TI-Nspire CX II (CAS): How Many Terms Are Needed to Calculate π?

TI-Nspire CX II (CAS): How Many Terms Are Needed to Calculate π?

Every One Wants a Slice of π

There are many ways of calculating the π, usually apply a formula or a sequence many, many, many times to get as many digits as possible.  The file pitests.tns tests (see below) five algorithms:

Test 1:  Zeta Formula

π^2 / 6 = Σ( 1 / n^2, n=1 to ∞ )

Test 2:  Pi Squared Over Eight Sum

π^2 / 8 = Σ( (2 * n + 1)^(-2), n = 0 to ∞)

Test 3:  Half of Pi Product

π / 2 = Π( (2 * k)^2 / ((2 * k - 1) * (2 * k + 1)), k = 1 to ∞)

Test 4:  Fibonacci Number Sum

π / 2 = Σ( arctan((F_[2*k+1])^(-1)), k = 0 to ∞ )

The angle is radians.  F_[2*k+1] is the [2*k+1]th Fibonacci number.

Test 5:  Limit Sequence

π = lim (n → ∞) [ a^2 / n ]

where 

a0 = 1

a_[n+1] = a_[n] * (1 + (2 * n + 1)^(-1))

The goal is to determine how many terms are needed to obtain a certain amount of digits.  

Certain algorithms take longer than others.     Here are some results:

2 places:  3.14

3 places:  3.141

Test 1:  

2 decimal places:  600 terms

3 decimal places:  1611 terms 

The zeta function converges super slowly for small arguments.

Test 2:

2 decimal places:  199 terms

3 decimal places:  537 terms

Test 3:

2 decimal places:  493 terms

3 decimal places:  1325 terms

Test 4:

2 decimal places:  9 terms

3 decimal places: 10 terms

The Fibonacci method is the fastest and probably is the quickest way to build the expansion of π.

Test 5:

2 decimal places:  94 places

3 decimal places:  1929 places

Download the file pitests.tns here:  https://drive.google.com/file/d/1XqHtPnoKvqfmFLNWHQ-IayFz_lM5zZyT/view?usp=share_link

Source

"List of formulae involving π"  Wikipedia.   Edited on February 11, 2023.   Accessed on March 12, 2023.  https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80

Note:   On June and July 2023, regular posts will be on Saturdays only, which will include the next Carnival of Math.

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

TI-Nspire CX and CX II: Finance Widgets

TI-Nspire CX and CX II:  Finance Widgets

Financial Widgets

The zip file has five widgets:

simple interest widget:  Calculates the total interest paid in a simple interest loan

pi payment widget:  Calculates the payment of a monthly mortgage without a balloon payment.  

piti widget:  Calculates the PITI (payment-interest-tax-insurance) of a mortgage.  Down payment, annual property tax, and annual property insurance are included.

qualified loan amount widget:  Determines the amount of purchase price a buyer can afford.  The standard 28/36 ratio test is used, compares the two methods, and uses the minimum payment to estimate the qualified amount.  

qualified income test widget:  Test whether the proposed PITI and debt payments against a buyer's income  

Instructions

1.  Download the zip file here:  https://drive.google.com/file/d/1cl-yMpcwRHypGa1kdi66ThZs2xf5pPaQ/view?usp=sharing

2.  Save the widget (tns) files to the MyWidgets folder.

3.  Any tns file in the MyWidgets can be opened as a regular document or be added to a document by Insert-Widget.  

4.  When operated as a widget, highlight the entire page by Ctrl+A, pressing [ menu ], selecting 1. Actions, 1.  Evaluate (Enter).  

