2:53 the reciprocal of ... [computing] ... phi!

@michaelroditis1952

4 жыл бұрын

That is literally the first time I hear him pause!

What I hated about this story when it came out is that all the headlines were like "Boy Genius Baffles MIT Scientists," when it should've been "Teen who Watches Numberphile Notices Sign Error."

@AshiqZaman3

8 жыл бұрын

+KapustaCuber Over Hyping any situation is the selling point in Journalism these days.

@varamonte

8 жыл бұрын

+Ashiq Zaman Nothing new. You could amend your comment to be more encompassing and still correct by taking out "these days".

@AshiqZaman3

8 жыл бұрын

well said.

@laurelindo9850

8 жыл бұрын

+KapustaCuber I see students on my Engineering Physics program who sometimes point out a mistake in the lecturers' calculations on the board, cleaarly those students should be awarded the Nobel Prize, lmao.

@ronalddump4061

4 жыл бұрын

phi = 1.61803....... = (sqrt5 + 1)/2 PHI = 0.61803....... = (sqrt5 - 1)/2 There was no "error" in any event.

Give this problem a try and when you're ready keep watching the video for the solution!

@GourangaPL

6 жыл бұрын

Did you figure it out?

@sabinrawr

4 жыл бұрын

... for "a" solution...

I prefer to call them "low phi" and "high phi"

@aaronbernal3189

8 жыл бұрын

+Copydot why phi?

@guiguspi

8 жыл бұрын

+Aaron Bernal Is the name of the greek letter used.

@guiguspi

8 жыл бұрын

+Aaron Bernal Φ

@RoboBoddicker

8 жыл бұрын

***** It took me way too long to get that joke ;)

@guiguspi

8 жыл бұрын

+Copydot Wi-Fi? if so took me till now.

Did you phigure it out?

@katzen3314

7 жыл бұрын

phigure out how to spell.

@jacksainthill8974

7 жыл бұрын

Katzen4u Phigure out what the video was about.

@katzen3314

7 жыл бұрын

Jack Sainthill ?

@jacksainthill8974

7 жыл бұрын

Katzen4u Hello, yes, I'm still holding.

@katzen3314

7 жыл бұрын

Jack Sainthill What was the joke?

It's like landscape vs portrait.

@thebammer5166

7 жыл бұрын

Now that you mentioned it, that sounds right to me.

@rv1111

6 жыл бұрын

denelson83 it is indeed

@mikaeelshah7205

6 жыл бұрын

Well sort of the ration from the sides of an normal piece of paper is 1:sqrt(2)

@sabinrawr

4 жыл бұрын

@@mikaeelshah7205 That only works on A-series paper (like A4). In the US, "Letter Size" is 8.5x11 inches, slightly different than A4.

@aarthykanagarajan7912

4 жыл бұрын

nailed it

One more note: PHI + 1 = phi. That's what makes the Golden Ratio so unique. You add one to the PHI ratio and you get its reciprocal, the phi ratio.

@filippo6157

2 жыл бұрын

I have a question: is this caused by the definition of tha golden section itself? (Middle proportional between the original segment and the part that's left)

@No_king1143

Жыл бұрын

1 + 1 = (1+1) thats so cool am i right

It makes sense since by definition the golden ratio is a number that satisfies the equation: 1 + (1/x) = x

@piedsniques

8 жыл бұрын

Indeed, and it would have been more interesting to have a video on this: giving more properties of this number at once, and demonstrating why the golden ratio has this value

@rabindranathghosh31

6 жыл бұрын

means x=(x+1)(x-1)

@user-nm4ni5us8j

6 жыл бұрын

x^2-xsqrt5+1=0 you get both 1.618 and 0.618

@angelmendez-rivera351

5 жыл бұрын

Read in reverse. That isn’t the equation of the golden ratio

@angelmendez-rivera351

5 жыл бұрын

That is not it. Historically, the golden ratio had to be positive by definition since geometric length ratios inherently are positive.

You should have mentioned that the two ratios are in fact different by exactly 1, not just approximately 1.

@potato-hj9nm

8 жыл бұрын

its kind of common sense that they are different by one though.

