Mathematical Constants III

from gogeometry

 
I will never forget my first encounter with the constant e. It was my first week of university. I was trying to complete the first of many Maths tutorials. I came accross this funny symbol: ln(...)

Since I couldn't find an explanation of what "ln" meant in my 700 page textbook (which was otherwise excellent, by the way), I went to my tutor. Our dialogue went something like this:

Me: What does ln (pronounced lin) mean?

Tutor: You don't know what lin means?

Me: That's right, I'm asking what ln means.

Tutor: Ln is the natural logarithm.

Me: Okay...

Tutor: It's just a logarithm with base e.

Me: What's e?

Tutor: It's a constant that forms the base of  the natural logarithm.

Me: But you said that the natural logarithm was a log with base e.

Tutor: Yes.

Me: And you defined e as the base of the natural logarithm.

Tutor: Yes.

Me:...

Needless to say, this dialogue continued around in circles for about ten minutes before we both gave up in disgust. Even though I learned how to use ln and e, they have remained largely mysterious to me.

It's gonna take a few posts...but I plan to correct that! Join me on my mission...

And just to lighten up your day (and its ALMOST on topic):

 

Famous Mathematical Constants II

---

3.14159265358979323846264338327950288419716939937510
  58209749445923078164062862089986280348253421170679
  82148086513282306647093844609550582231725359408128
  48111745028410270193852110555964462294895493038196
  44288109756659334461284756482337867831652712019091
  45648566923460348610454326648213393607260249141273
  72458700660631558817488152092096282925409171536436
  78925903600113305305488204665213841469519415116094
  33057270365759591953092186117381932611793105118548
  07446237996274956735188575272489122793818301194912
  98336733624406566430860213949463952247371907021798
  60943702770539217176293176752384674818467669405132
  00056812714526356082778577134275778960917363717872
  14684409012249534301465495853710507922796892589235
  42019956112129021960864034418159813629774771309960
  51870721134999999837297804995105973173281609631859
  50244594553469083026425223082533446850352619311881
  71010003137838752886587533208381420617177669147303
  59825349042875546873115956286388235378759375195778
  18577805321712268066130019278766111959092164201989
  38095257201065485863278865936153381827968230301952
  03530185296899577362259941389124972177528347913151
  55748572424541506959508295331168617278558890750983
  81754637464939319255060400927701671139009848824012
  85836160356370766010471018194295559619894676783744
  94482553797747268471040475346462080466842590694912
  93313677028989152104752162056966024058038150193511
  25338243003558764024749647326391419927260426992279
  67823547816360093417216412199245863150302861829745
  55706749838505494588586926995690927210797509302955
  32116534498720275596023648066549911988183479775356
  63698074265425278625518184175746728909777727938000
  81647060016145249192173217214772350141441973568548
  16136115735255213347574184946843852332390739414333
  45477624168625189835694855620992192221842725502542
  56887671790494601653466804988627232791786085784383
  82796797668145410095388378636095068006422512520511
  73929848960841284886269456042419652850222106611863
  06744278622039194945047123713786960956364371917287
  46776465757396241389086583264599581339047802759010

Tada! The first two thousand digits of Pi! And its growing everyday...*sniff*

Disclaimer - I did NOT calculate this myself. I got it from here.

To Infinity and (not) Beyond: Part II

photo from the nytimes archives

 
I'm sorry that this post is so belated, but the creative side of my brain has been dead, and understanding Maths  takes a lot of creativity.

everything maths

I promised you a discussion of Mr Cantor and his contribution to infinity... Cantor realised that there is more to understand about infinity than previously thought.

He started by thinking about different sets that we use everyday and think about as "infinite", such as the natural numbers (0,1, 2, 3, 4...). And the real numbers (0, 0.1, 0.0000000002, -0.0547829 etc...). In fact he thought about a lot of sets, and these are just two simple ones. But they will suffice for my highly simplified explanation.

Most of us would agree that there are infinitely many natural numbers. You could keep counting forever (though why you'd want to I'm not sure) , and you would never "run out" of natural numbers - even if what you were counting was the number of stars in the sky!

To briefly introduce some terminology, Cantor called the number of elements in a set the set's cardinality. He called the cardinality of the natural numbers "aleph null" (HINT: the squiggly N is the Hebrew letter aleph).

Well so far so good. All we've really done is make another label for infinity. Who cares?

The really exciting part comes now! Cantor went on to observe that the real numbers behave...differently...to the natural numbers. In fact, he demonstrated that there are infinitely many real numbers between every two natural numbers.

This is the core of the Cantor's insight, and it is demonstrated visually by Cantor sets, which you can learn how to draw here.

The idea is that if you keep dividing the interval between 0 and 1 into thirds, deleting the middle third each time, you could keep dividing forever. Since the real numbers between 0 and 1 essentially represent every possible fractional value between 0 and 1, they can in some sense be represented by the infinitely many "segments" you would have at the end of this process. If there was an end, which there obviously wouldn't be!

And yes, the Cantor set is a kind of fractal - go here for another (probably more rigorous) discussion!

Cantor went on to say (very reasonably in my view) that you can't say that the real numbers have the same cardinality (number of elements) as the natural numbers if there are infinitely many real numbers between 0 and 1 and further between any two natural numbers. It just doesn't make sense! So he labelled the cardinality of the real numbers aleph, with the rider that aleph is bigger than aleph null.

Um, aren't we forgetting something here??? We agreed that both the natural numbers and the real numbers had an infinite number of members. And do you remember Galileo: ...“it is wrong to speak of infinite quantities as being the one greater or less than or equal to the other”?

Cantor disagrees. According to him, there are at least two types of infinity. The first has the same cardinality as the natural numbers. This he called a "countable" infinity. The second has the same cardinality as the real numbers. This he called an "uncountable" infinity.

As you can imagine, this has some interesting consequences...but that's a story for another time.

from here