Ratios with algebra
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The Golden Ratio
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Ratio problem with basic algebra (new HD)
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Writing proportions
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More advanced ratio problem--with Algebra (HD version)
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Advanced ratio problems
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Alternate Solution to Ratio Problem (HD Version)
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Another Take on the Rate Problem
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Adding and Subtracting Rational Expressions
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Find an Unknown in a Proportion
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Find an Unknown in a Proportion 2
The Golden Ratio An introduction to one of the most amazing ideas/numbers in mathematics
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- What I want to explore in this video is given some length of string or line or some line segment right here B
- can I set up an A, so that the ratio of A to B
- is equal to the ratio of the sum of these two to the longer side
- so its equal to ratio of A plus B to A
- so I want to sit and think about this for a little bit
- I wanna see is, can I construct some A that's on this ratio, this perfect ratio
- I am somehow refering to here
- so that the ratio of the longer side, to the shorter side
- is equal to the ratio of the whole thing to the longer side
- and lets just assume we can find a ratio like that
- and we'll call it, we'll call it φ (Phi). We'll use the greek letter phi
- for that ratio over there
- so lets see what we can learn about this special ratio phi.
- Well, phi is equal to A over B, which is equal to A plus B over A
- we know that A plus B over A is the same thing as
- A over A plus B over A
- A over A is just one
- and B over A is just the inverse this statement right over here
- so B over A, this thing right over here is phi
- so B over A is just going to be 1 over phi. This is going to be 1 over phi.
- So this is interesting, we've now set up a number
- which is, we're gonna call this special ratio
- phi, is equal to 1 plus 1 over phi
- well just that is kind of a neat statement over there
- first of all is you subtract 1 from both sides of this
- you get phi minus 1 is equal to its inverse
- that seems to be a pretty neat property of any number
- that if I just subtract 1 from it, I get its multiplicative inverse
- And so that already that seems intriguing.
- but then even this statement over here is kind of interesting
- because you've redefined phi in terms of 1 plus 1 over phi
- so we can acutally think of it this way
- we could say, that phi is equal to 1 plus 1 over
- phi, but instead of writing phi, we say wait
- phi is just 1 plus 1 over, 1 plus 1 over
- and instead of saying phi. that's just 1 plus 1 over
- and I could just write phi again, or I could just keep on going
- i could just keep on going like this forever
- I could say that's 1 over, 1 plus 1 over
- and just keep on going on and on and on forever.
- and this is a recursive definition of a function
- were a func..were a recursive definition of a variable
- was defined in terms of itself
- but even this seems like a pretty neat property
- but we want to get a little bit further into
- we actually want to figure out what phi is
- what is the value of phi this weird number this weird ratio
- that we are beginning to explore
- so lets see if we can turn it into a quadratic equation that we can solve
- using fairly traditional methods
- and the easiet way to do that is to multiply both sides
- of this equation by phi
- and then you get phi squared
- plus, or phi squared is equal to phi
- phi plus 1
- phi squared is equal to phi plus 1
- and then, actually im going to take a little bit of a side here
- because even this is interesting
- because of then if we take the square rot of both sides
- of this, you get, scroll down a bit, you get phi is equal to
- the square root of, and ill just switch the order here
- the square root of 1 plus, the square root of 1 plus
- phi
- so once again we can set up another recursive definition
- phi is equal to the square root of 1 plus phi
- and i could write phi there, but hey
- phi is equal to the square root of
- 1 plus, is equal to the square root of 1 plus
- and i could write phi there, but hey phi is just equal to
- the square root of 1 plus
- square root of 1 plus
- and you could just keep on going on and on like this forever
- so even this is neat
- the same number that can be expressed this way
- the same number were if i just subtract one from it
- you get it's inverse
- it can also be expressed in this kind of recursive square roots
- underneath each other
- so this is already seeming, starting to get very very intriguing
- well lets get back to business
- lets actually solve for this, this magic number
- this magic, this magic ratio that we started thinking about
- and, and really from a very simple idea
- that the ratio of the longer side
- to the shorther is equal to reatio of the sum of the two
- to the longer side
- so lets just solve this as a traditional quadratic
- lets get everything on the left hand side
- so we are going to subtract phi plus 1 from both sides
- and we get phi squared
- minus phi, minus 1
- is equal to zero
- and we can solve for a phi now using the quadratic formula
- which we've proven in other videos
- you can prove using completing the square
- but the qaurdatic formula
- you say negative B, negative B is the coefficient
- on this term right here
- so let me just write down
- A is equal to 1
- that's the coefficient on this term
- B is equal to negative 1
- that's the coefficient on this term
