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Chapter 7: Problem 1

In Exercises \(1-10,\) plot each complex number and find its absolute value. $$ z=4 i $$

Short Answer

Expert verified
The complex number \(z=4i\) is plotted at the point (0,4) on the complex plane, and its absolute value is 4.

Step by step solution

01

Identify the real and imaginary components

For the complex number \(z=4i\), the real part \(a=0\) and the imaginary part \(b=4\).
02

Plot the complex number on the complex plane

On the complex plane, plot (0,4) as the complex number \(z=4i\). This point is directly above the origin at a distance of 4 units along the imaginary (y) axis.
03

Calculate the absolute value

Apply the formula to find the absolute value of \(z\), which is \(\|z\|=\sqrt{a^2+b^2}=\sqrt{0^2+4^2}=4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Complex Numbers
When it comes to understanding complex numbers, visual representation is crucial. Plotting complex numbers on a graph helps students comprehend their nature and behavior. To plot a complex number, one must first comprehend its structure: a complex number is expressed in the form of a + bi, where a is the real part and bi is the imaginary part.

Visualizing this on the complex plane, the horizontal axis (known as the real axis) represents the real part, while the vertical axis (known as the imaginary axis) represents the imaginary part. To plot the number 4i, which has a real part of 0 and an imaginary part of 4, you would place a point at the coordinates (0,4) on this plane. It will lie on the imaginary axis, four units above the origin.
Absolute Value of Complex Numbers
The absolute value, or modulus, of a complex number is a measure of its distance from the origin on the complex plane. To calculate it, one can use the Pythagorean theorem because a complex number and its components form a right-angled triangle with the origin.

The general formula for the absolute value of a complex number a + bi is |z| = sqrt(a^2 + b^2). For instance, given the complex number z = 4i with real part 0 and imaginary part 4, its absolute value is |z| = sqrt(0^2 + 4^2) = 4. This result represents the straight-line distance from the origin to the point (0,4) on the complex plane, reinforcing the geometric interpretation of a complex number's absolute value.
Complex Plane
The complex plane, also known as the Argand plane, is a two-dimensional plane used to represent complex numbers graphically. It's similar to the Cartesian plane used in real number geometry, but with the significant distinction that the 'y' axis, in this case, represents the imaginary component of complex numbers.

When students first encounter this concept, it's important to emphasize the analogy with the familiar coordinate system to aid understanding. A point in the complex plane has coordinates (a, b), where a is the real part and b the imaginary part of a complex number a + bi. Understanding how to navigate and utilize the complex plane is essential in advanced mathematics, especially when dealing with functions of complex variables or conformal mappings.

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Most popular questions from this chapter

In Exercises 9鈥�16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$ B=80^{\circ}, C=10^{\circ}, a=8 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. If I know the measures of all three angles of an oblique triangle, neither the Law of sines nor the Law of Cosines can be used to find the length of a side.

In Exercises \(81-86,\) solve equation in the complex number system. Express solutions in polar and rectangular form. $$ x^{6}-1=0 $$

Find the smallest interval for \(\theta\) starting with \(\theta \min =0\) so that your graphing utility graphs the given polar equation exactly once without retracing any portion of it. $$r=4 \sin \theta$$

In Exercises 17鈥�32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. $$ a=9.3, b=41, A=18^{\circ} $$
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