Friday, January 14, 2011

Irrational and transcendental

... continuing 'math fun'...

Numbers are fun.
Basic: natural numbers - 1, 2, 3, ...
... also known as positive integers.

I think I was 4 when I learned to count to 14. This was a huge achievement.
Apples, oranges, cars, fingers, toes, ... you can count 'em: 1, 2, 3, ...

In 3rd or 4th grade fractions come into play:
You have 2 apples and 2 friends.
How do you divide the apples among yourselves so's everyone gets an equal share?
... ah!
2/3 (two apples = 2; you + two friends = 3).
2/3 is a rational number: the ratio of two integers.

Sometime in the 6th century BCE, folks figured out that there were numbers that weren't rational.
This discovery is attributed to Pythagoras - he of the Pythagorean Theorem.
He noted that given a square with side = 1, the diagonal of that square (based on his famous theorem) was equal to √2...
... and he further deduced that √2 could NOT be represented as the ratio of two integers.
Irrational numbers were born.

A fellow named Georg Cantor - in the late 19th century (just over 100 years ago) - deduced that there are a WHOLE LOT MORE irrational numbers than there are rational numbers!
In fact, just about EVERY number is irrational.
[... there have been nay-sayers. Another 19th century mathematician famously declared, "God made the integers; all else is the work of man"... For Leopold Kronecker, irrational numbers did not exist!]

Still it was believed for a couple of thousand years that any number must be the solution of an algebraic equation.
-: 1 is the solution of x - 1 = 0
-: √2 is the solution of x2 - 2 = 0

Then, sadly, Gottfried Leibniz, deduced that for at least some values of x, sine(x) is NOT the solution of an algebraic equation.
... and our friend Leonhard Euler was the first to define a transcendental number as a number that is not the solution of a polynomial equation with integer coefficients.

Turns out - as Cantor proved - there are a WHOLE LOT MORE transcendental numbers than there are algebraic numbers. (For what it's worth: there are just as many algebraic numbers as there are rational numbers... tho' not every algebraic number is rational!... and, oh yeah - there are just as many algebraic numbers as there are integers!)

In fact, just about EVERY number is transcendental.
... what's really weird is...
We can NAME only a couple of 'em.
π is transcendental (first proved in 1882 by Ferdinand von Lindemann)
e is transcendental (first proved in 1873 by Charles Hermite)

It's not known if π+e is transcendental!

Again: Numbers are fun!


  1. Oh my all that convoluted figuring with numbers to come up with 0. Sounds like a lot of work for nothing.

  2. Is it generally accepted that one infinity can be larger than another infinity, or is it a matter of definitions?

  3. There are definitely different sized infinities. Two are countably infinite and uncountably infinite. The latter is bigger than the former. So, if you can count the objects without missing any (e.g. integers) then that is a smaller infinity than one where, no matter how you count, you can always find items "in between" the closest two.