@B3Band

8 жыл бұрын

He never said they were different by approximately 1. Literally never said anything like that in the entire video.

@jesusthroughmary

8 жыл бұрын

2:15

@linkinprk1981

8 жыл бұрын

Listened 5 times, he still never said they were different by approximately 1. Starting from 2:00 He said... The square root of 5 plus 1 over 2 is the ratio of the sides, or approximately 1.618. But you could also tell me ratio of the shorter side to the longer side. Which is the reciprocal. This is sometimes denoted by the capital letter Phi and that's equal to .618

@jesusthroughmary

8 жыл бұрын

He shows graphically that both amounts are approximate, and never states that the amounts are in fact different by precisely 1.

So the mind-blowing thing about the golden ratios are that the difference and the product of the two numbers are both 1. They are the only numbers where this happens.

@konstantinkh

8 жыл бұрын

It gets better. You can take an arbitrary power of φ to show that φⁿ = F(n) φ + F(n - 1), where F(n) is the nth Fibonacci number. This still holds true when you extend Fibonacci numbers to negatives and even the entire real line. It's a fun little number.

@aryamankejriwal5959

5 жыл бұрын

Konstantin Khitrin that’s so cool!

@aryamankejriwal5959

5 жыл бұрын

Konstantin Khitrin does it work for both phis?

@misternobody6798

4 жыл бұрын

@@aryamankejriwal5959 Doesn't hold for lower phi

@37rainman

4 жыл бұрын

Here is a much simpler but very interesting and elegant relationship concerning phi which "I discovered": Raise phi to ANY positive integer power (n), (except 1, which is a special case). Subtract the result from its nearest integer. The reciprocal of that difference is exactly the original phi^n. Thus, for example, if you raise phi to any ODD power, the reciprocal of the decimal portion of the result is exactly equal to the result. FI: phi^5 = 11.0901.... 1/(0.0901.....) = 11.0901..... Exactly. (Notice that this also works with the n =1 situation, which does not fit my original statement) If you raise phi to any EVEN power, and subtract that from the next larger integer, the reciprocal of the difference exactly equals the original power of phi. FI: phi^4 = 6.85410..... 7 - 6.85410.... = 0.14590..... 1/(0.14590....) = 6.85410..... The results are exactly accurate out to infinite digits

the thing is that while both formula are valid for the golden ratio only the one with the positive sign is valid for the golden number (phi) the museum just got it wrong but found a cheap way to avoid the public shame of getting it wrong. that's why, at first, they were forced to recognize their error and then, once they found a way off, they changed their minds.

The explanation using the ratios of the rectangle sides illustrates it well (long side / short side) vs (short side / long side). I never thought of it that way. That makes it a lot easier to make sense of.

Is anyone else freaked out that the reciprocal differs by exactly 1? Are there any other ratios that differ by exactly 1, or is this another reason why the golden ratio is what it is: so pleasing to the eye, prevalent in nature, harmonically significant, etc.?

@taemyr

7 жыл бұрын

No other ratio differ from it's reciprocal by exactly 1. If you learned about the golden ratio by exploration of golden rectangles this is not surprising.

@ExplosiveBrohoof

7 жыл бұрын

Consider the equation x(x-1) = 1. Note that x and x-1 are reciprocals of one another. Expanding this equation out, we get x^2 - x = 1 or x^2 - x - 1 = 0. This is the standard golden ration polynomial. Solving it via the quadratic formula gives us the solutions x = (1+sqrt(5))/2, x = (1-sqrt(5))/2. So technically, the number (1-sqrt(5))/2 also has a reciprocal that is one less than it. However, if you do the computation, you'll find that it's just -1/phi, and so its reciprocal would be -phi.

@yitzhakkornbluth2554

7 жыл бұрын

In fact, I think all the stuff that's true of the golden ratio boils down to the fact that it fulfills that particular quadratic equation.