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Spotlight: Jim Cullen - Mathematical Topics

Spotlight:  Jim Cullen - Mathematical Topics 

On today's blog I want to feature Jim Cullen, who has a mathematical web page:

Mathematical Topics Index Page - Associated Calculator Programs

Topics include:

Pi and several series to approximate π

Random Normal Number Generators

Generalized Fibonacci Sequences

A Comparison between Casio fx-115ES and Sharp EL-W516 (2 pdf files)

Diophantine Equations

Cullen also has a page of family's genealogy:  http://members.bex.net/jtcullen515/CullOrig.htm

Check it out,  

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Word Search Maker and Retro Review: Texas Instruments TI-18 SLR

Word Search Maker and Retro Review:  Texas Instruments TI-18 SLR

Word Search Maker

I was recently contacted by Phoebe and her website:  Word Search Maker.  Word Search Maker allows you to create word searches in one of four shapes:  diamond, circle, square, or spiral.  Word search maker automatically places the words in the search and also gives an answer.

Here is one I created, theme is math and calculators:  https://wordsearchwizard.com/puzzles/?id=1472

Here is a Milky Way themed word puzzle:
https://wordsearchwizard.com/puzzles/?id=1473

Create your own word searches here:  https://wordsearchwizard.com/maker/

General site:  https://wordsearchwizard.com

Setting up accounts are free.  Special thank you to Phoebe for sharing this wonderful website!


Retro Review:  Texas Instruments TI-18 SLR

Quick Facts:

Model:  TI-18 SLR
Company:  Texas Instruments
Type:  Scientific
Years:  1985-at least 1987
Display:  8 digits
Batteries:  Solar (calculator will work properly only in full light)
Original Retail Price:  Roughly $15 - $20
Logic:  Chain

Hybrid Between Four-Function and Scientific

The TI-18 SLR is a hybrid of a four-function calculator and a scientific calculator.

Four-Function Calculator:

*  Arithmetic (+, -, *, ÷)
*  Percent functions return the answer as soon as the percent key is pressed.
*  Operations are calculated in chain mode.  Order of operations are not followed.
*  Memory is one register:  1.   Keys regarding memory:

* M+:  calculate and add the result to memory
* M-:   calculate and subtract the result from memory
* MR:  memory recall
* MC:  memory clear

*  Square root function.  (takes the square root of the displayed number)

Scientific Calculator:

*  Square function (squares the displayed number)
*  Pi button
*  Reciprocal function (takes the reciprocal of the displayed number)
*  Parenthesis, see the next section

Parenthesis

The parenthesis key [ ( ) ] opens and closes one set of parenthesis.  There are no nested parenthesis, nor there is any indicator on the display when parenthesis are in effect.  Still, this key can aid in calculations.

Example:

Example 1:

(5^2 + 12^2) / ( 4 + 5 )

Key stroke:
 [ ( ) ] 5 [ x^2] [ + ] 12 [ x^2 ] [ ( ) ]
[ ÷ ] [ ( ) ] 4 [ + ] 5 [ ( ) ]
[ = ]

Result:  18.777777

Example 2:

π * ( 18^2 - 7^2 ) + ( π * 11^2 )

Key strokes:
[ π ] [ × ] [ ( ) ] 18 [ x^2 ] [ - ] 7 [ x^2 ] [ ( ) ]
[ + ] [ ( ) ] [ π ] [ × ] 11 [ x^2 ] [ ( ) ] [ = ]

Result: 1244.0706

Note results are truncated at the last digit, not rounded.

Display

The display is one line and it's small.  Still the display is serviceable.  The left slot has three vertical indicators:  M (memory indicator) on top, - (for negative numbers) in the middle, and E (error indicator) on the bottom.

Keyboard

The keys are responsive, but I wouldn't try to speed type on the TI-18SLR.

