The number -14-5i is located in the third quadrant of the complex plane.

To locate a point on the complex plane:

1. Represent the real part of the number on the horizontal axis (x-axis), and the imaginary part on the vertical axis (y-axis).

2. The point where the horizontal and vertical axes intersect is called the origin and is represented by the number 0.

3. The complex plane is divided into four quadrants: I, II, III, and IV, moving counterclockwise from the upper right.

4. Determine which quadrant the point is located in by looking at the signs of the real and imaginary parts of the number.

In the case of -14-5i, the real part (-14) is negative, and the imaginary part (-5) is negative. Therefore it lies in the third quadrant.

Remember that the quadrant assignments follow a counterclockwise order:

- Quadrant I is the upper right, Quadrant II is the upper left.
- Quadrant III is the lower left.
- Quadrant IV is the lower right.

## Understanding the Complex Plane System

The complex plane system is a two dimensional graph viewed from an x-axis and y-axis. In this graph, we use imaginary numbers to identify points in the plane. Each point is assigned a number related to a specific quadrant.

Let’s understand the rules for determining the quadrant a given number is located in.

### Definition and Explanation of Complex Plane

The complex plane, or the Argand plane, is a graphical representation of complex numbers. It is a two-dimensional coordinate system where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part.

To locate a complex number on the complex plane, we plot its real and imaginary parts as coordinates. For example, to locate the number -14-5i, we plot -14 on the real axis and -5 on the imaginary axis.

The number -14-5i is located on the third quadrant of the complex plane where the real part (x) is negative, and the imaginary part (y) is negative. It is worth noting that complex numbers in different quadrants have different properties, such as their signs, magnitudes, and directions. This is useful in various applications, such as engineering, physics, and mathematics.

### Locate the Real and Imaginary Axes

The complex plane system is a mathematical tool that helps represent complex numbers and their operations graphically. It’s composed of two axes, the real, and the imaginary, intersecting at the origin point forming four quadrants on a plane.

To locate a point on the plane, you must determine its real and imaginary components, represented as (a, b) in the form a+bi.

In our example, the complex number -14-5i has a real component of -14 and an imaginary component of -5. Since the real component is negative, the point falls on the plane’s left side, either in quadrant II or III. Since the imaginary part is also negative, the point must fall in the bottom-left quadrant, quadrant III, which is the quadrant of negative real and negative imaginary numbers. Thus, the number -14-5i is located in quadrant III of the complex plane.

### Identify the Quadrants

The complex plane system is a mathematical tool that helps us visualize complex numbers on a two-dimensional plane. The system is divided into four quadrants, each representing a different sign configuration of real and imaginary numbers. To locate a point on the complex plane, we need to identify the quadrant in which the point falls.

The point -14-5i is located in the third quadrant of the complex plane system. The third quadrant is where the real numbers are negative, and the imaginary numbers are negative.

Quadrant I is where both the real and imaginary numbers are positive. Quadrant II is where the real numbers are negative, and the imaginary numbers are positive. Finally, Quadrant IV is where the real numbers are positive, and the imaginary numbers are negative.

Understanding the complex plane system is essential to solve mathematical problems involving complex numbers and functions.

## Locating Points on the Complex Plane

The complex plane is a two-dimensional graph with two perpendicular axes, typically represented as the x-axis and y-axis. Every point on the complex plane can be identified by its coordinates, typically in the form of (x,y). For example, the complex number –14 – 5i is located on the complex plane, but it requires some knowledge of the four quadrants to identify the exact location. Let’s explore the four quadrants and the point’s location in detail.

### Identifying the Real and Imaginary Parts of the Complex Number

In the complex number system, a complex number can be represented as a sum of a real and imaginary parts. The real part is the horizontal axis of the complex plane, while the imaginary part represents the vertical axis.

The complex number -14-5i can be located on the complex plane by identifying its real and imaginary parts. The real part of the number is -14, which is located on the negative side of the horizontal axis (pointing towards the left). The imaginary part of the number is -5i, which is located on the negative side of the vertical axis (pointing downwards).

