In Exercises 1–10, plot each complex number and find its absolute value.z = −3 + 4i

Verified step by step guidance

1. Step 1: Identify the real and imaginary parts of the complex number. Here, the complex number is given as \( z = -3 + 4i \). The real part is \(-3\) and the imaginary part is \(4\).
2. Step 2: Plot the complex number on the complex plane. The horizontal axis (x-axis) represents the real part, and the vertical axis (y-axis) represents the imaginary part. Plot the point \((-3, 4)\) on this plane.
3. Step 3: To find the absolute value (or modulus) of the complex number \( z = a + bi \), use the formula \( |z| = \sqrt{a^2 + b^2} \).
4. Step 4: Substitute the real part \(a = -3\) and the imaginary part \(b = 4\) into the formula: \( |z| = \sqrt{(-3)^2 + (4)^2} \).
5. Step 5: Simplify the expression under the square root: \( |z| = \sqrt{9 + 16} \). This will give you the absolute value of the complex number.

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 this case, z = -3 + 4i has a real part of -3 and an imaginary part of 4. Understanding complex numbers is essential for visualizing them on the complex plane.

Plotting on the Complex Plane

The complex plane is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part of complex numbers. To plot the complex number z = -3 + 4i, you would locate the point at (-3, 4) on this plane. This visualization helps in understanding the geometric interpretation of complex numbers.

Absolute Value of a Complex Number

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 (0, 0) in the complex plane. For z = -3 + 4i, the absolute value is |z| = √((-3)² + (4)²) = √(9 + 16) = √25 = 5.