Routine Goodies

Absolute Value:  | n |
n  [ x^2 ] [ √x ]

Sign:  sgn(n) = -1 for negative, 1 for positive (does not work for n = 0)
[MC] n [ M+ ] [ x^2 ] [ √x ] [ ÷ ] [ MR ] [ = ]

Modulus:  n mod m  with n > m, n > 0, m > 0.
[ MC ] n [ M+ ] [ 1/x ] [ × ] m [ - ] (integer part of the result) [ × ] [ MR ] [ = ]

Example:
99 mod 16 = 3
[ MC ] 16 [ M+ ] [ 1/x ] [ × ] 99   (Display: 6.1875   integer(6.1875) = 6)
[ - ] 6  [ × ] [ MR ] [ = ]

Result:  3

Verdict

A collector's item.  I wish more basic calculators included x^2, π, and 1/x, which are very useful in many calculations.


Note: This post was scheduled for Sunday, 4/19/2020, but I decided to move it up to post this 12 hours early. 

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

DM 41L: A Song of Irrational Numbers

Tones of the first ten digits of the constants π, √2, Zeta(2), Phi (Golden Ratio constant), and e (Euclidean constant) using the Swiss Micros DM 41L

https://youtu.be/aCLT6lzvnS0

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Birthday Blog: Fun Facts About Me

Fun Facts About, me, Eddie Shore, the author of Eddie's Math and Calculator Blog:

1.  I have a Bachelor's Degree in Accounting and a Master's Degree in Mathematics, both from Cal Poly Pomona.

2.  I don't have a favorite sport.

3. My favorite video games are Super Mario Maker, Super Mario Brothers, Mario Kart, Millipede, and Joust.

4. My favorite colors are sky blue, denim blue, forest green, and gold.

5. For you zodiac fans, my zodiac sign is Pisces.  Both of my parents are Geminis, and most of my closest friends are Scorpios.

6. Chocolate chip cookies are my weakness.

7. My four favorite go-to music artists are Earth, Wind & Fire, Stevie Wonder, Janet Jackson, and Sheryl Crow.

8. At home, I have two dogs and three cats. 

9.  I'm Irish and Mexican.

10.  Every morning I write what I want to accomplish for the day.  I go through a lot of Post-Its.

11. The beach is my sanctuary.

12. The mathematics section of university library is to me what Disneyland is to most people, only I don't have to pay $60 for an admission ticket. ;)  My favorite library is the Honnold Mudd Library in Claremont (Claremont Colleges).

13. I can't live without music.  Or calculators.

14. My dream car is a Ferrari. 

15. I prefer tea over coffee.

16. My turn ons are honesty, warmth, kindness, intelligence, and a love for life.

17. My turn offs are dishonesty, arrogance, racism, sexism, and ageism.

18. My bucket list grows by the day.

19. I am very close to my family. 

20. My favorite vacation spot so far is Maui.

21. My favorite number is pi (π), partly because my birthday is March 14 (the day that this blog is posted).

22. I want to go to Greece, Italy, Ireland, and New Zealand.  I'd probably wouldn't return home. 

23. My guilty pleasure is the Real Housewives of New Jersey.  #TeamMargaretJosephs

24. I'm nearsighted and prefer glasses to contacts.

25. I prefer wine over beer, but I do enjoy a good ale.  My favorite shot is Fireball.

26. My newest favorite YouTube channel is Doctor Mike. Favorite of all time is both Cinemasins and Music Video Sins.

27. My favorite calculators are the HP Prime, HP 42S, HP 12C, TI-84 Plus CE and Casio fx-991EX.

28. I am a Press Your Luck addict. Big Bucks, no whammies!

29. I have two dogs and three cats.

30. My music tastes are kind of eccentric, but I learn towards rock, alternative, and some R & B.  I do have a Spotify account. 

31.  My favorite fruits of cherries, applies, and strawberries.

32.  My favorite holiday used to be Christmas, now it's Halloween.

33.  A hobby I like but don't do enough of is art.  I'm drawn to glass art, mythological art, and fantasy art. 

34.  My favorite movie is Ghostbusters - the 1984 original one.

35.  My favorite pie is apple. 

36.  My favorite fonts are Arial, Courier, Futura, and Banschrift.

37.  I lived in Southern California all my life.

38.  Starting and writing on this blog is one of the most fun things I am very fortunate to do, and I thank you for reading and supporting this blog. 

