In Exercises 1–10, plot each complex number and find its absolute value. z = 3 + 2i

Verified step by step guidance

1. Identify the complex number given: \(z = 3 + 2i\), where the real part is 3 and the imaginary part is 2.
2. Plot the complex number on the complex plane by marking the point with coordinates \((3, 2)\), where the x-axis represents the real part and the y-axis represents the imaginary part.
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 the real part \(a = 3\) and the imaginary part \(b = 2\) into the formula: \(|z| = \sqrt{3^2 + 2^2}\).
5. Simplify the expression under the square root to find the absolute value, which represents the distance of the point from the origin in the complex plane.

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. It can be represented as a point or vector in the complex plane, with the horizontal axis for the real part and the vertical axis for the imaginary part.

Plotting Complex Numbers

To plot a complex number, locate the point corresponding to its real part on the x-axis and its imaginary part on the y-axis. For z = 3 + 2i, plot the point at (3, 2) 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), representing the magnitude of the complex number.