If $c^2 = 12$, then show that $c$ is an irrational number.

Question

Show that, if $x$ satisfies $x^2 = 3$, then $c := 2x$ satisfies $c^2 = 12$. Using this fact, show that $c$ is an irrational number.

Sorry, I am fairly new to analysis. I will look at the question tagged above. — Christopher, Sep 15 '17 at 20:14

gen-ℤ ready to perish · Answer 1 · 2017-09-15T20:48:42.923

1

$$\begin{align} x^2 &= 3 \\ \left( \frac c2 \right) ^2 &= 3 \\ \frac{c^2}{4} &= 3 \\ c^2 &= 12 \\ c &= \pm2\sqrt 3 \\ \end{align}$$

Since $\sqrt 3$ is irrational, so is $\pm2\sqrt 3$ (this has already been proven on MSE).

edited Sep 15 '17 at 20:48

answered Sep 15 '17 at 20:14

Maybe add the fact that any non-zero rational number times an irrational number is irrational as well, which isn't hard to prove either., – WaveX Sep 15 '17 at 20:37
@WaveX I stated that in there, but I figured the OP could easily Google the proof. – gen-ℤ ready to perish Sep 15 '17 at 20:45
I reckon that $c=2\sqrt3$ (or $-2\sqrt3$). – Angina Seng Sep 15 '17 at 20:47
@LordSharktheUnknown Ah duh thank you – gen-ℤ ready to perish Sep 15 '17 at 20:47

1 Answers1