Pythagoras' Theorem

Years ago, a man named Pythagoras found an amazing fact about triangles:

If the triangle had a right angle (90°) and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

a2 + b2 = c2

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But it only works on right angled triangles.)

How Do I Use it?

Write it down as an equation:

Now you can use algebra to find any missing value.

Discovered by Khorshed

Before you do this, you need to know the following:
(a+b) (a-b) = a²- b<sup>2


Here's why.
(a+b) (a-b) = a+b (a-b)
                  = a (a-b) + b (a-b)
                  =</sup> a^{2 - ab + ba -} b^{2
                
                        =}  a^{2 -} b<sup>2


Therefore:
a2 - b2 = (a+b) (a-b)</sup>

Okay, let's start.
1.   Get two numbers that are the same.   Call them a and b.
 a = b

2.   Multiply by a
a<sup>2 = ba


3.   Subtract</sup> b^{2
a2 -  b2 = ba -} b<sup>2


4.  Swap the left side with (a+b) (a-b) and factorise the right side.
(a+b) (a-b) = b (a-b)


5.   Divide by (a-b)
(a+b) = b


6.    Open brackets.
a+b = b


7.   Subtract b from both sides.
a = 0


NOTE: You cannot divide by (a-b) because a-b = 0.   You cannot divide by 0, which it why it cannot be true.


Discovered by Richard</sup>

Circles

The circumference of a circle is the actual length around the circle which is equal to 360°. Pi (π) is the number needed to compute the circumference of the circle. 
π is equal to 3.14.
Pi is greek and has been around for over 2000 years!
In circles the AREA is equal to 3.14 (π) times the radius (r) to the power of 2. 
The formula looks like:
A= πr2.
In circles the circumference is 3.14 (π) times the Diameter. 
The formula looks like:
2πr or πd.

Example:

discovered by Andrew F

Graphing Lines

Line graphs provide an excellent way to map independent and dependent variables that are both quantitative. When both variables are quantitative, the line segment that connects two points on the graph expresses a slope, which can be interpreted visually relative to the slope of other lines or expressed as a precise mathematical formula. Scatter plots are similar to line graphs in that they start with mapping quantitative data points. The difference is that with a scatter plot, the decision is made that the individual points should not be connected directly together with a line but, instead express a trend. This trend can be seen directly through the distribution of points or with the addition of a regression line. A statistical tool used to mathematically express a trend in the data.



Example:


y=2x+1


discovered by Andrew F

Pascal

Pascal’s triangle is made up by adding the two numbers directly above e.g. the highlighted squares 3+1=4

Fibonacci Sequence
Try this: make a pattern by going up and then along, then add up the squares you will get the Fibonacci Sequence. 

(The Fibonacci Sequence starts "1, 1" and then continues by adding the two previous numbers, for example 3+5=8, then 5+8=13, etc)

Symmetrical
And the triangle is also symmetrical. The numbers on the left side have identical matching numbers on the right side, like a mirror image.





discovered by Eddy

Fibonacci Sequences in Nature

Fibonacci numbers are made of the two numbers before that by adding them up, that is : Add the last two numbers to get the next.

The first 10 Fibonacci numbers are 1,2,3,5,8,13,21,34,55.

Fibonacci sequence actually has a very unique characteristic. Dividing a number with previous number allows us to get approximately similar result. However, the result will be the same after the 13th number inside the series: 1,618 that is Golden Ratio.

Interestingly, Fibonacci numbers can be found in nature, like plants!

There are many examples of how Fibonacci numbers are found in nature.

Fibonacci numbers can be found on plants like Romanesco broccoli, sunflowers and daisies.

For example, the distribution of seeds in sunflower is spiral. The seeds of the sunflower spiral outwards in both clockwise and counterclockwise directions from the center of the flower. The numbers of clockwise and counterclockwise spirals are two consecutive numbers in the Fibonacci sequence.

However, you have to be careful that even though they are plants which are spirals, it has to be the numbers adjacent to each other in the Fibonacci sequence.

The shells of the snails also follow the Fibonacci sequence. In the same way, the shells of the nautilus follow the same rule. The only difference between these two is that nautilus’ shells grow in a three-dimensional spiral, whereas snails’ shells grow in a two-dimensional spiral.

Pine cones are one of the well-known examples of Fibonacci sequence. All cones grow in spirals, starting from the base where the stalk was, and going round and round the sides until they reach the top.

Another notable example is human body. In human body, the ratio of the length of forearm to the length of the hand is equal to 1.618, that is, Golden Ratio. Another well-known example on human body is The ratio between the length and width of face

- Ratio of the distance between the lips and where the eyebrows meet to the length of nose

- Ratio of the length of mouth to the width of nose

- Ratio of the distance between the shoulder line and the top of the head to the head length

- Ratio of the distance between the navel and knee to the distance between the knee and the end of the foot

- Ratio of the distance between the finger tip and the elbow to the distance between the wrist and the elbow
The same sequence exists on the leaves of poplar, cherry, apple, plum, oak and linden trees.

discovered by Qi Le