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Chapter 12: Problem 4

Locate the given numbers in the complex plane. $$-5+j$$

Short Answer

Expert verified
The number \(-5 + j\) is located at the point \((-5, 1)\) on the complex plane.

Step by step solution

01

Understand the Complex Number Format

Complex numbers are written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. In this case, the number given is \( -5 + j \), which can be interpreted as \( -5 + 1j \) since \( j \) (or \( i \)) represents the imaginary unit \( \sqrt{-1} \).
02

Identify the Coordinates in the Complex Plane

In the complex plane, the horizontal axis (real axis) represents the real part of the complex number, and the vertical axis (imaginary axis) represents the imaginary part. The complex number \( -5 + j \) can be represented by the point \( (-5, 1) \), where \( -5 \) is the real part and \( 1 \) is the imaginary part.
03

Plot the Point on the Complex Plane

To locate the number on the complex plane, plot the identified point \( (-5, 1) \). Start at the origin \((0,0)\), move left 5 units along the real (horizontal) axis to \( (-5, 0) \), and then move up 1 unit along the imaginary (vertical) axis to reach the point \((-5, 1) \).

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Key Concepts

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

Complex Plane
The complex plane is an essential tool for visualizing complex numbers. Similar to a 2D coordinate system, the complex plane consists of two axes.
The horizontal axis is called the real axis, and it represents the real part of the complex numbers.
The vertical axis is the imaginary axis, representing the imaginary part.
  • The position of complex numbers is determined like points in the Cartesian coordinate system.
  • The real part determines the left or right position along the real axis, while the imaginary part moves up or down along the imaginary axis.
Imagine each complex number as a kind of vector or arrow pointing from the origin to its specific location in this plane.
It's a powerful way to understand and perform operations on complex numbers.
Real and Imaginary Parts
Understanding the real and imaginary parts of complex numbers is crucial to working with them.
Each complex number is written as \(a + bi\), where \(a\) and \(b\) stand for:
  • \(a\): the real part of the complex number.
  • \(b\): the coefficient of the imaginary part, which includes the imaginary unit \(i\) or \(j\), representing \(\sqrt{-1}\).
For the complex number \(-5 + j\), we identify:
  • Real part: \(-5\)
  • Imaginary part: \(1\)
By understanding these parts, we can easily locate the number on the complex plane and perform calculations with ease.
Plotting Complex Numbers
Plotting complex numbers on the complex plane allows us to visualize what these numbers represent. Consider the number \(-5 + j\):
To plot it, follow these steps:
  • Start at the origin \((0, 0)\).
  • Move along the real axis: Since the real part is \(-5\), you move 5 units to the left, reaching \((-5, 0)\).
  • Move up the imaginary axis: The imaginary part is \(1\), so from \((-5, 0)\), move up by 1 unit to reach \((-5, 1)\).
This final position, \((-5, 1)\), represents the complex number \(-5 + j\) on the complex plane.
Plotting complex numbers gives you a spatial understanding of their relationships and operations.

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Most popular questions from this chapter

$$\text {solve the given problems.}$$ Show that \(-1-j\) is a solution to the equation \(x^{2}+2 x+2=0\)

solve the given problems. Refer to Example \(4 .\) In an alternating-current circuit, two impedances \(Z_{1}\) and \(Z_{2}\) have a total impedance \(Z_{T}\) of \(Z_{T}=\frac{Z_{1} Z_{2}}{Z_{1}+Z_{2}} .\) Find \(Z_{T}\) for \(Z_{1}=3.2+4.8 j \mathrm{m} \Omega\) and \(Z_{2}=4.8-6.4 j \mathrm{m} \Omega\)

Represent each complex number graphically and give the rectangular form of each. $$0.08\left(\cos 360^{\circ}+j \sin 360^{\circ}\right)$$

Express the given complex numbers in polar and rectangular forms. $$820.7 e^{-3.492 j}$$

Represent each complex number graphically and give the rectangular form of each. $$7.32 /-270^{\circ}$$
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