- c is equal to negative 1
- that's the coefficient that's really the constant term right over there
- so the solutions to this - phi
- we are actually only gonna care about the positive solution
- cause we're thinking about a positive
- both of, if we go to our original problem here
- we're assuming that these are both positive distances
- so we care about the positve value over here
- we get phi is equal to
- I'll do it in orange
- negative B, well negative negative 1 is 1
- plus or minus the square root of
- B squared, B squared is going to be 1
- minus 4AC
- A is 1, C is negative 1
- so negative 4 times negative 1 is positive 4
- so 1 plus 4 all of that over 2A
- so A is 1, so all of that over 2
- so phi is equal to 2, and once again we only care about
- the positve solution here
- this is going be the square root of 5
- you've have 1 minus the square of 5
- you're gonna get a negative in the numerator
- so we only care about the positive solution
- 1 plus the square root of 5
- over 2
- so this seems like a pretty
- pretty interesting, interesting number
- lets actually take a calculator out
- and see if we can get the first, the first few places
- of this magic number phi
- so let me get my calculator out
- and lets just actually evaluate it
- and you might recognize that square root of phi
- is an irrational number
- and so this actually, this whole thing is going to be an irrational number
- but I'll prove that in another video
- which means its never repeats,
- it goes on and on and on forever
- but lets actually evaluate it
- so its 1 plus the square root of 5
- 1 plus the square root of 5
- divided by 2
- so it this
- 1,6180339
- so let me put that aside, put that aside
- and let me write it down
- and this is where it starts to get really interesting
- and mysterious
- so this number right over here
- is 1,618033988....
- and it just keeps on going on and on and on
- keeps on never terminating, never repeating
- so that by itself, hey its this cool number
- its this ratio that has all of this neat properties
- which are pretty crazy
- any way that you express it
- but whats really neat is if we revisit this thing right over here
- because what is 1 over phi going to be?
- so, 1 over phi, 1 over phi. which we sometimes denote
- with a capital phi (Φ)
- we already know, 1 over phi
- is just phi minus 1
- so we actually, we can do this in our heads
- one over this is just going to be
- 0,618033988
- It don't know, there something wacky about that
- that the inverse of the number is really just the decimals
- left over after you get rid of the one
- that by itself is kind of a crazy idea
- but it gets even crazier!
- because this number is showing up everywhere
- and as you might imagine from the title of this video
- this phi right over here
- this is called the golden raito
- this is the golden ratio
- and it shows up everywhere
- "Golden Ratio"
- it shows up in art, it shows up in music
- it shows up in nature
- and just as to get an idea of were it shows up in nature
- it shows up in very pure ideas
- so if I were to just draw
- if I were to draw a perfect star
- if i were to draw a regular star like this
- let me just draw it like this
- draw it right over here
- so this is just a regular star
- all the lengths are equal
- all the, let me draw it a little bit better than that
- so if I just draw a star like this right over here
- or sometimes this would be called a pentagram
- some amazing things start to happen here
- the ratio of this pink side to this blue length
- right over here, thats the golden ratio
- the ratio of this magenta to this pink, is the golden raito
- as it should, by definition
- now the ratio of this magenta to this orange
- is also the golden raito
- it just keeps on showing up in a ton of different ways
- when you look at a pentagram like this
- if you look at something like a pentagon
- a regular pentagon
- were all the angles are the same and all the sides are the same
- a regular pentagon
- if you take any of the diagonals
- of a regular pentagon
- so right over here, so if you take this diagonal right over here
- the ratio of this green side to
- and what I am talking about the diagonals
- the ones that actaully aren't one of the edges
- the ratio of any of the diagonals
- to any of the sides, is once again
- this golden ratio
- so it keeps showing up, on and on and on
- and we can do interesting things
- with the golden ratio
- let's say we had a rectangle
- were the ratio of the width to the height is the golden ratio
- so lets try that out
- so lets say this is its height
- this is it's width
- and that the ratio
- so lets call this A, lets call this B
- and the ratio of A to B is equal to phi
- it's that 1,61.. so on and so forth
- and let me scroll down a little bit
- so that is going to be equal to phi
- so thats, you know maybe thats something
- interesting to do, maybe thats a nice looking rectangle
- of some sort
- but let me put out a square here
- so let me separate this into a B by B square
- so this is a B by B square right over here
- and then, actually let me do a little bit
- let me draw it a little bit differently
- this rectangle, it isn't exactly the way I would want to draw it
- so the ratio might look a little bit like this
- so the ratio of the width to the length
- or the width to the height, is going to be the golden ratio
- so A over B is going to be
- that golden ratio
- and let me separate out a little B by B square
- over here
- a little B by B square
- this has width B as well
- and so this distance right over here
- is going to be A minus B
- so now it is a B by A minus B square
- so we have a...actually I should say
- we have a B by B square
- right over here
- this is B by B
- and then were left with a B by A minus B rectangle
- now wouldnt it be cool if this is also
- this was also the golden ratio?
