Quotients of 3 Unequal Numbers are Unequal
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Theorem
Let $x, y, z \in \R_{\ne 0}$ be non-zero real numbers which are not all equal.
Then $\dfrac x y, \dfrac y z, \dfrac z x$ are also not all equal.
Proof
Aiming for a contradiction, suppose $\dfrac x y = \dfrac y z = \dfrac z x$.
\(\ds \) | \(\) | \(\ds \dfrac x y = \dfrac y z = \dfrac z x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds x^2 z = y^2 x = z^2 y\) | multiplying top and bottom by $x y z$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds x z = y^2, x y = z^2, y z = x^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds x y z = y^3, x y z = z^3, z y z = x^3\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds x^3 = y^3 = z^3\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds x = y = z\) |
This contradicts the assertion that $x, y, z$ are all unequal.
Hence the result by Proof by Contradiction.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $3$