Its also fun (and kinda important) that the irrational decimal portion of both golden ratio solutions is identical, its only the 1 at the front thats different. (And yes, I only know that because of numberphile)

Or just use the +/- symbol :/

It's the multiplicative inverse. (√5 + 1)/2 * (√5 - 1)/2 = 1

BitTorrent ratios on trackers are often given as a decimal number, which is assumed to be [uploaded bytes] divided by [downloaded bytes], though just using the word ratio is ambiguous as one can express the same information in the form of DL/UL. It is a bit easier to tell how well a user is behaving w/r/t to the tracker if the ratio is expressed as UL/DL because the desire is for users to seed as much as possible, remaining uploading after their torrent download completes.

I Figured it out by changing Phi's property (Phi^2 = Phi+1) to Phi^2 - Phi - 1 = 0, Solved it with the roots formula and came up with these two answers.

There had to be SOME pair of non-integer numbers, above and below 1, that when multiplied together made 1, that would have the same decimal digits. It's so cool that this also turns out to be the golden ratio. The next pair out is the silver ratio!

Please make a video on the silver ratio.

The properties of the golden ratio are that the ratio of the longer side to its shorter side is the same as the ratio of the sum of both sides to its longer side. In other words for 2 sides a and b where a is the longer side, (a+b)/a = a/b Letting x = a/b and substituting, you end up with the equation 1 + 1/x = x Which becomes a quadratic x^2 = x+1 Rearranging and solving this quadratic you get (1+sqrt(5))/2 or (1-sqrt(5))/2 The 2nd solution is a negative number and therefore ignored since only positive values are considered in ratios. This is NOT equal to the value that the museum wrote which is (sqrt (5) - 1)/2 And this value does NOT follow the properties of the golden ratio, so the museum is still wrong

The museum did the equivalent of saying "it's raining like dogs and cats" In Die Hard, John McClane shoots you for saying shit like that.

also noticing that 1/Golden ratio = Golden ratio -+1 then you can solve 1/x=x -+1 to get x= (sqrt(5) -+1)/2 where -+ means plus or minus

@Hydrastic-bz5qm

5 жыл бұрын

±±±±±±±±±±±±±±±±±±±±±

why he is screaming

@dannygjk

8 жыл бұрын

+robin van Sint Annaland Yes, use your indoor voice please. and btw isn't it normally written as 1.618...?

he sounds very angry

@LillianWinterAnimations

8 жыл бұрын

+Mister Dee He doesn't sound that way to me, at least.

Another way to see that the reciprocal of Phi is Phi-1, is to see that Phi = 1 + 1 / Phi, so if you subtract 1, you get 1 / Phi, the reciprocal

You can look at the golden ratio as x where x^2=x+1. (It's reciprocal is x where x^2=1-x.)

@Blaqjaqshellaq

7 жыл бұрын

The golden ratio is related to the Fibonacci series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... where the third and later entries are the sum of the previous two. As the series progresses, the ratio of an entry to the previous one gets closer and closer to the golden ratio.

@bonbonpony

7 жыл бұрын

The connection between the golden ratio and the Fibonacci series doesn't have to use any infinite limiting process and doesn't have to be only an approximate - there's a PRECISE connection between the golden ration and the Fibonacci (and Lucas) numbers, much more simple: goo.gl/43Gie3 The blue numbers are Fibonacci numbers, and the green ones are Lucas numbers. Connecting them in this formula will give you all the subsequent powers of Φ ;)

You can even use fibonacci sequence formula to derive the golden ratio

But why do they use the capital letter for the smaller value?

For a rectangle of golden ratio proportions, can still get 1.618 if consider the longer side first, namely the ratio of the longer side to the sum of the long & short sides. So if L = 1.618 and S=1, then (S+L)/L = (1 + 1.618) / 1.618 = 1.618 (this works because 1/Phi = Phi - 1)

It is the only where the multiplication and difference of both the risults are equal and that is 1. That is why it is golden ration.

