Illustrations of Formulas
Contents of this Page
What are Illustrations of  formulas?
Simple Formulas
Binomial Formulas
Pythagoras's Theorem 
Cubes
References
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What are Illustrations of  formulas?

Formulas are statements of algebra, which apply to numbers of a definition set.

They can be proved except the axioms. You can deduce new formulas by well-known formulas by logical reasoning. This procedure is called a proof.

The proof  ideas and also the proof ways can be described by pictures. In addition the formulas themselves become more alive.

You can find illustrations of well known formulas on this page.


Simple Formulas  top

Commutative law of multiplication (Axiom)

ab=ba

Distributive law (Axiom)


(a+b)c=ac+bc



Product of a difference and a number


(a-b)c=ac-bc



Product of two sums


(a+b)(c+d)=ac+ad+bc+bd



Product of two differences


(a-b)(c-d)=  ac+bd  -ad-bc



Product of a sum and a difference


(a-b)(c+d)=  ac+ad   -bc-bd



Looking for a parallelogram with the same area


a²=bx


Binomial Formulas   top

First binomial formula


(a+b)²=a² + 2ab + b²


 Second binomial formula


(a-b)²=  b²+a²  -2ab


Third binomial formula


a²-b²=(a+b)(a-b)



Tri-nomial formula

(a+b+c)²=a²+b²+c²+2ab+2ac+2bc



Difference of the squares of a sum and a difference


(a+b)²-(a-b)²=4ab


Pythagoras's Theorem  top

The Pythagorean theorem (Pythagoras or one of his students, Pythagoras of Samos, 580-500 BC)


a²+b²=c²


The Pythagorean theorem (Euklid, ~300 BC) 
Classical proof with triangles


a²+b²=c²



 The Pythagorean theorem (Euklid, ~300 BC)
Proof with four-sided figures


a²+b²=c²



Euklid's theorem (Euklid, ~300 BC)


a²=cp (You can show b²=cq in analogy.)



Height formula


a²=p²+h² (Pythagorean theorem), a²=pc=p²+pq (Euklid's theorem),
hence h²=pq



The Pythagorean theorem (Liu Hui, ~300, China)


a²+b²=c²



The Pythagorean theorem ("The bride's chair", ~900, India)


a²+b²=c²



The Pythagorean theorem (Atscharja Bhaskara, Indien,  ~1150)


c²=(a-b)²+2ab oder c²=a²+b²



The Pythagorean theorem (Leonardo da Vinci, 1452-1519)


a²+b²=c²



The Pythagorean theorem (Arthur Schopenhauer's case was a=b, 1788-1860)


a²+b²=c²



The Pythagorean theorem (James Garfield 1876, later on the 20th US President)


You use the formula of the area of a trapezium [A=mh, here h=a+b and m=(a+b)/2]
(a+b)²/2=c²/2+2*(1/2*ab) or a²+b²=c²



The Pythagorean theorem (Hermann Baravalle 1945)

.........

c²=a²+b²



The Pythagorean theorem

(a+b)²=c²+4*(1/2ab) oder a²+b²=c²


Cubes     top

Cube of a sum

(a+b)³=a³+3a²b+3ab²+b³

You can see both cubes and the six rectangular parallelepipeds in 3D-view:


Cube of a difference

The formula is (a-b)³=a³-3a²b+3ab²-b³. You convert it to (a-b)³=a³-3ab(a-b)-b³ for an illustration.
...... You take the drawing of the formula (a+b)³=a³+3a²b+3ab²+b³ from above and replace a by the difference a-b. 

Then the edges are (a-b)+b with different combinations (on the left).

The term (a-b)³ is illustrated by the blue cube (on the right).
 

......
You  recieve the blue cube, too, if you take away the three green bodies and the yellow cube from the red cube:


(a-b)³ = a³-3ab(a-b)-b³ = a³-3a²b+3ab²-b³


References    top
Alexander Bogomolny
Martin Gardner, Mathematisches Labyrinth, Vieweg Braunschweig 1979 (ISBN 3-528-08402-2)
Johannes Lehmann (Hrsg.): Rechnen und Raten, Köln 1987 (ISBN 3-7614-0930-3)
A.Schmid, I. Weidig: Lambacher Schweizer S8, Mathematisches Unterrichtswerk, Stuttgart 1995 (ISBN 3-12-730730-6)
A.Schmid, I. Weidig: Lambacher Schweizer S9, Mathematisches Unterrichtswerk, Stuttgart 1996 (ISBN 3-12-730740-3)
Peter Baptist: Pythagoras und kein Ende? Leipzig, 1998 (ISBN 3-12-720040-4)


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©  2002 Jürgen Köller

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