Repeated Presses of the Square Root Button

Repeated Presses of the Square Root Button

How many of you have ever done this?  You take an ordinary calculator (or even an older-style AOS calculator or an RPN calculator), enter a number, particularly greater than 1, and just kept pressing the square root button?  Eventually, the number would approach 1, and due to the number decimal points the calculator can hold, the display would be 1.

We can theorize on how many times we would have press the square root button before the displayed value falls below a limit level L. 

Taking repeated square roots of x, n times, will lead to:

√ … √ √ √ x

= √ … √ √ (x^(1/2)

= √ … √ (x^(1/4))

= √ … (x^(1/8))

…

= x^(1/(2^n))

The number of times the square root button needs to be pressed until it falls below a limit L (for the first time), the inequality is set up as:

x^(1/(2^n)) < L

Solving for n (x and L are given):

Take the logarithm of both sides:

ln (x^(1/(2^n))) < ln L

1/(2^n) ln x < ln L

1/(2^n) < (ln L)/(ln x)

Taking the reciprocal of both sides:

2^n > (ln x)/(ln L)

Again, take the logarithm of both sides:

ln (2^n) > ln (ln x/ln L)

n ln 2 > ln (ln x/ln L)

Solving for n:

n > ln (ln x/ln L)/ln 2

Implementing a short algorithm on an HP 42S:

01 LBL “BL08”

02 LN

03 X<>Y

04 LN

05 ÷

06 LN

07 2

08 LN

09 ÷

10 RTN

11 .END.

Input:  L followed by x.   (x>2)

I calculated a theoretical n for various limits (L = 2, L = 1.5, and L = 1.01) and various values of x.

Table of Repeated Square Root Button Presses:  Theoretical n vs. Actual n
(Excel, HP 42S)

From the table, it seems that a suitable formula for n (L ≥ 1, x ≥ 2):

n = int(ln (ln x/ln L)/ln 2) + 1  

Where int represents the integer part function.    

Try this out – and maybe revisit a small part of your childhood in the process.  Have a great day,

Eddie

This blog is property of Edward Shore – 2015.

Repeated Applications of Functions

Repeated Functions

If you have ever owned a calculator, chances are you done this: you enter a number and press the square root button many times in a row to see what happens.

For example, you enter say 200. And then you push the square root button. Here are the results for 10 applications of the square root function (to eight decimal places):

14.14213562
3.76060309
1.93922745
1.39256147
1.18006842
1.08630954
1.04226174
1.02091221
1.01040200
1.00518755

Eventually with any positive number, if you press the square root function enough times, you get 1 as a repeated answer.

The following graph shows three functions:

The function in red is the normal square root function.

The function in green shows what happens when the square root function is applied three times to each number.

The function in blue shows what happens when the square root function is applied five times to each number.

Each of the graphs on this blog have been produced using an TI nSpire CAS calculator. All photographs were taken using an iPad 2.

Finding a Closed Form for Repeating the Square Root

If we want to describe what a function would look like when applying the square root n times, we can use a nice, closed, and compact formula.

Let f(x) = √ x = x ^(1/2)

Then

f^n(x) = x ^ 1/2 ^ 1/2 ^ .... ^ 1/2 (n times)
= x ^ (1/ (2^n))
= x ^ (2 ^ -n)

Repeated Sine and Logarithm

Now let's apply common scientific functions and repeat them. First the sine function for x ∈ [-3 π, 3 π].

The graph in blue represents the normal sine function. (x is in radians)

The graph in green is sine repeated 25 times.

The graph in magenta is sine repeated 100 times.

Note with repeated applications of sine, the graph gets more flatter, towards 0, for each point. Could the approximation formula sin x ≈ x have something to do with this?


This next graph shows what happens when you repeatedly apply the natural logarithmic function:

The graph in red is the normal natural logarithmic function.

The graph in green represents the natural logarithm function applied three times.

Something interesting happens with repeating the natural logarithm function. Eventually, you will get a complex number (or an error message on a standard scientific calculator).

Cases in point:

When x=2,
ln 2 = 0.69314718 (to eight decimal places)
ln ln 2 = -0.36651292
ln ln ln 2 = -1.00372150 + π i

When x = 50
ln 50 = 3.91202301 (to eight decimal places)
Repeated applications of the natural logarithm yield:
1.36405463
0.31046161
-1.16969502
0.15674305 + π i

I hope you enjoy this blog, until next time,

Eddie




This blog is property of Edward Shore. © 2012