Another "stupid" question about the square root of a prime number

In a previous post it was shown by those obviously more learned than I that
the square root of a prime number is an irrational number. And again I am
curious, perhaps again revealing my ignorance, but this irrational number
that is the square root of a prime, we could actually start to compute it,
and I wonder, what if anything can we say about the actual stream of digits
that would be produced by the computation of the square root of a prime
number? Would they bear any formal similarity to the stream of digits
produced by, say, generating pi?

 

What do you mean by similarity? What is the difference between
ordinary similarity and formal similarity? Why are you limiting
yourself to the square root of prime numbers?

All irrational numbers, whether transcendental (such as pi), the
square root of a prime, the square root of any other non-perfect
square integer, etc, share certain properties. When expressed in
decimal (or other integer base): they never terminate and they never
form an infinitely repeating pattern. What more are you looking for?

 

Actually that was what I was looking for! However a point of confusion
remains. Wikipedia informs me that all transcendentals are irrational and
most irrationals are transcendentals.

Which other irrationals besides the square roots of prime numbers are _not_
transcendental?

 

Things like the solutions of x^7 - x + 2 = 0, for example. Look up
"algebraic numbers". Basically, if you have any polynomial p(x) with
integer coefficients and p(x) = 0 has no rational solutions,
then the solutions will be irrational and not transcendental.


Regards,

Rick

 

May I suggest that you try using google. I suggested in an earlier
post that
you seemed not to understand the meaning of transcendental. One might
think
that the suggestion would inspire someone truly interested in the
subject to
look up the answer using google. Do you plan on doing ANY study on
this subject?
Or do you just want answers handed to you?

A transcendental number is a number that is NOT the root of any
polynomial.
Equally clearly, any irrational number that IS the root of a
polynomial is NOT
transcendental. Such numbers form a set commonly referred to as the
algebraic
numbers. The algebraic numbers union the transcendentals form the
reals. They
are disjoint sets. This is all elementary, pre-calculus mathematics.

 

Perhaps you you don't notice when your students are studying and trying?
After your original post I did indeed read up to see how I had misused the
word
"transcendental" and as my post itself clearly states, I researched
irrational
numbers in the Wikipedia as well.

I hope it's not impertinent to ask: is there any way to estimate what
proportion of
irrationals are transcendental? The Wikipedia says "almost all." Are we
talking 99 percent?
Or 99.9999999999999999 percent?

 

Again, from Wikipedia:

http://en.wikipedia.org/wiki/Transcendental_number


However, transcendental numbers are not rare:
indeed, almost all real and complex numbers are
transcendental, since the algebraic numbers are
countable, but the sets of real and complex
numbers are uncountable. All transcendental
numbers are irrational, since all rational
numbers are algebraic. (The converse is not true:
not all irrational numbers are transcendental.)

From there, Google

cardinality countable uncountable

and take your pick of ~41000 hits.

Dale

 

... stuff deleted ...

Here, I ignored the fact that Mr. Wedel quoted Wikipedia
and referred him to Wikipedia yet again.

The relevant portion of my article should have been
to note that he had no appreciation for how "almost"
one gets with "almost all".

The sets (transcendental, algebraic) of numbers are
both infinite. However, the latter (algebraic) can
be put in 1:1 correspondence with the natural numbers,
while the former (transcendental) can be put in 1:1
correspondence with the reals.

In other words, if you're comparing "how many" or
"what proportion" of transcendentals there are,
it's like this:

#transcendentals : #reals :: 1:1

#transcendentals : #algebraics :: #reals : #naturals

 

This is lapidary. Thanks.

 

No. We are talking 100%. You need to learn some measure theory
to understand this. An alternative to measure theory is realizing
that
the algebraic numbers are COUNTABLE, while the transcendentals are
uncountable. Thus, the algebraic numbers form a subset of density
0 in the reals. While both sets are infinite, one set (the
transcendentals)
is "vastly" larger.

They can't be put into 1-1 correspondence. In a vague sense (and not
very precise) one
could say that for EACH algebraic number there are uncountably
infinitely
many transcendentals.

Look at it another way. You know that there are infinitely many
integers.
You know that there are infinitely many integers that are perfect
squares.
Now ask: what fraction of the integers are perfect squares?

The number of integers <= than N^2 is N^2. The number of squares is N.
The fraction of squares to integers is then N/N^2. Now let N go to
infinity in the
limit.

 

Hi ,

I think there is a difference between transcendentals and irrationals.

If I may boldly put, transcendental are more irrational than
irrational numbers.
I could discuss this further if anyone wishes to discuss more.

Jason Akitaishi

 

You may not. You don't know enough mathematics to have an opinion
on the subject.

There is a topic in mathematics known as "measures of irrationality".
I suggest that you look it up.

Liouville's theorem shows that the transcendentals are far
easier to approximate by rationals than the algebraics.

You don't know enough about the subject to conduct a meaningful
discussion.