Infinitely Nested Michael Jordans

Infinitely Nested Square Roots,
sqrt(20+sqrt(20+sqrt(20+...))),

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blackpenredpen | 曹老師

[blackpenredpen]

What if we had (x-5)(x-4)=0? Can we still do it?

[stevethecatcouch]

Are you saying Michael Jordan is a radical?

[benjaminparra]

12:34 my parrot knows you voice, the have heart you when I see your videos!

[balinttatai]

you are entertaining people with math, congratulations :)

[okaro]

You can do these easily with a calculator. First calculate sqrt(20). Then enter sqrt(20+Ans) and hit the = or EXE repeatedly.

[chimetimepaprika]

When life gives you dilemmas, make dilemmanade!

[sonicpawnsyou]

I just recommended you to my cat. She started purring afterwards. 😍

[kaushikmanna]

The process of generating this "nested square roots" helped me to get the answer (may be I'm not very sure it will work for all real nos.) to a long unanswered question that "for all real nos. there is a sequence of rationals and a sequence of irrationals that converges to that real no."

THANKS

[poutineausyropderable]

Induction:
x{1}=sqrt(20) <=5



If 0>=x{n)<=5,
x{n}+20<=25
sqrt(x{n}+20)<=5
x{n+1}<=25

.
We've already proven it's a fix point at x= 5(Video).

Now use indcution to prove x{n+1}>=x{n} if x{n} <=5 and show that it is decreasing if x> 5.

[aviralaryal]

sqwarut of twaanty, sqwarut of twaanty !!
So cute !!

[lilian]

Hi blackpenredpen. I juste want to say that I love yours videos and I understand everything. I'm French, and I'm only 15 years old. You're the best!

[luisaguinaga]

I am 1 year late but for me this video is new.
There are 2 roots for the expression (x-5)(x-4)=0, they can express as x=sqrt(-20+9*x). in the previous case we have a notable difference x= +-sqrt(-20+9*x) there are 2 different algorithms for 2 different cases therefore if we pick + or – we find the 5 or -4, but in this case we have one algorithm with 2 different solutions (Notice that if you pick the – root in the first case you change the x to –x in the second case is the almost the same but in the first case we have 20+5 and 20-4 both roots are at different sides of 20 in the real number line. Now we have 20+45 and 20+36 both at the same side from 20 in the real line). Also is obvious that we cannot do the infinite nested root we need to chop the root at some point, and we have something light this
x=sqrt(-20+9* sqrt(-20+9* sqrt(-20+9* and this is very, very important. This means that in order to calculate x we need an initial value of m. in the first case the algorithm converges for any real value of m at 5 or -4. (even if some roots are complex, the imaginary part goes to 0 and the real part to 5 or -4). But in the second case for a real m that is not 4 the algorithm converge at 5, the interesting case is that 5 is an stable root, any number and the system converge at 5, but 4 is the unstable root is like an inverse cone in equilibrium the minimal disturbance and instated of going back to 4 the algorithm converges at 5

[Dreamprism]

Good explanation. I like teaching this stuff. Infinite nested fractions, square roots, powers, etc. and talking about both what they mean as sequences and how we get solutions with algebra.

[rishadvance]

it was still working and for both 4 and 5 when i used x-4 instead of x+4. The equation 9x-20 gives a perfect square for both 4 and 5. Thank you

[Patrickhh]

Proof that all numbers are equal:
X=
Sqrt(x^2)=
Sqrt(1+x^2-1)=
Sqrt(1+sqrt((x^2-1)^2))
Apply the sequence to the inner sqrt

...
Sqrt(1+sqrt(1+...))=x
It doesn't matter what x is

[MyMusics]

It's much more intuitive from the other direction: We want to represent any real number x as an infinite nested square root.
x = sqrt(a + sqrt(a + ...))
= sqrt(a + x)
x^2 = a + x
a = x^2 - x

Otherwise it is entirely unclear why you would start off by multiplying with (x+4), even if it makes sense in the end.

[rob]

x = a,
(x-a)(x+a-1) = 0,
x^2- x - a(a-1) = 0,
x = sqrt(a(a-1) + x),
let a(a-1) = b,
a^2 - a - b = 0,
(a - 1/2)^2 = b + 1/4,
a = 1/2 + sqrt(b + 1/4),
sqrt(b + sqrt(b + sqrt(b + ...))) = 1/2 + sqrt(b + 1/4) ----> the answer to the nested Michael Jordan problem!

taking b = 1, we get 1/2 + sqrt(5)/2 = the golden ratio.

[Hexanitrobenzene]

"Nested root, not nasty root!" :D

[c.j.]

What about alternating between adding and subtracting?
sqrt(20 + sqrt(20 + sqrt(20...))) is 5, and sqrt(20 - sqrt(20 - sqrt(20...))) is 4
I thought sqrt(20 - sqrt(20 + sqrt(20 - sqrt(20...)))) would be somewhere between 4 and 5, but it's actually ~3.887

[Kurtlane]

Back in 1975, my math teacher gave the following problem:
x = sqrt(1+ sqrt(2 + sqrt(3 + … sqrt(1975))))

Find x.
Nobody in class could solve it. So the teacher showed how to solve it. I didn't get it.

Can anyone solve this?