The complex plane is a way to visualize complex numbers, which have both a real and an imaginary part. A complex number is generally written in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part, and $i$ is the imaginary unit ($i = \sqrt{-1}$). On the complex plane, the horizontal axis represents the real part ($a$), and the vertical axis represents the imaginary part ($b$). Plotting a complex number $a + bi$ is similar to plotting the point $(a, b)$ on the Cartesian plane.
The modulus of a complex number, often denoted as $|z|$, represents the distance from the origin to the point representing the complex number on the complex plane. For a complex number $z = a + bi$, the modulus is calculated using the Pythagorean theorem: $|z| = \sqrt{a^2 + b^2}$. The modulus is always a non-negative real number.
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Complex Number | A. The distance from the origin to the complex number on the complex plane. |
| 2. Real Part | B. A number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. |
| 3. Imaginary Part | C. The horizontal axis on the complex plane. |
| 4. Modulus | D. The coefficient of the imaginary unit $i$ in a complex number. |
| 5. Real Axis | E. The component of a complex number that does not include the imaginary unit $i$. |
Fill in the missing words in the paragraph below:
The complex plane is used to represent __________ numbers graphically. The horizontal axis represents the __________ part, and the vertical axis represents the __________ part. The __________ of a complex number is its distance from the origin.
Explain how finding the modulus of a complex number is similar to finding the magnitude of a vector in two dimensions. Provide an example to illustrate your explanation.
Please log in to post your answer.
Log InEarn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! 🚀