In Exercises 1–10, plot each complex number and find its absolute value. z = 4i

Verified step by step guidance

1. Step 1: Identify the complex number given, which is $z = 4i$. This means the real part is 0 and the imaginary part is 4.
2. Step 2: Plot the complex number on the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. Since the real part is 0, the point is on the imaginary axis at $4i$.
3. Step 3: To find the absolute value of a complex number $z = a + bi$, use the formula $|z| = \sqrt{a^2 + b^2}$. Here, $a = 0$ and $b = 4$.
4. Step 4: Substitute the values into the formula: $|z| = \sqrt{0^2 + 4^2}$.
5. Step 5: Simplify the expression under the square root to find the absolute value.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form z = a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i'. In the given example, z = 4i, the real part is 0 and the imaginary part is 4, indicating that the number lies purely on the imaginary axis in the complex plane.

Plotting Complex Numbers

To plot a complex number on the complex plane, the horizontal axis represents the real part, while the vertical axis represents the imaginary part. For z = 4i, the point is plotted at (0, 4), which shows that it is located 4 units above the origin along the imaginary axis, illustrating the geometric representation of complex numbers.

Absolute Value of Complex Numbers

The absolute value (or modulus) of a complex number z = a + bi is calculated using the formula |z| = √(a² + b²). This value represents the distance from the origin to the point (a, b) in the complex plane. For z = 4i, the absolute value is |4i| = √(0² + 4²) = 4, indicating that the distance from the origin to the point (0, 4) is 4 units.