In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −4i

검증된 단계별 안내

Identify the complex number given: \(-4i\). This means the real part is 0 and the imaginary part is \(-4\).

Plot the complex number on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. Since the real part is 0 and the imaginary part is \(-4\), plot the point at \((0, -4)\).

Calculate the magnitude (or modulus) \(r\) of the complex number using the formula \(r = \sqrt{a^2 + b^2}\), where \(a\) is the real part and \(b\) is the imaginary part. Here, \(r = \sqrt{0^2 + (-4)^2}\).

Determine the argument (or angle) \(\theta\) of the complex number. Since the point lies on the negative imaginary axis, the argument is the angle measured from the positive real axis to the point \((0, -4)\). Express \(\theta\) in degrees or radians accordingly.

Write the complex number in polar form using the formula \(r(\cos \theta + i \sin \theta)\) or \(r \operatorname{cis} \theta\), where \(r\) is the magnitude and \(\theta\) is the argument found in the previous steps.

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주요 개념

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Complex Numbers and the Complex Plane

A complex number is expressed as a + bi, where a is the real part and b is the imaginary part. The complex plane represents these numbers graphically, with the horizontal axis for the real part and the vertical axis for the imaginary part. Plotting a complex number involves locating the point (a, b) on this plane.

Polar Form of Complex Numbers

Polar form expresses a complex number using its magnitude (distance from the origin) and argument (angle from the positive real axis). It is written as r(cos θ + i sin θ) or r∠θ, where r is the modulus and θ is the argument in degrees or radians. This form highlights the geometric interpretation of complex numbers.

Calculating Magnitude and Argument

The magnitude r of a complex number a + bi is found using r = √(a² + b²). The argument θ is the angle formed with the positive real axis, calculated using θ = arctan(b/a), adjusted for the correct quadrant. These values are essential for converting from rectangular to polar form.