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Suppose that we have rational numbers $q_1$, $q_2$ such that $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{q_1}e^{-\frac{t^2}{2}} \,\mathrm{d}t=q_2.$$ Does this imply that $q_1=0$ and $q_2=\dfrac{1}{2}$?

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  • $\begingroup$ Two different questions here: (a) $\Phi(\Phi^{-1}(1/4)) = 1/4$ and $q = \Phi^{-1}(1/4) \approx -0.6744898$ is a number with rational $\Phi(q),$ with $q \ne 0$, so Yes to the title. (b) However, I would not want to claim that $q$ is rational, so the main question remains unanswered. $\endgroup$ – BruceET Jan 10 '18 at 23:29
  • $\begingroup$ I suspect $q_1$, $q_2$ can be even algebraic instead of rational (like in Lindemann-Weierstrass theorem showing that $\sin x$ is transcendental) $\endgroup$ – sdcvvc Feb 11 '18 at 1:48

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