Eddie
All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Review: Casio fx-55 Plus

Review:  Casio fx-55 Plus

General Information

Company:  Casio

Type:  Fractions, Elementary Math

Memory:  1

Battery:  Solar with battery backup (1 LR44)

Years:  2012 - current

Cost:  $10.00 - $20.00, I paid $12.99

Operating Logic:  Algebraic

Target Audience:

* Elementary and middle school students

* Anyone who wants a few more functions than the regular four-function calculator, but not a scientific calculator

* You want a calculator with an emphasis on fractions

Features

The major emphasis on the Casio fx-55 Plus is on fractions.  We’ll discuss this in the next section in detail.

The screen has a textbook display, meaning fractions and all other calculations are displayed as you would write them.  Fractions are simplified.  If MathO (Math Output) is selected, square roots and terms of pi (π) shown in exact form.  If LineO is selected, all answers (except fractions) are shown in decimal form.  No matter what the display mode is, you can always decimal approximation by pressing [SHIFT] [ = ].

Example (MathO):

√12 returns 2 √3

5^2 * π returns 25π

You enter expressions the way you would write them with the Casio fx-55 Plus.  The display is a mutli-line display, the expression on the top of the screen and the answer is on the bottom of the screen. 

The keyboard is simple where only four keys have shifted functions.  The OFF function is shifted as well. 

Other functions include reciprocal, random numbers, including random integers, and calculations in Degrees-Minutes-Seconds [ ° ‘ “ ]. 

The memory has four dedicated keys: store, recall, add to memory (M+), and subtract from memory (M-).

The percent keys work a bit differently from the normal four-function calculators (and is not very intuitive). 

[%] divides the argument by 100 and that’s it.  Its fine when you are working with multiplying or dividing percentages, but adding and subtracting percent will take an algorithm:

X + Y%:  enter X + X * Y %

X – Y%:  enter X – X * Y %

[>%] multiplies the result by 100 and displays the answer in percentage notation. 

Example:  3 [ = ] [ SHIFT ] (>%) returns 300%

Let’s Talk Fractions

The fraction keys are marked lime green on the Casio fx-55 Plus.  Let’s go over the keys in detail:

[ []/[] ] (1st key from the left):  fraction template

[ [] []/[] ] (2nd key from the left):  mixed number template

[ F←→D] (3rd key from the left): switches the result between exact (fractions, exact square roots, terms of π) and decimal answer

[ a b/c ←→ d/c ]:  (4th key from the left)

Manual Simplified mode:  This key does nothing.

Auto Simplified mode: This key converts the result between improper and proper fraction notation. 

[ Simp ]:  calls the simplify conversion to the entry.  In Manual Simplified mode, any fraction that is not reduced to its irreducible form will have a down arrow indicator ( ↓ ).  Also executing Simp in Manual Simplified mode not only returns the reduced fraction, but the factor used into reducing such fraction.

[x^-1]:  The reciprocal function

I also want to point a very handy key, the remainder division function ( [÷R]), which is located on the bottom right hand corner of the keyboard.  This function returns both the quotient and remainder. 

Example:  77 ÷R 13 returns 5, R = 12 

Only the quotient is stored in last answer (Ans).

Let’s compare this Casio fx-55 Plus with an older fraction oriented calculator, the Texas Instruments Math Explorer (aka TI-12). 

