Pythagoras
Pythagoras was born on Samos, an island in Greece, somewhere around 570 BC. As a young man he met Thales who encouraged Pythagoras' desire to learn about mathemathics, astronomy and science in general. During his younger years he gathered knowledge both from Egypt and Babylon.
When he returned, he set up a scool in Crotona, where he and his followers all lived and studied together. The society was quite secretive and had a variety of religious, moral and philosophical beliefs. The most important of which is that ”Everything is Numbers”. They considered each number to have it's own meaning and characteristics. For example: One wasn't thought of as a number as such, but as the essence of all numbers. Two represented women, and three represented men. Four, which was seen as a square with four equal sides and four equal angles represented equality. Five, being the sum of two and three (the representatives of man and woman respecively) represented marriage: the union between those two.
When he returned, he set up a scool in Crotona, where he and his followers all lived and studied together. The society was quite secretive and had a variety of religious, moral and philosophical beliefs. The most important of which is that ”Everything is Numbers”. They considered each number to have it's own meaning and characteristics. For example: One wasn't thought of as a number as such, but as the essence of all numbers. Two represented women, and three represented men. Four, which was seen as a square with four equal sides and four equal angles represented equality. Five, being the sum of two and three (the representatives of man and woman respecively) represented marriage: the union between those two.
The holiest number, or figure, however was the "tetractys". This represents the fourth triangular number, which simply means that it is the number derived, when you place first one dot (this is the first triangular number), then two below it to form a triangle, then three below those to form a third triangle and then finally four dots below that to form a fourth triangle. Counting the number of dots leads to the number ten. It represents the four classical elements fire, water, earth and air.
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Because of the way Pythagoras and his society of students lived and worked, sharing all discoveries amongs themselves, yet attributing them all to Pythagoras, it is uncertain which of the discoveries of the society actually came from Pythagoras himself and which might have been develloped by others.
Triangles
In the previous lesson we already started looking at triangles, but we didn't discuss how they were actually represented.
The angles in a triangle are named A, B & C, using capital letters. The sides of a triangle are named according to the angle opposite them, using lower case letters: a, b & c. A triangle that contains a 90° angle is called a right triangle. It is identified in drawings by making a little square at the corner. This angle is traditionally named C. Right triangles also have special name for the side opposite the right angle. This side is called the hypotenuse. The other two sides in a right triangle are called the Legs or the Catheti. |
Other angles are represented in drawings by a small rounded line. An angle that has two rounded lines rather than one has been specified and any angle in the same drawing that also has these two rounded lines is defined as being the same angle. This can be very useful when working with similar triangles and you can see an example of it in the previous lesson in the drawing of the ship.
One of Pythagoras' realisations was that if all the angles on the inside of a triangle were added together the total would always be the same: 180°. This makes it possible to calculate the last angle as soon as two are known.
This is fairly easily seen if we look at the triangle as half a square. A square has four right angles, adding up to 360°
If a square with equal sides is cut in half by a line going between two opposing corners, the two resulting triangles will contain one right angle (90°) and two halfed right angles (45°), resulting in the calculation: 90+45+45=180 For other rectangles the split may be different, but the rule still holds since both triangles are the same and are exactly half of the rectangle. |
It doesn't stop with triangles though. The sum of the interior angles of all polygones (that's shapes like triangles, rectagles, pentagons, etc.) can be calculated using a fairly simple formula.
180*(n-2) degrees = sum of interior angles n in this formula stands for the number of sides the polygon has. |
Let's try it on a triangle:
It has three sides, so n equals three. That gives us: 180*(3-2) = sum of interior angles 180*1 = sum of interior angles 180° = sum of interior angles |
Pythagoras' Theorem
The thing pythagoras is most well remembered for is of course the Pythagorean Theorem, which states that for right triangles this is true:
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Raising any number to the power of two means multiplying that number with itself. It can also be called squaring the number, because multiplying a number with itself is usually a way to find the area of a square where the sides of the square are the same. |
Finding the Square Root of a number means finding the number that, multiplied with itself, yields the number you are trying to find the square root of.
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Pythagoras wasn't the first to realise this truth however. It was already known in Babylon at the time, but he was the first to provide proof for it.
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Pythagorean Triples
When working with triangles and using pythagoras theorem to calculate the hypotenuse, you will soon realise that it sometimes comes out as a whole number – an integer – and sometimes it comes out as a decimal*.
A Pythagorean Triple however is a triangle where all three sides come out as integers. The best known is one where the legs of the triangle are 3 and 4 respecively, making the hypotenuse 5. These triangles are called Pythagorean triangles.
*And sometimes they come out as irrational numbers, but these will show as decimal numbers too since no calculater will be able to show the infinite amount of numbers it actually comes out to.
A Pythagorean Triple however is a triangle where all three sides come out as integers. The best known is one where the legs of the triangle are 3 and 4 respecively, making the hypotenuse 5. These triangles are called Pythagorean triangles.
*And sometimes they come out as irrational numbers, but these will show as decimal numbers too since no calculater will be able to show the infinite amount of numbers it actually comes out to.
How to use a Coordinate System
A Coordinate system is a system that allows us to plot in specific positions. The simplest coordinate system has two axes: the x-axis (horizontal) and the y-axis (Vertical). When listning a point in the coordinate system, the value for the x-axis is always listed first and then the value for the y-axis: (x,y).
In this example we can see that the value on the x-axis is 4 and the value on the y-axis is 2. The position for this point – or the coordinates for it – are therefore written as (4,2). |
Calculating the Distance Between Two Points
If you have two points in a coordinate system you can use Pythagoras' Theorem to find the distance between them. The reason this works is because of the right triangle we can imagine (or even draw in as in the example below). We can of course count how many squares the legs cover – in this case two and three respectively, but there is a more elegant way to calculate this, which will make it a lot easier once the distances – and thereby the triangles – become larger, or when we are only given the coordinates and would like to find the distance between them without actually drawing a coordinate system.
First we want to find the difference between the x-coordinates of both points, and then the difference between the y-coordinates of both points. Since we are only interested in the difference it pays to always choose the larger number first and subtract the smaller number.
In this case we have two points, called p1 and p2 for point 1 and point 2. Difference between the x-coordinates in p1 and p2: 4-1 = 3 Difference between the y-coordinates in p1 and p2: 4-2 = 2 Now we have the legs of the triangle and can treat it as any other right triangle and find the length of the hypotenuse – or the distance between the two points of the coordinate system. As a formula it could be written this way: |