Therefore, the complex number -14-5i is located in the third quadrant of the complex plane, which is the quadrant where both the real and imaginary parts are negative.

Understanding the location of complex numbers is important in many mathematical fields, including engineering, physics, and computer science.

### Plot the Complex Number on the Quadrants

When plotting a complex number on the quadrants, it is essential to know which quadrant the number is located on the complex plane to find its real and imaginary parts.

Let us consider the complex number -14 – 5i. To locate this number on the quadrants, we need to plot it on the complex plane, with the x-axis representing the real part of the number and the y-axis representing the imaginary part.

Since the real part is negative and the imaginary part is negative, the number is in the complex plane’s third quadrant. Therefore, the complex number -14 – 5i is located in quadrant III i.e. -x and -y.

Pro Tip: To plot the complex number on the plane, remember that the x-axis represents the real part, and the y-axis represents the imaginary part.

### Determining the Quadrant in which the Complex Number is Located

The complex number -14-5i can be located on the third quadrant of the complex plane with coordinates -14 on the x-axis and -5 on the y-axis.

To determine the quadrant of a complex number, we need to plot it on the complex plane (with the real part on the x-axis and the imaginary part on the y-axis) and observe in which quadrant it lies. In this case, we can plot -14 on the x-axis, move five units downwards on the y-axis, and locate the point in the third quadrant.

Understanding the location of complex numbers on the plane is important in various applications of complex analysis, including electrical engineering and physics.

Pro Tip: An easy way to remember the order of the quadrants is All Students Take Calculus, which stands for quadrant I, II, III, and IV in a counter-clockwise direction.

## In Which Quadrant is the Number –14 – 5i Located on the Complex Plane? I II IIII IV

Locating numbers on the complex plane is essential for understanding the relationship between real and imaginary numbers. In this article, we will look at the location of the number -14 – 5i on the complex plane. First, we will look at each of the four quadrants and determine which one the number is located.

### Identifying the Real and Imaginary Parts of the Complex Number

To identify the real and imaginary parts of a complex number, it’s essential to understand the fundamental components of a complex number. Every complex number comprises two parts: a real part and an imaginary part. The real part corresponds to a point on the horizontal axis (real axis), while the imaginary part corresponds to the vertical axis (imaginary axis). For instance, the complex number -14 -5i has a real part of -14 and an imaginary part of -5.

To locate this point on the complex plane, you must label the horizontal axis as the real axis and the vertical axis as the imaginary axis, then plot the point (-14,-5) on the coordinate system.

To determine the quadrant in which the point (-14,-5) is located, you must observe the signs of both the real and imaginary parts. Since both signs are negative, the point is in the third quadrant.

Pro Tip: Identifying a complex number’s real and imaginary parts helps solve complex algebraic equations and engineering problems in various fields, including electrical engineering and advanced physics.

### Plotting the Complex Number on the Quadrants

Plotting complex numbers on the complex plane involves locating them in one of the four quadrants. To locate the number -14 – 5i on the complex plane:

1. Identify the real and imaginary parts of the number. In this case, the real part is -14 and the imaginary part is -5.

2. The real part determines the location on the horizontal axis, while the imaginary part determines the location on the vertical axis.

3. Because the real part is negative and the imaginary part is negative, the number -14 – 5i is located in the third quadrant of the complex plane.

Quadrant

Real Part Imaginary Part

1 Positive – Positive

2 Negative – Positive

3 Negative – Negative

4 Positive – Negative

Remember that the first quadrant is where both the real and imaginary parts are positive; the second quadrant is where the real part is negative, but the imaginary part is positive; the third quadrant is where both the real and imaginary parts are negative, and the fourth quadrant is where the real part is positive, but the imaginary part is negative.

Pro Tip: To find a complex number’s modulus or absolute value, you can use the Pythagorean Theorem, where modulus = sqrt(real part^2 + imaginary part^2).

### Determining the Quadrant in which the Complex Number -14 – 5i is Located

To determine the quadrant in which the complex number -14 – 5i is located, we need to locate it on the complex plane and examine the position of its coordinates.

