements ''a'' and ''b'' of the ring, then ''R'' is commutative.  To see this, we apply the distributive law to the right-hand side of the original equation and get
'''source'''
:<math>a^2 + ba - ab - b^2\,\!</math>
:<math forcemathmode="source">E=mc^2</math> -->


and for this to be equal to <math>a^2 - b^2</math>, we must have
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


:<math>ba - ab = 0\,\!</math>
==Demos==


for all pairs ''a'', ''b'' of elements of ''R'', so the ring ''R'' is commutative.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==In geometry==
[[Image:Difference of two squares.png|right]]


The difference of two squares can also be illustrated geometrically as the difference of two square areas in a [[Plane (mathematics)|plane]]. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. <math>a^2 - b^2</math>. The area of the shaded part can be found by adding the areas of the two rectangles; <math>a(a-b) + b(a-b)</math>, which can be factorized to <math>(a+b)(a-b)</math>. Therefore <math>a^2 - b^2 = (a+b)(a-b)</math>
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is <math>a^2-b^2</math>. A cut is made, splitting the region into two rectangular pieces, as shown in the second diagram. The larger piece, at the top, has width a and height a-b. The smaller piece, at the bottom, has width a-b and height b. Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, the two pieces together form a rectangle, whose width is <math>a+b</math> and whose height is <math>a-b</math>. This rectangle's area is <math>(a+b)(a-b)</math>. Since this rectangle came from rearranging the original figure, it must have the same area as the original figure. Therefore, <math>a^2-b^2 = (a+b)(a-b)</math>.Any odd number can be expressed as difference of two squares.
==Test pages ==
[[Image:Difference of two squares geometric proof.png]]


==Uses==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
===Complex number case: sum of two squares===
*[[Displaystyle]]
The difference of two squares is used to find the linear factors of the ''sum'' of two squares, using [[complex number]] coefficients.
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


For example, the root of <math>z^2 + 5\,\!</math> can be found using difference of two squares:
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>z^2 + 5\,\!</math>
==Bug reporting==
:<math> = z^2 - (\sqrt{-5})^2</math>
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
:<math> = z^2 - (i\sqrt5)^2</math>
:<math> = (z + i\sqrt5)(z - i\sqrt5)</math>
 
Therefore the linear factors are <math>(z + i\sqrt5)</math> and <math>(z - i\sqrt5)</math>.
 
Since the two factors found by this method are [[Complex conjugate]]s, we can use this in reverse as a method of multiplying a complex number to get a real number. This is used to get real denominators in complex fractions.<ref>[http://www.themathpage.com/alg/complex-numbers.htm#conjugates Complex or imaginary numbers] TheMathPage.com, retrieved 22 December 2011</ref>
 
===Rationalising denominators===
The difference of two squares can also be used in the [[Rationalisation (mathematics)|rationalising]] of [[irrational number|irrational]] [[denominator]]s.<ref>[http://www.themathpage.com/alg/multiply-radicals.htm Multiplying Radicals] TheMathPage.com, retrieved 22 December 2011</ref> This is a method for removing [[Nth root|surds]] from expressions (or at least moving them), applying to division by some combinations involving [[square root]]s.
 
For example:
The denominator of <math>\dfrac{5}{\sqrt{3} + 4}\,\!</math> can be rationalised as follows:
 
:<math>\dfrac{5}{\sqrt{3} + 4}\,\!</math>
 
:<math> = \dfrac{5}{\sqrt{3} + 4} \times \dfrac{\sqrt{3} - 4}{\sqrt{3} - 4}\,\!</math>
 
:<math> = \dfrac{5(\sqrt{3} - 4)}{(\sqrt{3} + 4)(\sqrt{3} - 4)}\,\!</math>
 
:<math> = \dfrac{5(\sqrt{3} - 4)}{\sqrt{3}^2 - 4^2}\,\!</math>
 
:<math> = \dfrac{5(\sqrt{3} - 4)}{3 - 16}\,\!</math>
 
:<math> = -\dfrac{5(\sqrt{3} - 4)}{13}.\,\!</math>
 
Here, the irrational denominator <math>\sqrt{3} + 4\,\!</math> has been rationalised to <math>13\,\!</math>.
Any odd number can be expressed as difference of two squares.
 
===Mental Arithmetic===
The difference of two squares can also be used as a arithmetical short cut.  If you are multiplying two numbers whose average is a number which is easily squared the difference of two squares can be used to give you the product of the original two numbers.
 
For example: <math> 27 \times 33 = (30 - 3)(30 + 3) \!</math>
 
Which means by using the difference of two squares <math> 27 \times 33 \!</math> can be restated as
<math> a^2 - b^2 </math> which is <math>30^2 - 3^2 = 891. \!</math>
 
==See also==
*[[Conjugate (algebra)]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Difference Of Two Squares}}
[[Category:Elementary algebra]]
[[Category:Mathematical identities]]
[[Category:Articles containing proofs]]
 
[[ar:فرق مربعي عددين]]
[[sv:Konjugatregeln]]
[[zh:平方差]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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