Comparison:  Casio fx-55 Plus (2012-present) vs. Texas Instruments Math Explorer (1978-1998)

TI Math Explorer	Casio fx-55 Plus
1 line display	Multi-line, textbook display
No shifted functions	4 shifted functions
N/A	Degrees-Minutes Seconds calculations, random numbers and integers
Constant arithmetic storage function through the [ Cons ] key.	N/A
10^n is limited to integers 0-7	10^x allows for all real numbers between up to, but not including, 100.  Used to work with numbers in scientific notation
Int÷ works with only positive integers	÷R works with real, positive numbers (any non-positive numbers makes ÷R work like ÷)

Verdict

I like this calculator for middle school and elementary students. The Casio fx-55 Plus presents a few extra functions but without the many scientific functions that may intimidate some students.  (emphasis on some)  I wish the percent would work like we would write expressions when it comes to addition or subtraction (i.e. X + Y% instead of X + X * Y%), but this is understandable.  I also wish the proper/improper toggle worked in Manual Simplified mode.

The display is great because the user can see the expression and answer at the same time.  I really like that this calculator also returns exact answers in calculations involving square roots and π. 

I can see this calculator being used beyond the classroom, especially in everyday basic math calculations. 

In any case, the Casio fx-55 Plus serves a great introductory calculator.

Eddie

All original content copyright, © 2011-2018.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.  Please contact the author if you have questions.

Geometric Relationships: Circle, Sphere, and Equilateral Triangle

Geometric Relationships: Circle, Sphere, and Equilateral Triangle

Circle:  Relationship between Area and Circumference

We know that π is a constant (π ≈ 355/113, but more accurately, π ≈ 3.141592654).  And:

Circumference of a Circle:  C  = 2 * π * r

Area of a Circle: A = π * r^2

Observe that:

C = 2 * π * r

π = C / (2 * r)

And:

A = π * r^2

π = A / r^2

Hence:

C / (2 * r) = A / r^2

(2 * r) / C = r^2 / A

A * 2 * r = r^2 *C

A = C * r / 2

Sphere:  Relationship between Area and Circumference

Volume of a Sphere:  V = 4/3 * r^3 * π

Surface Area of a Sphere:  S = 4 * π * r^2

Solving for π:

V = 4/3 * r^3 * π

3 * V = 4 * r^3 * π

π = (3 * V) / (4 * r^3)

And:

S = 4 * π * r^2

π = S / (4 * r^2)

Then:

(3 * V) / (4 * r^3) = S / (4 * r^2)

Multiply both sides by 4 * r^2:

S = 3 * V / r

Equilateral Triangle:  Relationship between Perimeter and Area

Let a (small a) be the length’s side.   Then the area of the triangle:

A = 2 * (1/2 * a/2 * √3/2 * a) = a^2 * √3/4

With the perimeter:  P = 3 * a,

P = 3 * a

P^2 = 9 * a^2

a^2 = P^2 / 9

And

A = a^2 * √3 / 4

a^2 = 4 * A / √3

P^2 / 9 = 4 * A / √3

A = P^2 * √3 / 36

To summarize:

Circle:  Area and Circumference:  A = C * r / 2

Sphere:   Volume and Surface Area:  S = 3 * V / r

Equilateral Triangle:  Area and Perimeter:  A = P^2 * √3 / 36

The next blog will cover Platonic solids.  At least that’s the plan.  Have a great rest of the weekend.  

Eddie

This blog is property of Edward Shore, 2017.

TI-89 and HP Prime: Approximation Digits of π to Lots of Digits

TI-89 and HP Prime:  Approximation Digits of π to Lots of Digits 

Ways to find digits of π. 

Four steps with a calculator with CAS, which allows computation with large integers:

1.  Take a series and multiply each term by 10^n, where n is the number of digits desired. The exact form of each calculation 

2. Take the integer portion of the sum. Make any adjustments when necessary. 

3. Convert the sum to a string.  Insert a decimal point when appropriate. 

4. Return the answer as a string. 

There are three methods that are demonstrated, one routine is presented for the TI-89 and the HP Prime.  All programs presented takes two arguments: n number of digits, m is the number of iterations. Remember that m must be sufficiently high to get an accurate computation of π. 

