Don and his students Jay, Chris, and Elizabeth find a new number- the Dottie number 1

by Don Cohen

It all started like this: Jay, a 10th grader, put the following into a TI-84 Plus calculator: cos(cos(cos(2/5)))= 0.9998477415. Don's first reaction was hmmmm, that's interesting. And that started Don working very hard on what was happening. Don and Jay starting looking at results in radians and degrees.

On the TI-84 Plus, in degree mode:

cos(2/5) =                 0.9999756307

cos(cos(2/5)) =         0.9998477026

cos(cos(cos(2/5))) = 0.9998477415

In Mathematica when Don put in N[Cos[Cos[Cos[2/5]]],10] he got 0.8225159256, not  0.9998477415 as above. Later, Chris, another of Don's students, said we should use the Degree as below, then the outputs agreed.

In Mathematica in degree mode:

N[Cos[2/5 Degree],10] =                                               0.9999756307

N[Cos[Cos[2/5 Degree] Degree],10] =                        0.9998477026

N[Cos[Cos[Cos[2/5 Degree] Degree] Degree],10] = 0.9998477415

On a TI-84 Plus, in radian mode:

cos (2/5) =                   0.9210609940

cos (cos (2/5)) =          0.6049756873

cos (cos (cos (2/5))) = 0.8225159256

N[Cos[2/5],10] =                   0.9210609940

N[Cos[Cos[2/5]],10]=           0.6049756873

N[Cos[Cos[Cos[2/5]]],10]=  0.8225159256

COS(2/5) =                        0.921060994

COS(COS(2/5)) =            0.6049756872

COS(COS(COS(2/5))) = 0.8225159256

So finally, there was agreement using the calculator and 2 computer programs, as there should be, and different answers between radians and degrees as there should be, because the radian is the larger unit, 1 radian ≈ 57'

Don even tried the following in Mathematica

N[Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[Cos[

Cos[2/5]]]]]]]]]]]]]]]],10], to get 0.7385837973

An example of iterating a function:

Don realized this is a sequence of iterations, also called feedback, which he used in Chapter 8 in his Calculus books to solve a quadratic equation like

x2 ' 5x +6 = 0

}      add 5x and -6 to both sides

x2   = 5x - 6

}        divide both sides by x

x = 5 ' 6/x

Now we're going to get an infinite sequence by iterating the function 5 ' 6/x.  If we put 4, say, in for x we get

5 ' 6/4 = 5 ' 1 ' = 3 '

So our sequence starts with 4, 3 ',

The we'll put the output 3 ' in for x (feeback), to get  5 ' 6/(3 ') = 5 ' 6/(7/2) = 5 ' 12/7 = 5 - 1 5/7 = 3 2/7. Our sequence now looks like this: 4, 3 ', 3 2/7,

Now take the output 3 2/7 and put that in for x in 5 ' 6/(3 2/7) and so on. Using a calculator one can do this easily by doing 5-6/ANS  every time after putting in 4 to begin with. This infinite sequence approaches 3 as the limit, which is one of the roots of the original quadratic equation!

In Mathematica Don found the command NestList will do the same thing as above, so we define the function, 5 ' 6/x ,

f[x_]:=5 ' 6/x

then use the command Nestlist

Nestlist[f,4,2] which says start with 4 in for x in the function, and do 2 iterations to get {4,7/2,23/7}

To get decimals to 10 digits, use N[  ,  ,    ],10]

N[Nestlist[f,4,2],10] =

{4.000000000, 3.500000000, 3.285714286}

Don needed to see what was happening with iterating cos(x), so he figured out
how to make a graph with the commands below:

f[x_]:=Cos[x]

N[Nestlist[f,2/5,40],10]

{0.4000000000, 0.9210609940,0.6049756873, 0.8225159256, 0.6803795416,

0.7773340097, 0.7127859455, 0.7565429620, 0.7272133023, 0.7470298706,

0.7337101939, 0.7426950635, 0.7366486303, 0.7407241955, 0.7379800489,

0.7398290799, 0.7385837973, 0.7394227462, 0.7388576709, 0.7392383354,

0.7389819257, 0.7391546511, 0.7390383033, 0.7391166776, 0.7390638842,

0.7390994466,0.7390754914,0.7390916280,0.7390807582,0.7390880802,

0.7390831481, 0.7390864704, 0.7390842324, 0.7390857400, 0.7390847245,

0.7390854085, 0.7390849478, 0.7390852581, 0.7390850491, 0.7390851899,

0.7390850950}

ListLinePlot[%,PlotRange'{.3,.95}] ;  PlotRange tells the computer to go from

.3 to .95 on the y-axis; the resultant graph of the 40 iterations is:

Don decided he didn't want to see all the output numbers, when he did many iterations, like 200 in the examples below, then only look at the last one.

To do this in the examples below, Don used the Take command.

f[x_]:=Cos[x]                           -defines the function f = cos[x]

Take[N[NestList[f,2/5,200],10],-1]   says start with 2/5 in for x in the function f, do 200 iterations, get the outputs to10 decimal digits, then show us only the last output number with Take and -1. (Remember the starting number is in radians, except where it says Degree).

So here we go with different starting numbers in red:

f[x_]:=Cos[x]
Take[N[NestList[f,2/5,200],10],-1]
{0.7390851332}

f[x_]:=Cos[x]
Take[N[NestList[f,45*(Pi/180),200],10],-1]
{0.7390851332}

f[x_]:=Cos[x]
Take[N[NestList[f,4*(Pi/180),200],10],-1]
{0.7390851332}

f[x_]:=Cos[x]l
Take[N[NestList[f,(2/5) Degree,200],10],-1]
{0.7390851332}

f[x_]:=Cos[x]
Take[N[NestList[f,Pi/4,200],10],-1]
{0.7390851332}

f[x_]:=Cos[x]
Take[N[NestList[f,0,200],10],-1]
{0.7390851332}

f[x_]:=Cos[x]
Take[N[NestList[f,Pi/2,200],10],-1]
{0.7390851332}

f[x_]:=Cos[x]
Take[N[NestList[f,-500 Degree,200],10],-1]
{0.7390851332}

f[x_]:=Cos[x]
Take[N[NestList[f,.001,1000],10],-1
{0.739085}

So no matter what number you start with, you get the same answer for the limit of the infinite sequence, and that is 0.7390851332'

On 24 April, 2009 , Jay and Don discovered the new number. It was Jay's idea to Google 0.7390851332' One of the links was to The Encyclopedia of Integer Sequences where much information is given about this number.

On 25 April, 2009, Elizabeth, after Don went through with her what Jay, Chris, and he had done above, found on her TI-84 Plus calculator

cos(0.7390851332) = 0.7390851332

Wow, it works!!!

The graph Don made below, is in Eric Weisstein's World of Mathematics under: Dottie number.  The point of intersection of  y = cos(x) and y=x,  gives this number, 0.739085' the unique solution of the equation x=cos(x).

Great work Jay (who started the whole thing and took the giant step of googleing 0.7390851332), and Chris (who straightened out Don with how to write the degrees), and Elizabeth (who thought of finding the
cos (0.7390851332) and got 0.7390851332 on her calculator)!

What other questions can you ask now?

1 This number was named by Samuel R. Kaplan in 2007

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