Algebraic Number/Examples/3 - i

Example of Algebraic Number

$3 - i$ is an algebraic number.

Proof

\(\ds x\)

\(=\)

\(\ds 3 - i\)

\(\ds \leadsto \ \ \)

\(\ds x - 3\)

\(=\)

\(\ds -i\)

\(\ds \leadsto \ \ \)

\(\ds \paren {x - 3}^2\)

\(=\)

\(\ds \paren {-i}^2\)

\(\ds \leadsto \ \ \)

\(\ds x^2 - 6 x + 9\)

\(=\)

\(\ds -1\)

\(\ds \leadsto \ \ \)

\(\ds x^2 - 6 x + 10\)

\(=\)

\(\ds 0\)

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