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Chapter 8: Problem 9

Graph each complex number. In each case, give the absolute value of the number.\(-4-3 i\)

Short Answer

Expert verified
The complex number \\(-4 - 3i\\) is graphed at \\((-4, -3)\\) on the complex plane; its absolute value is \\5\\.

Step by step solution

01

Understand the Complex Number

The complex number \(-4 - 3i\) consists of a real part and an imaginary part. The real part is \(-4\) and the imaginary part is \(-3\).
02

Graph the Complex Number

To graph the complex number, plot the point \((-4, -3)\) on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
03

Calculate the Absolute Value

The absolute value (or modulus) of a complex number \(a + bi\) is given by the formula \(|a + bi| = \sqrt{a^2 + b^2}\). For \(-4 - 3i\), the absolute value is \(|-4 - 3i| = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\).

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Key Concepts

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

Graphing Complex Numbers
Graphing a complex number involves plotting it on a special type of coordinate system known as the complex plane. In this context, every complex number is represented by an ordered pair. Specifically, the point \((-4, -3)\) represents the complex number \(-4 - 3i\).

When graphing, use the horizontal axis for the real part, which is \(-4\) in this case, and the vertical axis for the imaginary part, \(-3i\).
  • First, move left from the origin by 4 units in the horizontal direction.
  • Then, move down 3 units in the vertical direction.
This gives you the precise location on the graph for the complex number \(-4 - 3i\).
Breaking down the process this way helps connect the abstract nature of complex numbers with visual representation, making it easier to understand.
Absolute Value of Complex Numbers
The absolute value of a complex number can be thought of as its "distance" from the origin in the complex plane.

This is similar to finding the length of the hypotenuse of a right triangle using the Pythagorean theorem.
The formula \(|a + bi| = \sqrt{a^2 + b^2}\) provides us with this magnitude.
  • Simplifying the calculation of the absolute value for \(-4 - 3i\):
  • Calculate \((-4)^2 = 16\), leading to the real contribution.
  • Then compute \((-3)^2 = 9\), from the imaginary part.
  • Add these, yielding \(|a + bi| = \sqrt{16 + 9} = \sqrt{25}\).
  • The result is simply \(|a + bi| = 5\), showcasing both simplicity and beauty of complex numbers.
Understanding the absolute value provides a larger context to the scale and effects of the complex part in a number.
Complex Plane
The complex plane is a two-dimensional plane where each complex number corresponds to a unique point. It is composed of a horizontal axis, representing the real part of the numbers, and a vertical axis denoting the imaginary part.

This way of representation helps us visualize operations on complex numbers. Here are some key aspects of the complex plane:
  • Origin: Where the real and imaginary axes meet, representing the complex number zero (0).
  • Each point on the plane corresponds to a unique complex number \(a + bi\).
  • Movements parallel to the axes represent addition of a real or imaginary number.
  • The complex plane makes it easier to understand complex conjugation, addition, and multiplication.
By treating complex numbers as points on this plane, we can gain a more intuitive understanding of their properties and behaviors. It's an essential tool for working with complex numbers in both mathematics and engineering.

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Most popular questions from this chapter

Show that \(x=a-b i\) is a solution to the equation \(x^{2}-2 a x+\left(a^{2}+b^{2}\right)=0\).

Use the formula for \(\cos (A+B)\) to find the exact value of \(\cos 75^{\circ}\).

Write each complex number in trigonometric form, once using degrees and once using radians. In each case, begin by sketching the graph to help find the argument \(\theta\).\(-2 \sqrt{3}+2 i\)

Simplify each expression to a single trigonometric function.\(\sin 18^{\circ} \cos 32^{\circ}+\cos 18^{\circ} \sin 32^{\circ}\)

Is subtraction with complex numbers a commutative operation?
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