In the TI-89 programs, (C) marks a comment.  All HP Prime programs on this blog entry are to be run in CAS mode (make sure CAS is checked when creating the program).  

Keep in mind the HP Prime is significantly faster than the TI-89.  

Newton/Euler - newpi

π = 2 * Σ ( 2^k * (k!)^2 / (2*k + 1)!, k, 0, infinity)

TI-89 Program newpi

newpi(n,m)

(C) Newton/Euler π 

(C) 2016-11-13

Local s,k,t

(C) initialize

0 → t

(C) loop

For k,0,m

t + exact( 10^n * 2^k * (k!)^2 / ( (2*k + 1)! ) ) → t

EndFor

iPart(2*t) → t

string(t) → s

left(s,1) & "." & right(s,n) → s

Disp s

EndPrgm

HP Prime CAS Program newpi

#cas

newpi(n,m):=

BEGIN

LOCAL t,k,s;

// Netwon/Euler π 

// 2016-11-13 EWS

t:=0;

FOR k FROM 0 TO m DO

t:= t + exact( 10^n * 2^k * (k!)^2 /  ( (2*k + 1)! ));

END;

t:= IP(2*t);

// string, left, right 

// must be in lower case

s:= string(t);

s:= left(s,1) + "." + right(s,n);

return s;

END;

#end

newpi(20, 175) returns "3.14159265358979323846"

Arctangent - arctanpi

π = Σ ( 2^(k+1) / ( COMB(2*k, k) * (2*k +1) ), k, 0, infinity)

TI-89 Program arctanpi

arctanpi(n,m)

(C) Arctan π 

(C) 2016-11-13

Local s,k,t

(C) initialize

0 → t

(C) loop

For k,0,m

t + exact( 10^n * 2^(k + 1) / ( nCr(2*k,k) * (2*k+1) ) ) → t

EndFor

iPart(t) → t

string(t) → s

left(s,1) & "." & right(s,n) → s

Disp s

EndPrgm

HP Prime CAS Program arctanpi

#cas

arctanpi(n,m):=

BEGIN

LOCAL t,k,s;

// Arctan π 

// 2016-11-13 EWS

t:=0;

FOR k FROM 0 TO m DO

t:= t + exact( 10^n * 2^(k+1)/(COMB(2*k,k) * (2*k + 1)));

END;

t:= IP(t);

// string, left, right 

// must be in lower case

s:= string(t);

s:= left(s,1) + "." + right(s,n);

return s;

END;

#end

arctanpi(25, 200) returns "3.1415926535897932384626433"

Bailey-Borwein-Plouffe Formula (BBP) - bbppi

π = Σ ( 1/16^k * ( 4/(8*k +1) - 2/(8*k + 4) - 1/(8*k + 5) - 1/(8*k + 6) ), k, 0, infinity)

TI-89 Program bbppi

bbppi(n,m)

(C) BBP π 

(C) 2016-11-13

Local s,k,t

(C) initialize

0 → t

(C) loop

For k,0,m

t + exact( 10^n/16^k * ( 4/(8*k + 1) - 2/(8*k + 4) - 1/(8*k + 5) - 1/(8*k + 6) ) ) → t

EndFor

iPart(t) → t

string(t) → s

left(s,1) & "." & right(s,n) → s

Disp s

EndPrgm

HP Prime CAS Program bbppi

#cas

bbppi(n,m):=

BEGIN

LOCAL t,k,s;

// Netwon/Euler π 

// 2016-11-13 EWS

t:=0;

FOR k FROM 0 TO m DO

t:= t + exact( 10^n/16^k * ( 4/(8*k + 1) - 2/(8*k + 4) - 1/(8*k + 5) 

- 1/(8*k + 6) ) );

END;

t:= IP(t);

// string, left, right 

// must be in lower case

s:= string(t);

s:= left(s,1) + "." + right(s,n);

return s;

END;

#end

bbppi(25,200) returns "3.1415925635897932846"

Source:  Wikipeida.  "Approximations of π"  https://en.m.wikipedia.org/wiki/Approximations_of_π. Retrieved November 12, 2016

This blog is property of Edward Shore, 2016. 

