- Imaginary Numbers
- The imaginary unit is and is denoted by i, and
- By definition,
i2 = -1.
- For any positive real number n,
- Complex Numbers
- Complex numbers have the form
a + bi
where a and b are real numbers.
- Examples:
- Adding and Subtracting Complex Numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
- Steps to adding and subtracting complex numbers:
- Change all imaginary numbers to bi form.
- Add (or subtract) the real parts of the complex numbers.
- Add (or subtract) the imaginary parts of the complex numbers.
- Write the answer in the form a + bi.
- Multiplying Complex Numbers
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
- Steps to multiplying complex numbers:
- Change all imaginary numbers to bi form.
- Multiply the complex numbers as you would multiply polynomials.
- Substitute -1 for each i2.
- Combine the real parts and the imaginary parts.
- Write the answer in the form a + bi.
- Complex Conjugates
- The complex conjugate of a complex number is a complex number having the same two terms
with the sign inbetween changed.
-
| Complex Number |
Complex Conjugate |
| 7 + 3i |
7 - 3i |
| 3 - 2i |
3 + 2i |
| -2 + 5i |
-2 - 5i |
| i |
-i |
- Dividing Complex Numbers
-
- Change all imaginary numbers to bi form.
- Write the division problem as a fraction.
- Rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
- Write the answer in the form a + bi.
| Example: Divide |
7 + 3i
7 - 3i |
7 + 3i
7 - 3i |
= |
7 + 3i
7 - 3i |
· |
7 + 3i
7 + 3i |
| |
= |
(7 + 3i)(7 + 3i)
(7 - 3i)(7 + 3i) |
| |
= |
49 + 21i + 21i + 9i2
49 + 21i - 21i - 9i2 |
| |
= |
49 + 42i + 9(-1)
49 - 9(-1) |
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