Pretty sure the spiral starts small in the center and becomes bigger outwards in nature (you should've flipped the ratio image over its horizontal axis on the nature image)

It's also 2cos36º, which is why it turns up in pentagrams.

the golden ratio also can be expressed as the roots of a quadratic equation x2 - x - 1 = 0 since every solvable quadratic has 2 roots , so does this one and hence the golden ratio can be represented these 2 ways (they're solutions to this quadratic btw)

@AA-100

10 ай бұрын

The 2nd solution is a negative number, which gets ignored when ratios are considered. This is not equal to the value the museum provided (which is a positive number), and that number given by the museum does NOT satisfy the equation x^2 - x - 1 = 0

people forget that the golden ratio comes from completing the square of x^2 - x - 1, giving you (root 5 + or - 1) /2 . The -1 comes from the square

@vonakakkola

2 жыл бұрын

but this equation gives you -0,618 and not 0.618

I think we should consider it as a percentage rather than a ratio , so it's ±61.8% ...... depending on whether we increment or decrement

that was a long way to simplify the multiplication... (a+b)(a-b) is always (a^2 - b^2) ; (5 -1) in our case.

i mean, it goes beyond art, even in maths it just says "kek idc it's going to work" lol. Pretty interesting how that golden ratio is the one taht can fit +/-1 that precise way. Maybe that's what makes it special.

I like to nickname 1-φ as the "pyrite ratio" as pyrite is fool's gold.

Good video but you should have included the Fibonacci sequence and how golden ratio is obtained from it ...and the two ways in which these values are obtained i.e ratio of bigger number to smaller and the ratio of smaller number to bigger.

they are reciprocals of each other.

3blue1brown made an excellent video about why the 1.618 was considered as the golden ratio and 0.618 wasn't

Am I the only one to spot the mistake. The golden ratio is (1 ± sqr(5))/2! This will not make a difference with the +, but 1 - sqr(5) is not equal to sqr(5) - 1!

You can also find the geometric mean = 1

(√6 + 2)/2 is a golden ratio? Please reply. As ur 1st method (√10 + 1) / 3, I checked & found it is golden ratio. when I kept it in equation x^2 - x - 1 = 0 but it is not satisfy. Please clarify me.

@AA-100

10 ай бұрын

It doesnt satisfy because its not the golden ratio

Theres also the negative root of 5 plus 1 over 2

@sabinrawr

4 жыл бұрын

True, but that results in a negative ratio. Since the Golden Ratio generally refers to lengths, and we cannot have a negative length, having a negative ratio is an extraneous solution.

So is the golden ratio then (√5±1)/2

Presh was very passionate about this one.

Actually it isn't the same... The golden ratio is defined by the ratio of the sum of two number to the larger number or the ratio of larger number to smaller number?...?.. So it will be greater the one only?

1. The smaller one is negative!! 2. The golden ratio match X^2=1+1/X 3. The 1+ part changes between the other ratios silver etc.

Let's assume the golden ratio is the greatest of the reciprocals :)

Whats trippy is minusing 1 flips phi over!

I ran into this in a Brilliant.org quiz question (if x + 1/x = 5, what is X^2 + 1/x^2 ?). If you do this by brute force (setting up quadratic equation, solving for X, you get fraction with sqrt(23)+ rational in numerator. Plugging this into X for second equation, algebra balloons into a nightmare. But I recognized right away that 1st equation has 2 roots, (which have + and - rational terms, just like the golden ratio) and these HAD TO BE INVERSES OF ONE ANOTHER (by "algebraic symmetry"). So rather then inverting X value for the 2nd term (which now has sqrt(23) + rational in denominator), I summed the squares of the 2 roots. Rational terms neatly cancelled out, and (sqrt(23)^2) was the answer. This is the same answer as doing it "the easy way", which is simply squaring (X + 1/X), setting equal to 5^2). A 2 term appears on left side, subtracting from the 25.

Can phi be (1-sqrt(5))/2 or -0.68...?

@carultch

2 жыл бұрын

That's what 3blue1brown calls "phi's little brother". They are both solutions to the same quadratic equation, but only one of them is positive.

I'm pretty sure the golden ratio is money over bitches

@bestredditstories1158

7 жыл бұрын

Jacob Vogt Well, damn. You're right.

And oddly close to the ratio of 1 mile to 1 kilometer.

