Algebraic Number/Examples/One Third (1 + i Root 2)

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