Can a rational number be represented by a combination of irrational numbers?

Question

Any real number is some power of $e$ (because $\ln(x)$ has values in the range $(-\infty , + \infty)$.

Say, $5$ is a rational number. So there is some $x$ which makes $\exp(x)= 5$.

What is $x$? rational , irrational?

A power of $e$ leads to $5$, Will it be an infinite series converging to the value $5$?

Can a combination of irrational number result into a rational number?

I don't really know what the question is, but $\ln(5)$ is irrational, a consequence of the fact that $e$ is transcendental. If $e^{p/q}=5$, then $e^p=5^q$. Slightly more generally, if $e^x$ is rational, then $x$ is irrational. Related question where you'll find a much more general statement: http://math.stackexchange.com/questions/15285/is-the-natural-log-of-n-rational — Jonas Meyer, Jan 11 '13 at 06:22
What @Jonas said. Regarding Can a combination of irrational number(s) ... result into a rational number?, consider $\pi+(42-\pi)$. — Did, Jan 11 '13 at 06:30
Also, if x is rational, then e^x is irrational. This statement is different to Jonas'. — Adam Rubinson, Jan 11 '13 at 08:53

score 2 · Answer 1 · answered Jan 11 '13 at 07:15

(When he was 12) George Bergman proved that any integer can be represented by a finite sum of integral powers of the golden ratio (irrational number), http://en.wikipedia.org/wiki/Golden_ratio_base. And any rational can be written as the ratio of two integers.

score 1 · Answer 2 · edited Apr 13 '17 at 12:58

What precisely do you mean by a "combination of irrational number[s]"?

Obviously $\sqrt{2}$ and $-\sqrt{2}$ are both irrational, and if "combining" them allows for the operation of addition, then just observe the sum is the rational number $0$.

If you are more interested in questions surrounding exponentiation, logarithms, and the such, you might check out this recent MO question in a similar spirit.

score 0 · Answer 3 · answered Jan 11 '13 at 07:09

Also Rationals are dense in Reals and Irrationals are dense in reals

which implies that you

can always find a sequence of irrational numbers that converges to a given rational
can always find a sequence of rational numbers that converges to a given irrational

Can a rational number be represented by a combination of irrational numbers?

3 Answers3