Ok. When is it going to hit 50 billion? Put your bets down.
Old posts from when Apple was at 25 billion app downloads.
More questions:
When will it hit 100 billion? 1 trillion? How many apps will be sold by December 25th? Any more?
Edit 5/11/13
More info:
Neat link: Team Maker
Allows you to paste in a roster and make random groups. No friction, no mess. Cool stuff.
Intro to Computer programming worked at calculating digits of pi today. The actual algorithms aren’t too bad, but getting more than the standard number of digits from a double is a bit trickier. Here’s a program that calculates pi using:
Bailey–Borwein–Plouffe formula
Bellard’s formula
and
Chudnovsky algorithm
Holy smokes is Chudnovsky algorithm’s fast!
Plouff Bellard Chudnovsky Iteration number 1 3.133333333333333333333333 3.141765873015873015873017 3.141592653589734207668453 Iteration number 2 3.141422466422466422466422 3.141592571868390306374053 3.141592653589793238462642 Iteration number 3 3.141587390346581523052111 3.141592653642050769944284 3.141592653589793238462642 Iteration number 4 3.141592457567435381837004 3.141592653589755368080514 3.141592653589793238462642 Iteration number 5 3.141592645460336319557021 3.141592653589793267843377 3.141592653589793238462642 Iteration number 6 3.141592653228087534734378 3.141592653589793238438852 3.141592653589793238462642 Iteration number 7 3.141592653572880827785241 3.141592653589793238462664 3.141592653589793238462642 Iteration number 8 3.141592653588972704940778 3.141592653589793238462644 3.141592653589793238462642 Iteration number 9 3.141592653589752275236178 3.141592653589793238462644 3.141592653589793238462642 Iteration number 10 3.141592653589791146388777 3.141592653589793238462644 3.141592653589793238462642 Iteration number 11 3.141592653589793129614171 3.141592653589793238462644 3.141592653589793238462642 Iteration number 12 3.141592653589793232711293 3.141592653589793238462644 3.141592653589793238462642 Iteration number 13 3.141592653589793238154767 3.141592653589793238462644 3.141592653589793238462642 Iteration number 14 3.141592653589793238445978 3.141592653589793238462644 3.141592653589793238462642 Iteration number 15 3.141592653589793238461733 3.141592653589793238462644 3.141592653589793238462642 Iteration number 16 3.141592653589793238462594 3.141592653589793238462644 3.141592653589793238462642 Iteration number 17 3.141592653589793238462641 3.141592653589793238462644 3.141592653589793238462642 Iteration number 18 3.141592653589793238462644 3.141592653589793238462644 3.141592653589793238462642 Iteration number 19 3.141592653589793238462644 3.141592653589793238462644 3.141592653589793238462642
from decimal import *
#Sets decimal to 25 digits of precision
getcontext().prec = 25
def factorial(n):
if n<1:
return 1
else:
return n * factorial(n-1)
def plouffBig(n): #http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula
pi = Decimal(0)
k = 0
while k < n:
pi += (Decimal(1)/(16**k))*((Decimal(4)/(8*k+1))-(Decimal(2)/(8*k+4))-(Decimal(1)/(8*k+5))-(Decimal(1)/(8*k+6)))
k += 1
return pi
def bellardBig(n): #http://en.wikipedia.org/wiki/Bellard%27s_formula
pi = Decimal(0)
k = 0
while k < n:
pi += (Decimal(-1)**k/(1024**k))*( Decimal(256)/(10*k+1) + Decimal(1)/(10*k+9) - Decimal(64)/(10*k+3) - Decimal(32)/(4*k+1) - Decimal(4)/(10*k+5) - Decimal(4)/(10*k+7) -Decimal(1)/(4*k+3))
k += 1
pi = pi * 1/(2**6)
return pi
def chudnovskyBig(n): #http://en.wikipedia.org/wiki/Chudnovsky_algorithm
pi = Decimal(0)
k = 0
while k < n:
pi += (Decimal(-1)**k)*(Decimal(factorial(6*k))/((factorial(k)**3)*(factorial(3*k)))* (13591409+545140134*k)/(640320**(3*k)))
k += 1
pi = pi * Decimal(10005).sqrt()/4270934400
pi = pi**(-1)
return pi
print "\t\t\t Plouff \t\t Bellard \t\t\t Chudnovsky"
for i in xrange(1,20):
print "Iteration number ",i, " ", plouffBig(i), " " , bellardBig(i)," ", chudnovskyBig(i)
2^86 is largest known pwr of 2 that does not contain 0.What about other digits?Does each non-zero digit occur inf often among the pwrs of 2?
— James Tanton (@jamestanton) March 6, 2013
Interesting. The Intro to Programming Students gave a crack at this one; and after 30 minutes all were close and half had it. Some worked in Python (easier large numbers and string manipulation) and some in Java (muuuuch faster). Here’s my Java (BlueJ) code. It checks all powers of 2 up to 2^10,000, and then it prints out the counts of all the digits 0-9. The digits are spread out pretty evenly.
import java.math.BigInteger;
public class Powersof2
{
public int[] digits = new int[11];;
public Powersof2()
{
BigInteger number = new BigInteger ("1");;
int power = 1;
for (int i = 0; i < digits.length; i++)
{
digits[i]=0;
}
while (power <= 10000)
{
number = number.multiply(new BigInteger ("2"));
if (HasAZeroInIt(number))
{
System.out.println("Next power of 2 is " + number + " with an exponent of " + power);
}
power += 1;
}
for (int i = 0; i < digits.length-1; i++)
{
System.out.println("index " + i + " and number of digits is " + digits[i]);
}
}
public boolean HasAZeroInIt(BigInteger n)
{
String strn = String.valueOf(n);
int index = 0;
while (index < strn.length())
{
digits[Character.getNumericValue(strn.charAt(index))] += 1;
index += 1;
}
index = 0;
while (index < strn.length())
{
if (strn.charAt(index) == '0')
{
return false;
}
index += 1;
}
return true;
}
}
Number theory nugget: 357686312646216567629137 is a prime number which remains prime after removing any number of digits from the left side!
