A complex number is a number of the form a + bi , where i = and a and b are real numbers. For example, 5 + 3i , - + 4i , 4.2 - 12i , and - - i are all complex numbers. a is called the real part of the complex number and bi is called the imaginary part of the complex number. In the complex number 6 - 4i , for example, the real part is 6 and the imaginary part is -4i .
To add two complex numbers, add their real parts and add their imaginary parts: (a _{1} + b _{1} i) + (a _{2} + b _{2} i) = (a _{1} + a _{2}) + (b _{1} + b _{2})i .
Examples:
(12 + 6i) + (11 + 5i) = (12 + 11) + (6 + 5)i = 23 + 11i
(5 - 7i) + (4 + i) = (5 + 4) + (- 7 + 1)i = 9 - 6i
.
(2 - 4i) + (- 6 - 5i) = (2 - 6) + (- 4 - 5)i = - 4 - 9i
.
To subtract two complex numbers, subtract their real parts and subtract their imaginary parts: (a _{1} + b _{1} i) - (a _{2} + b _{2} i) = (a _{1} - a _{2}) + (b _{1} - b _{2})i .
Examples:
(4 + 5i) - (2 + 3i) = (4 - 2) + (5 - 3)i = 2 + 2i
.
(3 - 7i) - (4 + 6i) = (3 - 4) + (- 7 - 6)i = - 1 - 13i
(- 4 + 2i) - (3 - 11i) = (- 4 - 3) + (2 - (- 11))i = - 7 + 13i
(6 - 9i) - (- 3 - 4i) = (6 - (- 3)) + (- 9 - (- 4))i = 9 - 5i
To multiply a complex number by a scalar, multiply the real part by the scalar and multiply the imaginary part by the scalar: c(a + bi) = ca + cbi .
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 + 14i
^{2} = 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
i
^{2}
term (
i
^{2} = - 1
) and
combine like terms.
(a _{1} + b _{1} i)(a _{2} + b _{2} i) | = | a _{1} a _{2} + a _{1} b _{2} i + a _{2} b _{1} i + b _{1} b _{2} i ^{2} | |
= | a _{1} a _{2} + (a _{1} b _{2} + a _{2} b _{1})i + b _{1} b _{2}(- 1) | ||
= | (a _{1} a _{2} - b _{1} b _{2}) + (a _{1} b _{2} + a _{2} b _{1})i. |
Examples:
(2 + 3i)(5 + 2i) =
?
= | 10 + 4i + 15i + 6i ^{2} | ||
= | 10 + 19i - 6 | ||
= | 4 + 19i. |
= | 18 + 3i - 24i - 4i ^{2} | ||
= | 18 - 21i + 4 | ||
= | 22 - 21i. |
= | 42 - 14i - 12i + 4i ^{2} | ||
= | 42 - 26i - 4 | ||
= | 38 - 26i. |
= | (2 + 3i)(2 + 3i) | ||
= | 4 + 6i + 6i + 9i ^{2} | ||
= | 4 + 12i - 9 | ||
= | -5 + 12i. |
= | 25 - 16i ^{2} | ||
= | 25 + 16 | ||
= | 41. |