Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

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Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 546

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 548

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 546

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 548

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 931

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 931

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

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Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 644

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 613

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 614

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 617

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 625

Deprecated: preg_replace(): The /e modifier is deprecated, use preg_replace_callback instead in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 629

Warning: Cannot modify header information - headers already sent by (output started at /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php:613) in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 1489

Warning: Cannot modify header information - headers already sent by (output started at /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php:613) in /home/timfahlberg/mathcasts.org/mtwiki/pmwiki.php on line 1489

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(:title Absolute Value:) >>nav_bar<<%reg%Mathcasts         %exa%Examples      %pra%Practice%%    %mor%More      %up%[[glossary/A|A]] >><< [[<<]] [table border=1 cellpadding=3 width=100%] [row] []>>&<< %ref b%Definition:%% The %red b%absolute value%% of a number is the distance between that number and 0. >>&<< %reg b%Rule:%% To show that we want the absolute value of a number, we put straight lines around it. >>&<< (:showhide init=show div=div111 lshow="+" lhide="-":) Interactivity: Absolute Value %sol s9%(:showhide init=hide div=div112 lshow="+" lhide="-":) Directions for Interactivity (:div111 id=div111 style="margin-left:20px":) >>&<< (:div112 id=div112 class="s9" :)   Click and drag the %ref%big blue point%%. The %up%absolute value of the number%% is the '''length''' of the %up%purple line segment%%. >>&<< (:div112end:) (:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/dev/unpacked/geogebra.jar" width="700" height="150" filename="http://mathcasts.org/mtwiki/uploads/GlossaryT/absolute_value1.ggb" showResetIcon="true" showToolBar="false" showMenuBar="false":) >>&<< (:div111end :) >>&<< [tableend] >>-<< [table border=1 cellpadding=3 width=100%] [row] [] (:showhide init=show div=div1 lshow="+" lhide="-":) Examples                          (:div1 id=div1 style="margin-left:20px":) *The distance between two points is always a positive value. *%exa%This means, the absolute value of a number is always positive.%% >>&<< Attach:GlossaryT./ab_val1.jpg In the above image, there is a number line and 4 points %navy b%Ѕ%% %ref b%W%% %red b%Y%% %pra b%Z%% and in the image below is shown the way to calculate the absolute value *The point %pra b%Z%% represents the number 26. The distance between %pra b%Z%% and 0 is 26 (see below) (:div style="margin-left:40px":) >>&<< So the absolute value of 26 is 26. We write this sentence: {$ |26| =26 $}. (:divend:) *Look at the point %red b%Y%%. Do you see that {$ |19| =19 $} ? >>&<< ---- >>&<< Now, let's look at the points to the left of 0. *The point %ref b%W%% represents the number -7, but the distance between %ref b%W%% and 0 is 7. (:div style="margin-left:40px":) >>&<< So the absolute value of –7 is 7 or {$ |-7| =7 $}. (:divend:) *Look at the point %navy b%Ѕ%%. Do you see that {$ |-25| =25 $}? Attach:ab_val6.jpg >>&<< (:div1end :) [tableend] >>-<< [table border=1 cellpadding=3 width=100%] [row] [] (:showhide init=hide div=div2 lshow="+" lhide="-":) The graph of {$ y=|x| $} (:div2 id=div2 :) >>&<< [table border=1 cellpadding=10] [row] [c valign=middle width=230] [table border=1 cellpadding=2] [row] [c]{$ x $} [c]{$ y=|x| $} [row] [c]0 [c]|0|=0 [row] [c]1 [c]|1|=1 [row] [c]-2 [c]|-2|=2 [tableend] [c width=260] {gg width=200; height=200; xmin=-5; xmax=5; xscl=1; ymin=-2; ymax=8; yscl=1; axes(); stroke="#009900"; strokewidth=3; plot("abs(x)"); strokewidth=2; stroke="#660000"; dot([-2,2],"closed"); a=text([-2,2],"(-2,2)",left); dot([1,1],"closed"); b=text([1,1],"(1,1)",right); stroke="#660099"; dot([0,-6],"open"); c=text([0,-6],"y-пресек",belowleft) gg} >>&<< [tableend] (:div2end :) >>-<< (:showhide init=hide div=div3 lshow="+" lhide="-":) The graph of {$ y=2|x|+1 $} (:div3 id=div3 :) >>&<< [table border=1 cellpadding=10] [row] [c valign=middle width=230] [table border=1 cellpadding=2] [row] [c]{$ x $} [c]{$ y=2|x|+1 $} [row] [c]0 [c]2|0|+1=0+1=1 [row] [c]1 [c]2|1|+1=2+1=3 [row] [c]-2 [c]2|-2|+1=4+1=5 [tableend] [c width=260] {gg width=240; height=200; xmin=-6; xmax=6; xscl=1; ymin=-1; ymax=9; yscl=1; axes(); stroke="#009900"; strokewidth=3; plot("2*abs(x)+1"); strokewidth=2; stroke="#660000"; dot([-2,5],"closed"); a=text([-2,5],"(-2,5)",left); dot([1,3],"closed"); b=text([1,3],"(1,3)",right); stroke="#660099"; dot([0,-6],"open"); c=text([0,-6],"y-пресек",belowleft) gg} >>&<< [tableend] (:div3end :) >>-<< (:showhide init=hide div=div4 lshow="+" lhide="-":) The graph of {$ y=-|x|+2 $} (:div4 id=div4 :) >>&<< [table border=1 cellpadding=10] [row] [c valign=middle width=230] [table border=1 cellpadding=2] [row] [c]{$ x $} [c]{$ y=-|x|+2 $} [row] [c]0 [c]-|0|+2=0+2=2 [row] [c]1 [c]-|1|+2=-1+2=1 [row] [c]-3 [c]-|-3|+2=-3+2=-1 [tableend] [c width=260] {gg width=240; height=200; xmin=-6; xmax=6; xscl=1; ymin=-6; ymax=4; yscl=1; axes(); stroke="#009900"; strokewidth=3; plot("-1*abs(x)+2"); strokewidth=2; stroke="#660000"; dot([-3,-1],"closed"); a=text([-3,-1],"(-3,-1)",left); dot([1,1],"closed"); b=text([1,1],"(1,1)",right); stroke="#660099"; dot([0,-6],"open"); c=text([0,-6],"y-пресек",belowleft) gg} >>&<< [tableend] (:div4end :) >>-<< [tableend] ---- >>&<< %rel%Related Topics >>&<< ---- %popwin%[[standards/ACTgr20?action=popopen|ACT GR 20-23]] (:showhide3 init=hide div=box4 lshow='Add comment' lhide='Close':) >>id=box4<< (:commentbox:) >><< [table width=100%] [row] [][[glossary/A| Attach:main/tri_purple_up_a.gif ]] [[glossary/A| Up one level]] [r](:html:) (:htmlend:) [tableend]