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What is the golden ratio? Well it can be defined like this: a/b = (a+b)/a
So if you set b = 1 you get
a = a/a + 1/a
Also a = a/b,
let’s call this number Φ
Φ = 1 + 1/Φ (Where Φ>1)
This is a nice recursive definition for Φ
Here is a solution to this equation that defines Φ
Check this using the definition of Φ I just gave you:
Φ=1 + 2/(1+√5)
here we do this neat trick mathematicians implement sometimes, it’s called multiplying by one. Except we don’t call it 1 we call it something else like x/x (x= 1-√5, so x can’t be zero, what I just did wouldn’t work if 1 was equal to √5 but 1²≠5 so we’re safe). Also I used the result 1+2=3.
=(3 -5 + √5 - 3√5)/(1-5)
do some boring algebra stuff.
=-2(1 + √5)/(-4) = (1 + √5)/2
Φ ≈ 1.6180339887
This definition can also be thought of as a quadratic, like this: Φ²-Φ-1=0
we now have a polynomial, and it’s degree two so we know we it will have at most two solutions because of this thing called the fundamental theorem of Algebra (although proving it for degree two polynomials is pretty easy).
The other solution is (1-√5)/2 you can go check this yourself. But this number isn’t Φ because we defined Φ > 1.
This is actually my third favourite number, and my favourite algebraic number.
This number is also related to the Fibonacci sequence.
Let’s say take two consecutive Fibonacci numbers a, b.
You can get the next Fibonacci number by summing up the last two: a+b So let’s divide Fibonacci numbers by the previous ones:
Let’s use an example:
610 and 987 are both Fibonacci numbers.
987/610 = 1.61803278688…
These two are pretty close, and in fact not bad rational approximations of Φ.
So for pretty big consecutive Fibonacci numbers a and b then you get:
a/b ≈ (a+b)/a
And as they approach infinity, they get more exact and you get ever closer to exactly Φ.
And so that’s the golden ratio and where it comes from. Apparently it comes up in nature and shit but I’m not talking about that because it isn’t maths so I don’t really care. Anyway that’s most of what I know about this thing I hope you enjoyed this post, if you have any questions or didn’t understand something you can feel free to ask me I guess.
“f(x)” (pronounced as “eff-of-eks”).
f(unction) of x
My sister just unfollowed me wow rude.
What if Jessie’s girl is Stacy’s mom?
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people who feel the need to add their commentary to every post
I know omg I hate it when people do that
Is expression of opinion a bad thing now?
Ask me maths questions
I’m bored, I’ll try to answer them.
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