What a discovery you just did...lol

There is a clarification needed to be made here. The golden ratio is also know to be the solution to "finding a number that to square, you have to just add 1". The value of 0.6180339.. is not a solution to that. But the value of 1.6180339... is. Geometrically, then both are right, but if we go by this polynomial definition, 1.6180339.. is the answer. The other solution to the quadratic is -0.6180339.. not the positive number.

That's really cool. I also noticed that the golden ratio is the only solution to x-x^-1=1 Basically, it is the only number where the number and the reciprocal are exactly 1 apart. I'm not sure why, but I suspect that has to do with something.

@ffggddss

7 жыл бұрын

x - 1/x = 1 has two solutions; although only one of them is positive. Multiply through by x, and collect terms, keeping in mind that this can introduce extraneous solutions: x² - x - 1 = 0 x = ½(1 ± √5) = {φ, -1/φ} Check (from the vid, we already know that φ - 1 = ½(√5 + 1) - 1 = ½(√5 - 1) = 1/φ.): x = φ: . . . . . . φ - 1/φ = φ - (φ-1) = 1 . . √ x = -1/φ . . . -1/φ - (-φ) = φ - 1/φ = 1 . . √ Note that the opening equation gives you both: x⁻¹ = 1/x = x - 1 and x² = x + 1 Another interesting connection of the Fibonacci numbers with φ (besides that the limit, as i→∞, of Fᵢ₊₁/Fᵢ is φ) is that integer powers of φ are made of consecutive Fibonacci numbers acting on φ and 1: φª = Fₐφ + Fₐ₋₁ . . . note that this works with negative indices, too - you can continue the Fibonacci sequence "to the left" by using the usual recursion: Fₐ₊₁ = Fₐ + Fₐ₋₁ to get Fₐ₋₁ = Fₐ₊₁ - Fₐ These negative-index Fibonacci numbers are just the positive-index ones, but with alternating signs. ... , -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, ... And of course, from the powers of φ equation, you can get both 1/φ and φ².

@MitchBurns

7 жыл бұрын

ffggddss Yes, but the negative solution is the negative reciprocal of the golden ratio, so still the golden ratio.

@AA-100

10 ай бұрын

Yep, but definitely not equal to the value the museum showed. Museum made a mistake. Period.

I read about the golden ratio first in a book by Dan Brown. If you multiply any number from the Fibonacci sequence and round it off, you get the next number of the sequence.

@NPCrash

8 жыл бұрын

The marvellous thing about the golden ratio is that its conception was made with Geometry, hundreds of years before Fibonacci applied it to his sequence.

@ffggddss

7 жыл бұрын

+Nick Crash Yes, the division of a segment, AB, into so-called, "mean and extreme ratio." Placement of point C on segment AB, such that AC:BC = AB:AC The ratio that satisfies that proportion is φ:1.

This video is wrong. Yes, there are two ways of writing a ratio, but the values those two ratios represent are definitely different. The golden ratio actually has a definition, and when we calculate the numerical value based on that definition, the golden ratio is (1 + SqRt5)/2. (SqRt5 - 1)/2 is the RECIPROCAL of the golden ratio, not a different expression for the same ratio, much in the same way that 2 & 1/2 are different ratios even if represented by the same dividends.

@andychess4606

5 жыл бұрын

Angel Mendez-Rivera Definitions can be differently understood just like there are 2 metric systems.

@chanderule605

5 жыл бұрын

@@andychess4606 2 metric systems? u sure

So the golden ratio is what the ratio of the Fibonacci numbers approaches. If you divide the larger number by the previous smaller number, you get 1.618... If you divide the smaller number by the next larger number you get .618...

@ffggddss

7 жыл бұрын

Or if you take the sequence infinitely in both directions, then to the right, the ratio of one term to its predecessor goes to φ; while to the left, the ratio goes to -1/φ. These are just the two solutions of x² - x - 1 = 0 or, equivalently, of 1/x = x - 1.

I finally new something before i watched a MYD vid about it

Square route of phive plus one, over 2

it's just the same ratio turned around isn't it?

you're the best. keep it up love your vids.

But the Capital should be Phi and the Lower Case - the Reciprocal.

tl;dw version. It's the golden *ratio*. Doesn't matter if it's longer : shorter or shorter : longer. It's still a ratio and it's golden.

