Angles

Types of angles.                                    Acute angle:

An angle whose measure is less than 90 degrees. The following is an acute angle.

Right angle:

An angle whose measure is 90 degrees. The following is a right angle.

Obtuse angle:

An angle whose measure is bigger than 90 degrees but less than 180 degrees. Thus, it is between 90 degrees and 180 degrees. The following is an obtuse angle.

Straight angle

An angle whose measure is 180 degrees.Thus, a straight angle look like a straight line. The following is a straight angle.

Reflex angle:

An angle whose measure is bigger than 180 degrees but less than 360 degrees.The following is a reflex angle.

Adjacent angles:

Angle with a common vertex and one common side. <1 and <2, are adjacent angles.

Complementary angles:

Two angles whose measures add to 90 degrees. Angle 1 and angle 2 are complementary angles because together they form a right angle.

Note that angle 1 and angle 2 do not have to be adjacent to be complementary as long as they add up to 90 degrees

Supplementary angles:

Two angles whose measures add to 180 degrees. The following are supplementary angles.

Vertical angles:

Angles that have a common vertex and whose sides are formed by the same lines. The following(angle 1 and angle 2) are vertical angles.

When two parallel lines are crossed by a third line(Transversal), 8 angles are formed. Take a look at the following figure

Angles 3,4,5,8 are interior angles

Angles 1,2,6,7 are exterior angles

Alternate interior angles:

Pairs of interior angles on opposite sides of the transversal.

For instance, angle 3 and angle 5 are alternate interior angles. Angle 4 and angle 8 are also alternate interior angles.

Alternate exterior angles:

Pairs of exterior angles on opposite sides of the transversal.

Angle 2 and angle 7 are alternate exterior angles.

Corresponding angles:

Pairs of angles that are in similar positions.

Angle 3 and angle 2 are corresponding angles.

Angle 5 and angle 7 are corresponding angles

Formula

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=baDistributive 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

History of mathematics

History of mathematics

A proof from Euclid's Elements, widely considered the most influential textbook of all time.^[1]

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.

Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian c. 1900 BC),^[2]the Rhind Mathematical Papyrus(Egyptian c. 2000–1800 BC)^[3] and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

The study of mathematics as a demonstrative discipline begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα(mathema), meaning "subject of instruction".^[4] Greek mathematicsgreatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.^[5] Chinese mathematics made early contributions, including a place value system.^[6][7] The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and were transmitted to the west via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī.^[8][9]Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.^[10] Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.

From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day.