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Basically you can represent a number by adding a series together (or in the second on mulitplying) to infinity. The series is the first bit i posted
Σ ((-1)^(r-1) x 4)/(2r-1) = pi
By subbing subsequent numbers, starting with 1 and continuing to infinity, the number represented is pi (3.14.... yadda yadda yadda).
so you sub 1 in and get 4/1 = 4
then you sub in 2 and 'add' that to the first value (I say 'add' because the presence of (-1)^r-1 means that every second term will be negative, therefore you need to take)
so you add -4/3 to the first value, and continue doing so to infinty
One of my favorites is where half something times something else squared equals the energy in joules. For instance, half the capacitance times the voltage squared, half the inductance times the current squared, half the mass times the velocity squared, half the spring rate times the displacement squared, etc. This same equation (the result of integrating a linear function) gets a lot of milage.
Another one I like (though I never found a use for it) is where the differential of u (u is typically used as a function of x) is equal to the doubly iterated function of y.
one thing that is awesome is that if you firt get a series by taking any two numbers and add them together, then take the new number and add it to the previous. (i.e. fibonacci series: 1, 1, 2, 3, 5, 8, 13.....)
Next thing to do is divide any number in that series by the previous number, and as greater values in the series (number further along) are used the value will get closer and closer to the golden ratio, φ (phi).which ≈ 1.618.
e.g. say you take the numbers 31 and 17. add them together and you will get 48. The next number in the series would be 48 + 31 = 79, and then the next 79 + 48 = 127.
so the series is 17, 31, 48, 79, 127 and so forth.
and when you divide a number in that series by the previous the golden ratio is expressed to a certain degree of accuracy e.g 127 ÷ 79 = 1.607
I think this is amazing as it doesn't amtter what two number you start with, as long as you follow the rules for the series, you will always find the values converge to phi.
incidentally phi is called the golden ratio because it is a ratio that is commonly found in nature, from the arrangement of stems on a plant to geometry of chemicals.It is also a hypothesis that leonardo da vinci utilised this ratio to properly proportion the human features in many of his paintings.