@bonbonpony

7 жыл бұрын

TL;DR: you're an idiot. Turning the ratio upside down makes it a different number. 1:2 = 0.5, 2:1 = 2, and 2 is not exactly the same as 0.5, is it? :P But if you think it's the same, then I'll give you half a dollar and you'll give me "the same": 2 dollars. Deal? ;>

@rainydeestar4806

4 жыл бұрын

@@bonbonpony Why are you replying to a comment 2 years after it was posted ?

@bonbonpony

4 жыл бұрын

@@rainydeestar4806 Why? Is this suddenly not allowed or something? The comment I was replying to will be wrong even 100 years from now (provided that KZread still exists then, because judging by its constant decline, I highly doubt it).

@rainydeestar4806

4 жыл бұрын

@@bonbonpony You missed the joke buddy

@bonbonpony

4 жыл бұрын

@@rainydeestar4806 More like you thought you made a joke, but nobody was laughing :q What is it that so many people these days excuse their brain farts as "jokes"? Jokes are supposed to be funny, and usually are clever.

And this is again Fibonacci. Fibonacci is trying to settle down to the golden ratio at the infinity.

φ ≈ 1.618 Φ ≈ 0.618

Golden ratio is considered to be greater than 1 then it is the left one but if we take it is less than onethen the right is correct.

the second phi squared doesn't equal itself +1 as the golden ratio does

@lagomoof

8 жыл бұрын

+Yoav Shati phi^2 = 1+phi. PHI^2 = 1-PHI

@AA-100

10 ай бұрын

Exactly, the museum is still wrong, and so is this video that is trying to justifty the museums value as correct

@onyong2766

2 ай бұрын

If you take the value of the second phi to be -0.618, then it does. -0.618 x -0.618 = 0.382, and -0.618 + 1 = 0.382. 🙏🙏🙏

Is it merely coincidence that the difference between the 2 values is 1?

This is just so commonplace, by definition, either is good, so the one who asked and the one who answered are both meaningless.

You also have the nice relation between this Golde ratio #2 (0.61803...) and the Golden ratio #1 (1.61803...) just by adding the number 1 to #2 to get #1! Or was that too obvious? :p

@ash82286

8 жыл бұрын

+MisterrLi yeah, the golden ratio is the only number 1 more than its reciprocal

@MisterrLi

8 жыл бұрын

Etaash Katiyar If you only count real numbers of course. Complex reciproxals adds a few...

@GordonHugenay

8 жыл бұрын

+MisterrLi you're both wrong: there is another real number, which is one more than it's reciprocal: -0.618... and there can't be another complex number, which has this property. I mean: we're looking for a solution for x=1/x+1 x^2=1+x x^2-x-1=0, which has exactly two solutions, 1.618... and -0.618...

@MisterrLi

8 жыл бұрын

GordonYoung Why can't there be more complex solutions? If you work with complex numbers, the formula is this: (a+bi)^2-(a+bi)-1=0 Now try to solve this for a (or b), you will get the real (2 or 4) solutions for b=0 and some more solutions with b0. Hint: there are infinitely many complex number solutions!

@GordonHugenay

8 жыл бұрын

So can you give me a non-real complex solution? I won't believe there is one, until I see it: take for instance b=1: the formula then is: (a+i)^2-(a+i)-1=0 a^2+2ai-1-a-i-1=0 (a^2-a-2)+(2a-1)*i=0 a^2-a-2=0 and 2a-1=0 a^2-a-2=0 and a=1/2 (1/2)^2-1/2-2=0 -9/4=0, which clearly is wrong, so there is no solution for b=1. I claim, unless you disprove it with a counter-example, that for any real b≠0, you'll get something similar to the above.

So... lower case phi is the golden ratio and upper case Phi is the conjugate.

