Final answer:
To plot a complex number like 3 + 4i on the complex plane, place a point at the coordinates (3,4). The polar form is (5, tan⁻¹(4/3)), and the exponential form is 5e^(i tan⁻¹(4/3)), using magnitude and angle calculated with the Pythagorean theorem and tangent function respectively.
Step-by-step explanation:
To plot a complex number on the complex plane, you need to understand that it has a real part and an imaginary part. For example, let's consider the complex number 3 + 4i, where '3' is the real part and '4i' is the imaginary part. To plot this on the complex plane, you would place a point at the coordinates (3,4).
In polar form, this complex number is represented by its magnitude (r) and the angle it makes with the positive x-axis (θ). The magnitude can be calculated using the Pythagorean theorem as r = √(3² + 4²) = 5. To find the angle θ, we use the arctan function (tan⁻¹(opposite/adjacent)), which in this case is θ = tan⁻¹(4/3). Therefore, the polar form of the complex number is (5, tan⁻¹(4/3)).
For the exponential form, we use Euler's formula, which states e^(iθ) = cos θ + i sin θ. The complex number in exponential form is given by r*e^(iθ), which in our example is 5e^(i tan⁻¹(4/3)).