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#1 Re: Maths Teaching Resources » Solving puzzles » 2021-03-24 09:08:33
In your question on how to end a set of equations :
Most people are lazy and simply use QED
but if its at the end of a mathematical statement QEF is an alternative. If Euclid used it who are we to disagree.
"Q,E.F." is an abbreviation for the Latin phrase "quod erat faciendum" ("that which was to be done"). It is a translation of the Greek words used by Euclid to indicate the end of the justification of a construction, while "Q.E.D." was the corresponding end of proof of a theorem Latin abbreviation for quod erat demonstrandum: "Which was to be demonstrated." Q.E.D. may appear at the conclusion of a text to signify that the author's overall argument has just been proven.
QED and QEF are normally less formally written omitting the full stops
#2 Re: Maths Teaching Resources » Solving puzzles » 2021-03-24 08:35:05
When I was learning Dutch the first thing I did after buying Woordenboeken was buy W W Sawyer's book translation 'Wiskunde zonder Omslag' and I also bought 'Vademecum van de wiskunde' by Otto Teller .
The first one I had already read a few times as 'Mathematicians Delight' . I had a good Maths Teacher who recomended it. The language is a bit dated ( 1943 ) but it is widely available now in English as a free PDF.
Having W W Sawyer help me learn Dutch as I reread a favorite book was erg Gezellig
#3 Maths Teaching Resources » Cosine Rule simultaneous equations handout » 2021-03-23 11:50:53
- wim
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A nice example of simultaneous equations and substitutions
corrected errors, sorry for that
#4 Formulas » Elegant Cosine, Heron's and Half Angle Formulae using semiperimeter » 2021-03-23 02:27:04
- wim
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The Cosine Rule with its squares of sides a , b and c cries out for a fusion with the semiperimeter.
First lets turn the semiperimeter into a few substitutions for later use
Where semiperimeter = s = ( a + b + c ) / 2
( a + b + c ) = 2 s
( a + b - c ) =2 ( s – c )
( a - b + c ) =2 ( s – b )
( b + c - a ) =2 ( s – a )
Then our Cosine rule
a² = b² + c² – 2bcCos A
Cos A = ( b² + c² - a² ) / 2bc
1 – Cos A = ( 2bc - b² - c² + a² ) / 2bc
= ( a – b + c )( a + b - c ) / 2bc
= 2 ( s - b )( s – c ) / bc
1 + Cos A = ( 2bc + b² + c² - a² ) / 2bc
= ( a + b + c )( b + c - a ) / 2bc
= 2 s ( s – a ) / bc
Sin² A = ( 1 – Cos a )( 1 + Cos A )
= 4 s ( s – a )( s – b )( s - c ) / b²c²
_______________________
Sin A = 2 ( √ s ( s – a )( s – b )( s - c ) ) / bc
Area of Triangle = ½ . bc Sin A
______________________
= √ s ( s – a )( s – b )( s - c ) Heron's Area Formula
A much more elegant way of deriving Heron's Formula and there is still the Half angle Formula to go :-
______________
Cos ( A / 2 ) = √ ( 1 + Cos A ) / 2
______________
= √ s ( s – a ) / bc
______________
Sin ( A / 2 ) = √ ( 1 - Cos A ) / 2
___________________
= √ ( s – b )( s – a ) / bc
& Sin A = 2 Sin ( A / 2 ) Cos ( A / 2 ) gives the same Sin A result as above
Tan ( A / 2 ) = Sin ( A / 2 ) / Cos ( A / 2 )
_______________________
= √ ( s – b )( s – c ) / s ( s – a )
corrected mistake in half angle formulae
#5 Re: Formulas » Modern Algebra Symbols » 2021-03-20 09:42:23
You have probably solved the problem by now.
Its the geometric symbol for congruent too , and not in open office function editor, which calls the equivalance symbol congruent .
The best stop gap solution would be to screen shot your formula and edit in a photoshop type editor.
I use gimp which is a free editor and just as powerful .
If you then search Google images for ' congruent symbol ' you can copy the one you prefer to your clipboard and paste it in.
There are a lot of formula editors on the market all have limitations but you are using the best of the bunch I think.
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