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Complex Numbers
Examples:
4(2 + 5i) = 4(2) + 4(5i) = 8 + 20i
(6 - 9i) = (6) + (- 9i) = 2 - 3i.
-2(11 - 2i) = - 2(11) + (- 2)(- 2i) = - 22 + 4i.
2i(5 + 7i) = 2i(5) + 2i(7i) = 10i + 14i2 = 10i + 14(- 1) = - 14 + 10i.
To multiply two complex numbers, use the FOIL method and treat each complex
number as an ordinary binomial. Then simplify the i2 term (i2 = - 1) and
combine like terms.
| (a1 + b1i)(a2 + b2i) | = | a1a2 + a1b2i + a2b1i + b1b2i2 | |
| = | a1a2 + (a1b2 + a2b1)i + b1b2(- 1) | ||
| = | (a1a2 - b1b2) + (a1b2 + a2b1)i. |
Examples:
(2 + 3i)(5 + 2i) = ?
| = | 10 + 4i + 15i + 6i2 | ||
| = | 10 + 19i - 6 | ||
| = | 4 + 19i. |
| = | 18 + 3i - 24i - 4i2 | ||
| = | 18 - 21i + 4 | ||
| = | 22 - 21i. |
| = | 42 - 14i - 12i + 4i2 | ||
| = | 42 - 26i - 4 | ||
| = | 38 - 26i. |
| = | (2 + 3i)(2 + 3i) | ||
| = | 4 + 6i + 6i + 9i2 | ||
| = | 4 + 12i - 9 | ||
| = | -5 + 12i. |
| = | 25 - 16i2 | ||
| = | 25 + 16 | ||
| = | 41. |
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