- and so lets try it out
- lets find the ratio of B to A minus B
- So the ratio of B to A minus B
- well that is going to be equal to
- 1 over the ratio of A minus B to B
- I just took the reciprocal of this right over here
- and this is just going to be equal to
- 1 over A over B minus 1
- right, I just rewrote this right there
- and that is just going to be equal to 1 over
- phi, the ratio of A to B we said by definition was phi
- phi minus 1
- but what is phi minus 1?
- Well phi minus 1 is 1 over phi!
- It's this cool number
- So this is equal to 1 over
- 1 over 1 over phi
- which is once again just equal to phi
- So once again, the ratio of this smaller rectangle
- of it's height to it's width
- is once again, this golden raito
- this number that keeps showing up
- and then we can do the same thing again
- we can separate this into an A minus B by A minus B square
- Just like that
- and then we'll have another "Golden Rectangle"
- sometimes it's called, right over there
- and then we can separate that in to
- a square
- and another golden rectangle
- and we can separate that in to a square
- and then another golden rectangle
- actually, let me do it like this
- it will be better
- so let me separate, let me do the square up here
- so this is an A minus B by A minus B square
- and then we have another golden rectangle right over here
- I could put a square right in there, we'll have another golden rectangle
- that we can put in another square right over there
- you have another golden rectangle...I think you see
- were this is going
- another square and another golden rectangle
- Which by itself starts to create a cool design that we can keep
- kind of circling in and in and in
- and then if we actually draw an arc here
- something kind of cool happens
- if we have an arc that kinda traces these things out
- We have something, a pattern that you might have seen
- many times before
- and that pattern does not look too different from
- what you might see in something like
- a nautilus shell
- and it shows up all over the place in nature
- and that makes sense
- because just the way cells construct themselves
- it kinda makes sense to be the same
- at different scales, and the ratio from one scale
- to the next is maybe the same as
- kinda the constituent ratios
- This right here, and it shows up all over art
- a lot of Leonardo da Vinci's paintings
- He never, well, not explicity stated it but there's a lot of
- interesing ratios in them.
- But Salvador Dali, this painting right here
- "The sacrement of the last supper"
- he explicity used the golden ratio
- So the actual ratio of the width to the height
- is the golden ratio
- so this is a golden rectangle
- And also there's all sort of ratios and I'll invite you
- to explore it, the ratios of the different parts in the tables
- to were it sits in the painting
- is the golden ratio
- it shows up a ton in this painting
- and then he does have the pentagons over here
- and we know that the ratio of the diagonal to the sides of the pentagon
- are also the golden ratio
- and so he just tought it was a really cool thing
- and theres all sort of neat things
- that if you find, you know where these two
- guys are bowing down are
- if you draw that line right over here
- this is the golden ratio
- the ratio of this length right over here
- to that length over there, once again the golden ratio
- It just keeps showing up in this painting
- So it's a really really really cool thing
- and I really encourage you to explore this further
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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