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

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.