Algebraic Number/Examples/One Third (1 + i Root 2)
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Example of Algebraic Number
- $\dfrac 1 3 \paren {1 + i \sqrt 2}$ is an algebraic number.
Proof
\(\ds x\) | \(=\) | \(\ds \dfrac 1 3 \paren {1 + i \sqrt 2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 x - 1\) | \(=\) | \(\ds i \sqrt 2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {3 x - 1}^2\) | \(=\) | \(\ds 2 i^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 9 x^2 - 6 x + 1\) | \(=\) | \(\ds -2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 x^2 - 2 x + 1\) | \(=\) | \(\ds 0\) |
Hence the result by definition of algebraic number.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): algebraic number
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): algebraic number