Algebraic Number/Examples/2 - Root 2 i

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: $158 \ \text{(b)}$