Rational or Irrational?

Question

Here is my question: can an Irrational number; like $e$, be equal to a second irrational number, $\pi$, times an integer, then divided by a second rational number? Such as: $e = \frac{\pi a }{b} $ , where $a$ and $b$ are some integers. If not, can you explain why and include a proof? Thank you for reading.

Answer 1

Short answer:

$$\sqrt8=2\sqrt2.$$

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More generally, any irrational times a rational gives another irrational.

Answer 2

Let $x,y$ be irrational numbers, and $a,b$ be some integers different from $0$. Consider: $$ \frac{x}{y}=\frac{a}{b} $$ which is clearly equivalent to your question. If $ \frac{x}{y}$ is rational, then the answer is yes, you can find some integers $a,b$ as above; otherwise, the answer is no. For instance, let $x=\sqrt{2}$, $y=3\sqrt{2}$. Then, you can write: $$ x= \frac{1}{3} y $$ Regarding your particular case, namely $\frac{e}{\pi}$, I think it is not yet known whether this number is irrational or not.

Answer 3

An irrational number (called $a$) times rational number (called $b$) always gives a irrational number, except when $b=0$. Similar to the answer above (More generally, any irrational times a rational gives another irrational).

Now, irrational multiplied by irrational can give a rational (example $xy$ where $x=\sqrt{2}$ and $y=2\sqrt{2}$). But an irrational multiplied by a irrational can also give a irrational (such as if $r=\sqrt{2}$ and $d=\sqrt{5}$) then $rd$ is irrational. So a irrational number multiplied by irrational number is undetermined.

Regarding $\pi$ and $e$, I don't know the answer although this may be related to the topic of transcendental numbers as someone else has said above.