Video: The Pi Episode by Danica McKellar and Nerdist

Major thanks to Dr. Cliff Pickover for posting the link to this awesome video on my Twitter feed.  On January 9, 2014, Danica McKellar premiered her YouTube show "Math Bites".  The first show is all about pi (π).  OK, they talk about pie too, but the main focus is one the famous number. 

From memory, I know π = 3.14159265353897923846...  (I think that's right).  Anyway, click on the link and enjoy!

http://www.youtube.com/watch?v=8-cazxAL_tU

And not only my favorite number is π, my favorite pie is apple, especially if it has cinnamon with a crust on top.

If you are on Twitter (who isn't these days?):

Follow Danica McKellar, author of "Math Doesn't Suck", "Hot X: Algebra Exposed", "Kiss My Math: Showing Pre-Algebra Who's Boss" and "Girls Get Curves: Geometry Takes"  - @danicamckellar

Follow Chris Hardwick, head of Nerdist and supremely funny guy.  He also has his Nerdist podcasts.  Too bad I don't have a burrito around to enjoy while I'm typing this. Oh, he hosts Talking Dead (The show which discusses "The Walking Dead") and @midnight  - @nerdist

Follow Dr. Cliff Pickover, scientist, mathematician, and writer.  His books include "The Math Book" and "The Physics Book" - @pickover

And I have a Twitter account too, mine is @edward_shore. 


Eddie

Next post will be post the 300th of this blog!  Again, I thank you for your support - much appreciated.

 

Distribution of Digits for π, e, Euler's Constant, and √2

Here are four famous numerical constants, extended to 50 decimal places: π, e, γ, and √2. For each constant, the approximation will be listed and a histogram of each of the 50 decimal places and the integer part (51 digits in total) will be presented.

Pi
π ≈ 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

Digit Distribution: (digit: number of occurrences)
0: 2
1: 5
2: 5
3: 9
4: 4
5: 5
6: 4
7: 4
8: 5
9: 8

e

e ≈ 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995

Digit Distribution:
0: 3
1: 3
2: 8
3: 4
4: 5
5: 6
6: 4
7: 7
8: 5
9: 6

γ (Euler's Constant)

γ ≈ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992

Digit Distribution:
0: 9
1: 5
2: 7
3: 5
4: 4
5: 6
6: 5
7: 2
8: 2
9: 6

√2

√2 ≈ 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694

Digit Distribution:
0: 5
1: 5
2: 4
3: 4
4: 5
5: 4
6: 6
7: 6
8: 7
9: 5

Of the four constants presented, √2 has the most even distribution, at far as the first 50 decimal points are concerned.

Fun Fact: 99/70 gives √2 accurate to four decimal places.
99/70 = 1.41428 571428...

Resources

π, e, γ

Zwillinger, Daniel. CRC - Standard Mathematical Tables and Formulae 32nd Edition. CRC Press, Boca Raton, FL. 2012

√2

√2 Wikipedia: Retrieved October 2, 2012.


Enjoy! This is something I wanted to do for a while, as I am fascinated by numerical constants.

Eddie


This blog is property of Edward Shore, 2012.

My Favorite Number

Even though I am a fan of numbers in general, my favorite number is π. π is the famous irrational constant used in measuring the area and circumference in circles, plays an important role in trigonometry, and the Gamma of 1/2 is the square root of π. Every March 14 is designated π Day, celebrated around the world, particularly in mathematical circles, schools, and universities. It also happens to be the birthday of Albert Einstein, Billy Crystal, Les Brown, Quincy Jones, and yours truly (Eddie - the writer on this blog).


Going by memory, the first 15 digits of π is 3.14159 26535 89793.

So, what is your favorite number?