Prove: Irrational Numbers Have Rational Numbers Between

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need to prove that : between any two irrational numbers there is at least one rational number .

TNX .
 
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It's not true, there is none between \sqrt{2} and \sqrt{2}.

Also, what would you like to use for a proof? Analysis? Topology?
 
CompuChip said:
It's not true, there is none between \sqrt{2} and \sqrt{2}.

Also, what would you like to use for a proof? Analysis? Topology?


Analysis, for x and y that x<y
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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