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

Verified step by step guidance

1. Identify the complex number given: \(z = 4i\). This means the real part is 0 and the imaginary part is 4.
2. Plot the complex number on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. For \(z = 4i\), plot the point at \((0, 4)\).
3. Recall that the absolute value (or modulus) of a complex number \(z = a + bi\) is given by the formula \(|z| = \sqrt{a^2 + b^2}\).
4. Substitute the values of \(a = 0\) and \(b = 4\) into the formula to express the absolute value as \(|z| = \sqrt{0^2 + 4^2}\).
5. Simplify the expression under the square root to find the absolute value of \(z\) (do not calculate the final numeric value here).

Key Concepts

Complex Numbers and the Complex Plane

A complex number is expressed as z = a + bi, where a is the real part and b is the imaginary part. The complex plane represents these numbers graphically, with the horizontal axis for the real part and the vertical axis for the imaginary part. Plotting a complex number involves locating the point (a, b) on this plane.

Imaginary Unit and Pure Imaginary Numbers

The imaginary unit i is defined by i² = -1. A pure imaginary number has no real part and is written as bi, where b is a real number. For example, z = 4i lies on the imaginary axis at (0, 4) in the complex plane.

Absolute Value (Modulus) of a Complex Number

The absolute value or modulus of a complex number z = a + bi is the distance from the origin to the point (a, b) in the complex plane. It is calculated as |z| = √(a² + b²). For z = 4i, the modulus is |4i| = √(0² + 4²) = 4.