# Difference between transcendental and irrational numbers.

#### burt

##### New member
As far as I can tell, a transcendental number goes on forever without being able to be expressed in terms of any sort of rational expression. How is this different than the term irrational number?

#### Subhotosh Khan

##### Super Moderator
Staff member
As far as I can tell, a transcendental number goes on forever without being able to be expressed in terms of any sort of rational expression. How is this different than the term irrational number?
Do a little research using google search!

Please let us know what you find - or need further clarification.

#### burt

##### New member
Do a little research using google search!

Please let us know what you find - or need further clarification.
My google etc research is how I came up with this definition to begin with... Am I misunderstanding it?

#### LCKurtz

##### Junior Member
My google etc research is how I came up with this definition to begin with... Am I misunderstanding it?
Both transcendental and irrational numbers "go on forever", meaning they have non-terminating non-repeating decimal expansions. That is the equivalent of the definition of an irrational number, but not a transcendental one. Google "transcendental number". You will find lots of information.

#### burt

##### New member
I think I found the difference - irrational numbers may be able to be expressed as a root of a rational number ie: $$\displaystyle \sqrt{2}$$ but transcendental numbers cannot.

#### burt

##### New member
Are there any other known transcendental numbers besides for $$\displaystyle e$$ and $$\displaystyle \pi$$?

#### topsquark

##### Full Member
Are there any other known transcendental numbers besides for $$\displaystyle e$$ and $$\displaystyle \pi$$?
Here's a list of a few. It's very hard to prove that a number is transcendental. You have to show that the number is not the zero of any polynomial with integer coefficients.

It's not just square roots. For example, $$\displaystyle \sqrt{ 1 + \sqrt{3} } + 3$$ is irrational but not transcendental.

Here's a list of a few.

-Dan

#### Romsek

##### Full Member
I think I found the difference - irrational numbers may be able to be expressed as a root of a rational number ie: $$\displaystyle \sqrt{2}$$ but transcendental numbers cannot.
you want to look at the definition of "algebraic" numbers. Irrational numbers are simply that, not rational. Both the sets of
algebraic numbers and transcendental numbers are subsets of the irrationals.

By "known" I think you mean labeled. The golden ratio is another transcendental number given a name. The vast bulk of numbers are transcendental.
Both the set of algebraic numbers and the set of rationals have measure zero.

#### burt

##### New member
you want to look at the definition of "algebraic" numbers. Irrational numbers are simply that, not rational. Both the sets of
algebraic numbers and transcendental numbers are subsets of the irrationals.

By "known" I think you mean labeled. The golden ratio is another transcendental number given a name. The vast bulk of numbers are transcendental.
Both the set of algebraic numbers and the set of rationals have measure zero.
Does this mean more numbers are transcendental than not? How is that?

#### Subhotosh Khan

##### Super Moderator
Staff member
you want to look at the definition of "algebraic" numbers. Irrational numbers are simply that, not rational. Both the sets of
algebraic numbers and transcendental numbers are subsets of the irrationals.

By "known" I think you mean labeled. The golden ratio is another transcendental number given a name. The vast bulk of numbers are transcendental.
Both the set of algebraic numbers and the set of rationals have measure zero.
The golden ratio is NOT a transcendental number. It is root of x2 - x - 1 = 0

#### LCKurtz

##### Junior Member
Does this mean more numbers are transcendental than not? How is that?
Yes. The rationals and the algebraic numbers are both countable infinity. The transcendentals are an uncountable infinity.

#### Romsek

##### Full Member
The golden ratio is NOT a transcendental number. It is root of x2 - x - 1 = 0

Staff member

#### Jomo

##### Elite Member
Yes. The rationals and the algebraic numbers are both countable infinity. The transcendentals are an uncountable infinity.
Can we please a proof of this. I'm not say you are wrong, I just want to see this proof.

#### Subhotosh Khan

##### Super Moderator
Staff member
Can we please a proof of this. I'm not say you are wrong, I just want to see this proof.
For that you have to read a paper by George Cantor. Internet is full of reference to that paper.