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Chapter 8: Problem 7

Graph the complex number and find its modulus. $$-2$$

Short Answer

Expert verified
The complex number -2 is plotted on the real axis at -2, and its modulus is 2.

Step by step solution

01

Understand the Complex Number

The number given is $$-2.$$ In complex number notation, it can be written as $$-2 + 0i.$$ This means the real part is $$-2$$ and the imaginary part is $$0.$$ Thus, the complex number is located on the real axis at $$-2$$ in the complex plane.
02

Plot the Complex Number on the Complex Plane

The complex plane consists of a horizontal real axis and a vertical imaginary axis. Plot the point $$(-2, 0)$$ on this plane. This represents the complex number $$-2$$ since there is no imaginary component.
03

Calculate the Modulus

The modulus of a complex number $$a + bi$$ is calculated using the formula: \[\sqrt{a^2 + b^2}\] For the complex number $$-2 + 0i,$$ we have \(a = -2\) and \(b = 0\). Thus, the modulus is \[\sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2.\]

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Key Concepts

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

Modulus of Complex Numbers
Understanding the modulus of a complex number can be quite enlightening in comprehending its properties. This value, often referred to as the magnitude, represents the distance from the origin (0,0) to the point located by the complex number on the complex plane.
To calculate this, we use the formula \ \sqrt{a^2 + b^2} \ where \( a \) and \( b \) are the real and imaginary parts of the complex number, respectively.
  • If a complex number is \(-2 + 0i\), the real part \(a\) is \(-2\) and the imaginary part \(b\) is \(0\).
  • Plugging these values into the formula, \ \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2 \.
This shows that the modulus is always a non-negative number, representing the distance in a geometrical sense, akin to the length of a hypotenuse in a right triangle.
Complex Plane
The complex plane is an essential concept for visualizing complex numbers. It consists of two axes: a horizontal real axis and a vertical imaginary axis.
A complex number such as \(-2 + 0i\) can be plotted on this plane:
  • The number is depicted as a point, and you can think of this plane as an extension of the regular coordinate plane used in algebra but designed for complex numbers.
  • On the complex plane, the point \((-2,0)\) is located on the horizontal axis, illustrating its real component.
This visualization helps in better understanding operations on complex numbers like addition, subtraction, and notably, finding the modulus, as it translates geometrical concepts into algebraic operations.
Real and Imaginary Parts
In any complex number \(a + bi\), the real part is \(a\), and the imaginary part is \(b\), where \(i\) is the imaginary unit, satisfying the equation \(i^2 = -1\).
Knowing these parts allows a complex number to be expressed in terms of Cartesian coordinates on the complex plane:
  • For \(-2 + 0i\), the real part is \(-2\), and the imaginary part is \(0\).
  • This means it lies purely on the real axis, reflecting a purely real number with no imaginary component.
This understanding clarifies the structure of complex numbers and their placement on the complex plane, laying the foundation for further operations involving complex numbers in various mathematical and engineering applications.

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Most popular questions from this chapter

Find the indicated roots, and graph the roots in the complex plane. The fourth roots of \(-1\)

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$-20$$

Compare the polar equation of the circle \(r=2\) with its equation in rectangular coordinates. In which coordinate system is the equation simpler? Do the same for the equation of the fourleaved rose \(r=\sin 2 \theta .\) Which coordinate system would you choose to study these curves?

Sketch a graph of the rectangular equation. [ Hint: First convert the equation to polar coordinates.] $$\left(x^{2}+y^{2}\right)^{2}=x^{2}-y^{2}$$

Find the indicated roots, and graph the roots in the complex plane. The square roots of \(4 \sqrt{3}+4 i\)
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