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Chapter 7: Problem 9

Plot the complex number and find its absolute value. $$-5-12 i$$

Short Answer

Expert verified
The absolute value of the complex number -5 - 12i is 13. The point (-5,-12) represents this number in the complex plane.

Step by step solution

01

Identify the Real and Imaginary Parts

In the complex number -5 - 12i, -5 is the real part and -12 is the imaginary part.
02

Plotting in the Complex Plane

Plot the complex number on the complex plane which is similar to a Cartesian plain. The x-axis (horizontal) represents the real part and the y-axis (vertical) the imaginary part. Mark the point (-5, -12) which represents the complex number -5 - 12i.
03

Calculate the Absolute Value

The absolute value of a complex number is the distance from the origin (0,0) to this point in the complex plane. Use the formula \( \sqrt{Re^2 + Im^2} \). Substitute the values into the formula as \( \sqrt{(-5)^2 + (-12)^2} \) which simplifies to \( \sqrt{25 + 144} \) that equals \( \sqrt{169} = 13 \).

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Key Concepts

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

Absolute Value of Complex Numbers
The absolute value of a complex number is similar to the distance from the origin to the point that represents the number on the complex plane. It provides a measure of how far the complex number is from the origin of the plane.
This concept is crucial for understanding the magnitude or "size" of complex numbers.
  • The formula to find the absolute value, also known as the modulus, is \( \sqrt{Re^2 + Im^2} \).
  • Here, \( Re \) represents the real part of the complex number, while \( Im \) stands for the imaginary part.
  • In our exercise, the complex number \(-5 - 12i\) has the real part \(-5\) and the imaginary part \(-12\).
  • To find the absolute value, substitute these into the formula: \( \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = 13 \).
Thus, the absolute value of \(-5 - 12i\) is 13, meaning the point is 13 units away from the origin on the complex plane.
Complex Plane
To fully grasp complex numbers, visualize them on the complex plane, which acts much like a coordinate system. The complex plane allows us to clearly see and understand the position and magnitude of complex numbers.
  • The horizontal axis (x-axis) represents the real part of the complex numbers.
  • The vertical axis (y-axis) is associated with the imaginary part.
In our example, to plot the complex number \(-5 - 12i\), move 5 units to the left on the x-axis and 12 units down on the y-axis. This places the number at the point \((-5, -12)\).
The complex plane provides a visual way to interpret the behavior and properties of complex numbers, aiding in the understanding of operations like addition, subtraction, and finding absolute values.
Imaginary Part
The imaginary part of a complex number is essential because it defines the number's position in the imaginary dimension of the complex plane. In a complex number such as \( a + bi \), the term \( bi \) is the imaginary part where \( b \) is a real number and \( i \) is the imaginary unit.
  • It's crucial to note that the imaginary part is not a real number. Instead, it multiplies the imaginary unit \( i \), which satisfies the equation \( i^2 = -1 \).
  • In our specific complex number \(-5 - 12i\), the imaginary part is \(-12\), indicating 12 units down on the imaginary axis.
Understanding the imaginary part helps with operations involving complex numbers, such as addition, subtraction, and especially visualizing them on the complex plane. By knowing the imaginary part, you can determine how far up or down a point is on the imaginary axis, indicating the complex number's position in this extra dimension.

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Most popular questions from this chapter

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$[2(\cos 1.25+i \sin 1.25)]^{4}$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[3\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)\right]^{4}$$

Find the magnitude and direction angle of the vector v. $$\mathbf{v}=8\left(\cos 135^{\circ} \mathbf{i}+\sin 135^{\circ} \mathbf{j}\right)$$

Find the component form of v given its magnitude and the angle it makes with the positive \(x\) -axis. Sketch v. Angle:\begin{aligned}&\theta=0^{\circ}\\\&\theta=45^{\circ}\\\&\theta=120^{\circ}\\\ &\theta=135^{\circ}\\\&\theta=150^{\circ}\\\&\theta=90^{\circ}\\\&\mathbf{v} \text { in the direction } \mathbf{i}+3 \mathbf{j}\\\&\mathbf{v} \text { in the direction } 3 \mathbf{i}+4 \mathbf{j} \end{aligned}. Magnitude:$$\|\mathbf{v}\|=3$$

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $$\mathbf{v}=\langle-2,2\rangle$$
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