Subtraction of Complex Numbers
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Theorem
Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers.
The subtraction operation on $z_1$ and $z_2$ is:
- $z_1 - z_2 = \paren {a_1 - a_2} + i \paren {b_1 - b_2}$
Proof
\(\ds z_1 - z_2\) | \(=\) | \(\ds z_1 + \paren {- z_2}\) | Definition of Complex Subtraction | |||||||||||
\(\ds \) | \(=\) | \(\ds a_1 + \paren {-a_2} + i \paren {b_1 + \paren {-b_2} }\) | Inverse for Complex Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a_1 - a_2} + i \paren {b_1 - b_2}\) | Definition of Complex Subtraction |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $6.3$: Subtraction of Complex Numbers
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Fundamental Operations with Complex Numbers: $2$. Subtraction