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

Verified step by step guidance

1. Step 1: Identify the real and imaginary parts of the complex number. For the complex number \( z = 3 + 2i \), the real part is 3 and the imaginary part is 2.
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. Plot the point (3, 2) on this plane.
3. Step 3: To find the absolute value of the complex number \( z = 3 + 2i \), use the formula \( |z| = \sqrt{a^2 + b^2} \), where \( a \) is the real part and \( b \) is the imaginary part.
4. Step 4: Substitute the values of \( a \) and \( b \) into the formula. Here, \( a = 3 \) and \( b = 2 \), so the expression becomes \( |z| = \sqrt{3^2 + 2^2} \).
5. Step 5: Simplify the expression under the square root to find the absolute value. Calculate \( 3^2 \) and \( 2^2 \), then add the results and take the square root.

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, 'b' is the coefficient of the imaginary unit 'i', and 'i' is defined as the square root of -1. In the given example, z = 3 + 2i, 3 is the real part and 2 is the imaginary part.

Plotting Complex Numbers

Complex numbers can be represented graphically on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. For the complex number z = 3 + 2i, it would be plotted at the point (3, 2) on this plane, allowing for a visual understanding of its position relative to the origin.

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 of the point (a, b) from the origin in the complex plane. For z = 3 + 2i, the absolute value would be |z| = √(3² + 2²) = √(9 + 4) = √13.