— Maths World UK (@MathsWorldUK) March 4, 2013
Otherwise known as left-truncating primes with every suffix prime and no digit are zero. Here is the whole sequence. It was proven that 357686312646216567629137 is the last number of the sequence in 1999. Cool that math understandable to me is still being done in my lifetime. Although I’d bet that the actual proof is beyond my pay grade.
Here’s a recursive python program to print out a list of these numbers under 1 million. Unfortunately it doesn’t work on the number in question, my prime checker isn’t efficient enough.
from math import sqrt
#function to check to see if a number is prime
def prime(n):
for i in xrange(2,int(sqrt(n))+1):
if (n%i == 0):
return False
return True
#recursive function to check to see if a number continues to be prime
#as the left digit gets taken away
def reducingprime(n):
#Stopping condition
if n == 0:
return True
else:
string_n = str(n)
if n < 10:
return prime(n)
#check recursive step only if second digit isn't 0
elif string_n[1]!='0' and prime(n):
#this takes away the left digit
next = n % (10**((len(str(n))-1)))
return reducingprime(next)
else:
return False
for a in xrange(2,1000000):
if reducingprime(a):
while a > 0:
print a," ",
a = a%(10**((len(str(a))-1)))
print " "
Are Double Stuf Oreos really double stuffed? Some disagreement here: Chris Danielson in part 1 2 and 3 versus Chris Lusto in part 1 and 2.
Enter in the Mega Stuf Oreo.
Is the Mega Stuf double stuffed, triple stuffed, more???? And who was correct, Chris or Chris?
10 Original Oreos:
10 Double Stuf Oreos:
10 Mega Stuf Oreos:
5 Wafers:
I’ll leave the math to you. Or if you’re terrifically lazy, spoilers found here.
How far away is the meteor from this security camera when it explodes?
As animals get bigger, from tiny shrew to huge blue whale, pulse rates slow down and life spans stretch out longer, conspiring so that the number of heartbeats during an average stay on Earth tends to be roughly the same, around a billion.
And …
… the surface area a creature uses to dissipate the heat of the metabolic fires does not grow as fast as its body mass.
To see this, consider a mouse as an approximation of a small sphere. As the sphere grows larger, to cat size, the surface area increases along two dimensions but the volume increases along three dimensions. The size of the biological radiator cannot possibly keep up with the size of the metabolic engine.
Might be fun to start with the theory that all animals get a billion heartbeats and build up some charts to see if its true. Also a surface area of various mammals vs age graph might be interesting to calculate and plot.
Does this mean that I should stop exercising?
Link from Kottke.
Spirograph – @geogebra Tube bit.ly/XkaDL4 full frontal trochoid features. Maybe purges the obsession.
— John Golden (@mathhombre) February 6, 2013
Questions for students:
Results to Favorite Proof:
What's your name/twitter? | Name of proof | Link to proof somewhere online... | Why do you like this proof? | Additional testaments of love for this proof | Additional testaments of love for this proof | ... | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| @dandersod | Euclid's Proof of Infinte Primes | link | Love proof by contradiction. Simple, powerful, easy to understand. | From the book | ||||||||||||
| @sophgermain | Why square root of 2 is irrational | link | It's the first proof i show students and it's simple. | |||||||||||||
| @nik_d_maths | Induction for sum of squares | link | Proof by induction is fun, and this involves enough algebra for students to feel they have done something, while being short enough to use early on | I love induction? | ||||||||||||
| @stwwilkinson | Uncountability of the real numbers (Cantor) | link | very simple to explain, yet challenges a lot of intuitive notions about infinity | The set up for this proof -- the countability of the rational numbers -- is also mind blowing. | ||||||||||||
| @MrHonner | Midline of a Triangle | link | Elementary, but powerful idea in geometry; several different ways to think about why it's true | Great way to demonstrate the power of coordinate geometry for proof | Leads to beautiful results like Varignon's Theorem | |||||||||||
| @Moko58 | Product of 2 odd number is an odd number | link | Even beginners can attempt this proof. | this is a great proof. | ||||||||||||
| @mpershan | Incompleteness of axioms of arithmetic | link | It's problably the hardest mathematical idea that I even sort of understand, which is definitely part of why I like it, but the self-referential sentence is brilliant and all sorts of cool things follo wform it. | |||||||||||||
| @mathymcmatherso | Geometric Proof of the Pythagorean Theorem | link | Has a visual and algebraic component that go hand-in-hand. Personally: most convincing and direct proof of pythagorean theorem I've ever seen | |||||||||||||
| @j_lanier | There exists an irrational number that when raised to an irrational power results in a rational number. | link | I like this proof because it is simple; it yields a positive result, rather than a "there exists no..."; and it's concrete but so delightfully non-constructive--we find out that there exists such a thing without having a single example of it! So cheeky. | I can't promise that it's my favorite ever, but it's a dang neat proof. | ||||||||||||
| @DanielPearcy | Can't pick between irrationality of root(2), infinitely many primes, uncountability of reals | Looks like I'm a sucker for proof by contradiction. | Funnily enough I don't know many recent (last 100 years) proofs | |||||||||||||
A pretty common theme: Simplicity in presentation or Visual Component.
<snark>Note: No 2 column geometry proofs are found in this list.</snark>