That is not a lower case phi

-(sqrt(5)-1)/2 duh you forgot the minus sign

@deldarel

8 жыл бұрын

+zboodles2 not at all. I think you got both ways to write big phi confused. This is your equation: -(sqrt(5)-1)/2 = -0.6180... and these are two different equations that are both big phi: (sqrt(5)-1)/2=0.6180... -(1-sqrt(5))/2=0.6180... either that, or you have phi-1 (big phi) and 1-phi (negative big phi/your equation) confused. I admit that it's confusing. I had to look it up to be sure as well. It's just phi's property to do all sorts of kinky stuff with the number 1 and still make sense.

@Double-Negative

8 жыл бұрын

+PrimaPunchy i just found the second zero to x^2-x-1

A significant thing about the golden ratio is that phi - 1 = 1/phi. It is the only (NOTE - see edit) real number that has this characteristic. If you use this feature to try to figure out what it is as a decimal number, it turns into a formula: x-1=1/x , which turns into x^2 - x - 1 = 0. Solving this using the quadratic equation results in both of the formulas in the video. EDIT: I was wrong here. Firstly, the formulas in the video are NOT quadratic solutions to the same formula. The quadratic equation would give (1 + root5)/2 and (1 - root5)/2. One of the formulas in the video is (root5 - 1)/2 which is clearly not the same. Of course there are two solutions to the quadratic - I was only thinking of positive numbers. Interestingly, the second quadratic solution is negative PHI, that is, the negative of the reciprocal of phi. (Which corresponds to the reversal of the order of operations in the video compared to the quadratic - they were talking about the positive reciprocal.)

@BlueCosmology

8 жыл бұрын

+losthor1zon It is incredibly obvious that that is not the only real number that has that characteristic. x^2 - x - 1 is obviously continuous over the whole real line, x^2 - x - 1 obviously goes below 0 x^2 obviously increases faster both as x tends to -inf and +inf than - x - 1 hence there are obviously two real solutions.

@losthor1zon

8 жыл бұрын

+BlueCosmology - Sorry... you're correct. I was only thinking of positive numbers. (It's been a while since I did anything with a quadratic equation! lol) Oh, and.. thanks!

Hi Presh, Great Vid .... But albeit that instructors in the 80's and 90's had different opinions.... Wouldn't it make sense to have Capital Phi as the greater notation, (1.6180339 .....) and minor Phi as the reciprocal. (0.6180339 ....) ??? Kinda like Big C calories and little C calories. What say you ?? Stay Safe.

And so is 2/(sqrt.5-1). And phi-Phi is 1

They're NOT _equivalent_, they're *INVERSES* of each other: φ = 1/Φ, or Φ = 1/φ, or φ·Φ = 1. The positive one is the golden ratio, and the negative one is the inverse golden ratio. They both are conjugated solutions to the quadratic equation x² - x - 1 = 0 which can be derived from the definition of the golden ratio: x = (a + b) / b = b / a (the mean proportional between the whole and its parts).

@lukapopovic5802

7 жыл бұрын

Bon Bon They are both positive. I love to use ,,the smaller one" because it's the actual place where I would have to draw a line if a had a picture that has length 1, in such a way that a : b= b : ( a+b )

@lukapopovic5802

7 жыл бұрын

Bon Bon Where a,b are the sections of the picture

Pausing at 1:23... Wouldn't both answers be correct simply because the second root could be positive or negative..? When positive, use +1, when negative, use -1... Don't get the controversy, but I'll watch the rest and see if it makes sense.

@acediamond5399

8 жыл бұрын

Ohhh, okay, I get it now. BOTH cases are using the positive second root of 5, but the two differing positive numbers are reciprocals of each other, so they're the "same" ratio. Makes sense!

knock knock you can also consider a rectangle with length 2 and width sqrt(5) + 1 and you could say the same thing.

But (sqrt of 5-1)÷2 = 2÷2 = 1

@ospero7681

4 жыл бұрын

Sqrt(5)-1, not sqrt(5-1), obviously.

This is why I pronounce Φ "fai" but φ "fee".

@paulingersoll9197

7 жыл бұрын

Both of the letters he used are phi, one upper case and one lower case. The second letter you used is psi.

@djenni12

7 жыл бұрын

Sorry, copy and paste error (my keyboard doesn't do Greek). You're right. Φ and ϕ.

