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Chapter 7: Problem 16

Graph each complex number, and find its absolute value. $$-4-4 i$$

Short Answer

Expert verified
Plot at (-4, -4). Absolute value is 4鈭�2.

Step by step solution

01

- Understand the Complex Number

A complex number is of the form \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part. For the complex number \(-4-4i\), the real part \(a\) is \(-4\) and the imaginary part \(b\) is \(-4\).
02

- Plot the Complex Number

To graph the complex number, use the complex plane where the x-axis represents the real part and the y-axis represents the imaginary part. Plot the point \((-4, -4)\).
03

- Determine the Absolute Value

The absolute value (or magnitude) of a complex number \(a + bi\) is given by \(|a + bi| = \sqrt{a^2 + b^2}\). Substitute \(a = -4\) and \(b = -4\) into the formula: \(|-4 - 4i| = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\).

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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 like finding the distance from that number to the origin \((0, 0)\) on the complex plane. We use the formula \(|a + bi| = \sqrt{a^2 + b^2}\) where \(a\) is the real part and \(b\) is the imaginary part.

Let's look at the example \(-4 - 4i\). Here, both \(a\) and \(b\) are \(-4\).
  • First, square each part: \( (-4)^2 = 16 \).
  • Then, add them together: \( 16 + 16 = 32 \).
  • Finally, take the square root: \( \sqrt{32} = 4\sqrt{2} \).
This tells us that the absolute value of \(-4 - 4i\) is \(4\sqrt{2} \), which is the distance from the point \((-4, -4)\) to the origin.

By understanding this concept of distance, it becomes clear how the absolute value helps describe the size or magnitude of a complex number.
graphing complex numbers
Graphing complex numbers is just like plotting points on a regular graph, but here we use a special plane called the complex plane. The x-axis represents the real part, and the y-axis represents the imaginary part.

For the complex number \(-4 - 4i\):
  • The real part is \(-4 \) and the imaginary part is also \( -4\).
To graph it:
  • Find \(-4\) on the x-axis (the real part).
  • Then move \(-4\) units down on the y-axis (since \( -4i \) represents moving 4 units downward).
  • You'll place your point at \((-4, -4)\).
This point is now graphically representing the complex number \(-4 - 4i\) on the complex plane. It鈥檚 a straightforward way to visualize complex numbers!
complex plane
The complex plane is a handy tool for understanding complex numbers. It's like a coordinate system specifically designed for complex numbers. Instead of just an x-axis and y-axis, the complex plane uses a real axis and an imaginary axis.

The horizontal axis (x-axis) represents the real part of complex numbers, while the vertical axis (y-axis) represents the imaginary part.

For example:
  • Any real number is located on the real axis.
  • Any imaginary number is located on the imaginary axis.
  • A complex number like \(a + bi \) is a point plotted with the real part \(a\) on the x-axis and the imaginary part \(b\) on the y-axis.
By using the complex plane to plot numbers like \(-4 - 4i \), we get a visual feel for their relationships and distances, making complex numbers easier to understand.

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Most popular questions from this chapter

Find the angle to the nearest tenth of a degree between each given pair of vectors. $$\langle- 2,-5\rangle,\langle 1,-9\rangle$$

Solve each problem. Given that \(z=\sqrt{3}+i,\) find \(z^{4}\) by writing \(z\) in trigonometric form and computing the product \(z \cdot z \cdot z \cdot z\)

Find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\) for each pair of complex numbers, using trigonometric form. Write the answer in the form \(a+b i\). $$z_{1}=-\sqrt{3}+i, z_{2}=4 \sqrt{3}-4 i$$

Find the product of the given complex number and its complex conjugate in trigonometric form. $$2\left(\cos 7^{\circ}+i \sin 7^{\circ}\right)$$

Prove that scalar multiplication is distributive over vector addition, first using the component form and then using a geometric argument.
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