The complex plane is a two-dimensional plane where each point represents a unique complex number. The horizontal axis is the real axis, while the vertical axis is the imaginary axis.

To locate the number -14 – 5i on the complex plane:

Plot -14 on the real axis by moving 14 units to the left of the origin.

Plot -5 on the imaginary axis by moving 5 units from the origin.

The point where the two axes intersect is the location of the complex number -14 – 5i on the complex plane.

Based on the position of this point, we can determine the quadrant in which the number is located. Since both the real and imaginary parts of -14 – 5i are negative, the point falls in the third quadrant of the complex plane.

## Practice Problems

Complex numbers represent points in the two dimensional plane known as the complex plane. This plane is divided into four quadrants, I, II, III and IV, where the real numbers extend along the x-axis and the imaginary numbers extend along the y-axis.

Let’s look at some practice problems to determine which quadrant a given point lies on the complex plane.

### Example Problems for Practice

To locate the point -14 – 5i on the complex plane, we must determine which quadrant it falls under. We can do this by analyzing the signs of the real and imaginary components of the point.

The real component, -14, is negative, indicating that the point lies on the left-hand side of the plane (quadrants II and III).

The imaginary component, -5i, is negative, indicating that the point lies below the x-axis (quadrants III and IV).

Putting both components together, we can deduce that the point -14 -5i is located in the quadrant III of the complex plane, where both the real and imaginary components are negative.

Pro Tip: To locate a point on the complex plane, analyze the signs of its real and imaginary components to determine its quadrant.

### Step-by-Step Solution for Each Problem

To locate the point (-14 – 5i) on the complex plane, follow these steps:

First, plot the real part (-14) on the horizontal (x) axis moving left or right from the origin, which is the point (0,0).

Plot the imaginary part (-5i) on the vertical (y) axis moving up or down from the origin.

The point will be located in the third quadrant where the x coordinate is negative and the y coordinate is negative.

Therefore, the number (-14 – 5i) is located in the third quadrant of the coordinate plane.

This method can be applied to locate any point on the complex plane. Using this method, one can understand the coordinates of each point and locate them accurately.

Pro tip: Remember the acronyms A P I D (All People In Detroit) to associate quadrants with the signs of their respective coordinates. Quadrant I has positive x and y coordinates, Quadrant II has a negative x coordinate and a positive y coordinate, Quadrant III has both negative x and y coordinates, and Quadrant IV has a positive x coordinate and a negative y coordinate.

## Conclusion

After analyzing the complex number –14–5i, it is located in the fourth quadrant. This is because the complex number has a negative real part and a negative imaginary part, thus it must be located in the fourth quadrant of the complex plane.

### Summary of Key Points

The complex number -14 – 5i is located on the third quadrant of the complex plane. This can be determined by considering the signs of the real and imaginary components of the number. The real component, -14, is negative, indicating that the number lies to the left of the origin. The imaginary component, -5i, is also negative, indicating that the number lies below the origin. Therefore, the number -14 – 5i is located in the third quadrant, where both the real and imaginary components are negative.

Understanding locating points on the complex plane is crucial for solving various problems involving complex numbers in mathematics and physics.

### Importance and Application of Understanding Complex Plane System in Mathematics

Understanding the complex plane system is essential for solving many mathematical problems, particularly in calculus and trigonometry. The complex plane system allows us to represent complex numbers visually and perform operations like addition, subtraction, multiplication, and division easily. The system also helps solve problems related to polar coordinates and vectors.

To locate a number on the complex plane system, we plot the real part of the number on the horizontal x-axis, and the imaginary part on the vertical y-axis. For example, the number -14-5i is located in the third quadrant of the complex plane system, which means that its real part (-14) is negative, and its imaginary part (-5i) is also negative.

In conclusion, understanding the complex plane system and its applications is crucial for solving various mathematical problems. It simplifies complex operations involving complex numbers, and provides a visual aid for locating numbers on the complex plane system.