Transcendental numbers

W@t3R

What exactly are they?

Reply 1

visesh

17

Wikipedia has a decent explanation I think.

Reply 2

cms271828

Transcendental numbers are basically NON-Algebraic numbers
And to answer your your next question, "what are Algebraic numbers?"
Algebraic numbers are basically solutions of polynomials equations
degree n>0 with integral coeficients.
So algebraic numbers include all integers because of the equation
x-k=0.
Also includes all rational numbers because of ax-b=0 gives x=b/a.
And also all numbers with surds in, such as cube root of (1+rt2)

pi and e are excellent examples of transcendentals, since they
will never be solutions of a polynomial(n>0) with integral coeffs.

A few more things to say:
Algebraic numbers are countable, so have 0 measure,and take up 0%
or the real line.
Equivalently the set of transcendental numbers are uncountable, and
take up 100% of the real line.
Ironically then that you can say that 100% of the reals cannot be
written down!!!!!!

Reply 3

J.F.N

1

cms271828

Transcendental numbers are basically NON-Algebraic numbers
And to answer your your next question, "what are Algebraic numbers?"
Algebraic numbers are basically solutions of polynomials equations
degree n>0 with integral coeficients.
So algebraic numbers include all integers because of the equation
x-k=0.
Also includes all rational numbers because of ax-b=0 gives x=b/a.
And also all numbers with surds in, such as cube root of (1+rt2)

pi and e are excellent examples of transcendentals, since they
will never be solutions of a polynomial(n>0) with integral coeffs.

A few more things to say:
Algebraic numbers are countable, so have 0 measure,and take up 0%
or the real line.
Equivalently the set of transcendental numbers are uncountable, and
take up 100% of the real line.
Ironically then that you can say that 100% of the reals cannot be
written down!!!!!!

Worse yet, even though we know they exist, its difficult to 'find' these transcendental numbers. Liouville has a quite accessible method for constructing transcendental numbers. Look it up if you're interested.

Reply 4

cms271828

i havent seen it, is it just sums of 10^-n for n=j! or sumthing?

Reply 5

RichE

15

Yes that's the famous one given his name.

However he had a general method to show certain numbers, which could be too quickly approximated by rational numbers (in a certain technical sense), are transcendental.

The Gelfond Schneider Theorem and similar results produce a lot of transcendental numbers as well.

Reply 6

J.F.N

1

I recently used the 'rationality measure' Riche spoke about to prove that f_3:R-->R defined by f_3(x)=0, if x=0, 1/q^3 if p/q in lowest terms, is differentiable at x=sqrt(2). Its testimonial to the breadth of mathematics. See attached if you're interested.

f3.pdf71.7KB

Transcendental numbers

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