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

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Identify the complex number given: \(z = 3 + 2i\), where the real part is 3 and the imaginary part is 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.

Recall that the absolute value (or modulus) of a complex number \(z = a + bi\) is given by the formula \(|z| = \sqrt{a^2 + b^2}\).

Substitute the values of the real part \(a = 3\) and the imaginary part \(b = 2\) into the formula: \(|z| = \sqrt{3^2 + 2^2}\).

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.

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주요 개념

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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.