@bonbonpony

7 жыл бұрын

+Paul Ingersoll nope, they're both phi (upper case and lower case): Φ φ. It's just that the lower case can be written in two different ways depending on the font: φ or ϕ. For psi, here are the upper case and lower case versions: Ψ ψ (they look like a fork).

@TigruArdavi

3 жыл бұрын

which is total BS, because you don't pronounce the upper case representation of a letter different from its lower case representation. 'A' is not different from 'a'. In Greek it's pronounced like 'fee'.

The bigger error is that you rounded it to 1.6 when it is universally viewed that the golden ratio is rounded to 1.618. Similar to pi or e.

Both Φ are the capital letter of phi. The lower case is written φ.

Just Rationalise them

Gold Ratio = 2×sin54° & 2×sin18°^

1,1,2,3,5,8,13,21...……….is known as the Fibonacci Series and as the n th and (n+1)th terms become very large,the ratio of the (n+1)/n th terms tends to (sqrt5+1)/2= large phi.This can be proved analytically.

did yall know that golden ratio is actually equal to -2sin(666 degrees)?

also 1-phi=-1/phi

This is just a semantic game playing post! The Golden ratio is defined a certain way i.e. +1. The other definition which mathematically true is not the golden ratio according to convention! That is like saying the definition of pi is wrong! While pi could be expressed as its inverse it is not the same by definition. Pi and Pi inverse are not the same number although in operations they may give the same result with operational changes to reflect their differences.

@sabinrawr

4 жыл бұрын

Also, 1/pi doesn't have many of the interesting properties that 1/phi does. That aside, if you want to say that 1/pi is the ratio of a circle's diameter to its circumference, I'll support you.

Its just the inverse... Love it when people who dont have a clue about maths talk about maths

They got it wrong. The boy was right. I should have expected better from the Boston Museum of Science. The golden ratio is the limit as n tends to infinity of F[n+1]/F[n] where F[n] are the terms in the Fibonacci series. You could also say the ratio is F[n]/F[n+1] too but this is uncommon. We normally think of it as the number to multiply the very high Fibonacci numbers with in order to approximate the next one. There is a simple way to determine this ratio from the above information. For this limit, F[n+1]/F[n] = F[n+2]/F[n+1] = (F[n+1]+F[n])/F[n+1] = 1 + F[n]/F[n+1] so, making this ratio be x gives x = 1 + 1/x or x^2 - x - 1 = 0 (x - 1/2)^2 = 5/4 x - 1/2 = ±√5 / 2 x = (1 ± √5)/2 now, since x must be positive, the correct solution is (1 + √5)/2 or (√5 + 1)/2. If we'd taken the other definition for the golden ratio (the uncommon one), we'd be solving 1/x = 1 + x or x^2 + x - 1 = 0 (x + 1/2)^2 = 5/4 x + 1/2 = ±√5 / 2 x = (-1 ± √5)/2 now, since x must be positive, the correct solution is (√5 - 1)/2.

@paulingersoll9197

7 жыл бұрын

F[n+1}/F[n] is always > 1, so the boy was wrong, as was the museum. In the original problem solved thousands of years ago, the question was the ratio of a line segment a to the remainder b such that a/b = b/(a+b), where a is the shorter segment. The relationship to FIbonacci sequences came way later.

@lukapopovic5802

7 жыл бұрын

Paul Ingersoll Still you will get b = ( -1 -+ sqrt(5)) / 2 the line can't have negative length so b = ( -1 + sqrt(5)) / 2

It is the only formula with fractal properties 1,618 and 0,618 1/0,618=1,618

@bonbonpony

7 жыл бұрын

What is fractal about it? :q

Actually it's the square root of 1.25+0.5

Actually the capital letter is the positive and the lowercase is the negative, and that’s not the correct form of lowercase phi, that’s the simpler form because of the difficulty of drawing the uppercase phi, that’s just the shorthand way of writing the uppercase phi. This is the lowercase phi: φ

Of course that is also the golden ratio. One number is ~8/5, the other is ~5/8.

@AA-100

10 ай бұрын

It's not, the other number is ~ -5/8 and not ~5/8, that other number is negative. The museum made an error