Complex numbers are numbers of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $i^2 = -1$. Operating with complex numbers involves addition, subtraction, multiplication, and division, all while carefully handling the imaginary unit. These operations build upon basic algebra with the added consideration of simplifying expressions involving $i$. Understanding these operations is crucial for more advanced mathematics and engineering applications.
When adding or subtracting complex numbers, you combine the real parts and the imaginary parts separately. Multiplication involves using the distributive property (FOIL method), and division requires multiplying both the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. Let's put your skills to the test!
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Complex Conjugate | A. A number of the form $a + bi$ |
| 2. Imaginary Unit | B. A number of the form $a - bi$, where $a + bi$ is a complex number |
| 3. Real Part | C. The 'a' in the complex number $a + bi$ |
| 4. Imaginary Part | D. The 'b' in the complex number $a + bi$ |
| 5. Complex Number | E. $i$, defined as $i^2 = -1$ |
Complete the following paragraph using the words: real, imaginary, conjugate, distributive, $i^2$.
When multiplying complex numbers, you use the ___________ property. Remember that ___________ = -1. When dividing complex numbers, multiply the numerator and denominator by the ___________ of the denominator to eliminate the ___________ part from the denominator and leave only a ___________ number.
Explain, in your own words, why multiplying a complex number by its conjugate results in a real number. Provide an example to illustrate your explanation.
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