Infinite Square Roots

If you square root a number infinite amount of times, will it eventually become 1, or some sort of whole number? Can you please give proof as to why you got your answer? Thanks!

#Algebra #InfiniteDescent #Infinity #SquareRoot

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
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Comments

Let y = ... sqrt ( sqrt ( sqrt ( x

Square both sides:
y**2 = ... sqrt ( sqrt ( sqrt ( x 
OR
y**2 = y

y**2 - y = 0
y( y - 1 ) = 0
y = 0 or 1

y cannot be 0 since the input number (x) is >0
so y=1

Vincent Miller Moral - 5 years, 9 months ago

You need to show that the sequence converges to $1$ . Finding a fixed point is not sufficient for convergence, e.g., instead of taking square roots, if we square the numbers, for any initially chosen number $N>1$ , the sequence diverges to $\infty$ . But you still have the equation $y=y^2$ , hence $y=0$ or, $y=1$ .

Abhishek Sinha - 5 years, 9 months ago

If the initially chosen number is $N\geq1$ the sequence $\{x_n\}$ defined as $x_1=N, x_{n+1}=\sqrt{x_n}, \forall n\geq 1$ converges to $1$ . This follows from Banach's Fixed Point Theorem because the map $f:[1,N] \to [1,N]$ defined by $f(x)=\sqrt{x}$ is a contraction map, with the unique fixed point $1$ , in the metric space $[1,N]$ , with the usual euclidean metric.

Abhishek Sinha - 5 years, 9 months ago

It's some kind of elvish. I can't read it. :)

Vincent Miller Moral - 5 years, 9 months ago

Not really a proof, but if we look at $\sqrt{x}$ , we know the result will be less than $x$ (unless of course $x$ is less than or equal to 1.

When we square-root a number an infinite number of times, the number $x$ must lower in value an infinite number of times, until 1 is reached.

Interestingly, when we square-root a number greater than -1 but less than 1 an infinite number of times, we get back to 1. When we square (not square-root) a similar number an infinite number of times, we get back to 0.

For negatives, squaring gets us infinitely high (or infinitely low), and square-rooting gets us imaginary numbers.

Clive Chen - 5 years, 9 months ago

sqr(sqr(sqr(...sqr(n) = x , X^2= sqr(sqr(...sqr(n) , X^2=X , X= 1 or 0 . I think that would be it

Caio Garcez - 5 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$