Algebraic Number/Examples/2 - Root 2 i
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Example of Algebraic Number
- $2 - \sqrt 2 i$ is an algebraic number.
Proof
Let $x = 2 - \sqrt 2 i$.
We have that:
\(\ds x - 2\) | \(=\) | \(\ds \sqrt 2 i\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x - 2}^2\) | \(=\) | \(\ds 2 i^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2 - 4 x + 4\) | \(=\) | \(\ds -2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2 - 4 x + 6\) | \(=\) | \(\ds 0\) |
Thus $2 - \sqrt 2 i$ is a root of $x^2 - 4 x + 6 = 0$.
Hence the result by definition of algebraic number.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $158 \ \text{(b)}$