Problem 47 - Algebra 2 and Trigonometry

Question

In \(44-51,\) locate the point that corresponds to each of the given complex numbers. $$ -4-2 i $$

Step-by-Step Solution

Answer

The point for -4-2i is (-4, -2) on the complex plane.

Step 1: Identify components of the complex number

The complex number provided is -4-2i. In a complex number, the part before the 'i' is the real part and the part with 'i' is the imaginary part. Here, the real part is -4 and the imaginary part is -2.

Step 2: Understand the complex plane

The complex plane is a two-dimensional plane where the horizontal axis represents the real part of the complex number, and the vertical axis represents the imaginary part. Points on this plane are given by the coordinates (x, y), where x is the real part and y is the imaginary part.

Step 3: Plot the complex number

Locate the point for the complex number -4-2i on the complex plane. The real part is -4, moving 4 units to the left from the origin on the horizontal axis. The imaginary part is -2, moving 2 units down from the origin on the vertical axis. Thus, the point is (-4, -2) on the complex plane.

Key Concepts

Complex Plane

Imagine the complex plane as a big graph paper where each point corresponds to a complex number. It has two axes, much like the one in regular algebraic graphs:

• Horizontal Axis: This represents the real numbers. On the complex plane, this is where you measure the real part of complex numbers.
• Vertical Axis: This represents the imaginary numbers. This is where the imaginary component leads you on the plane.

To determine the position of a complex number on this plane, consider it as a point offered by a pair of values. For instance, for the complex number you'd find the "real" value on the horizontal axis and the "imaginary" on the vertical axis, allowing you to plot its exact location neatly. It transforms what might appear as abstract numbers into a visual representation that can greatly aid in understanding complex mathematics.

Real Part

The real part of a complex number is akin to the familiar numbers we already know and work with every day. For any complex number written in the standard form, the real part would be the number that doesn’t rely on the imaginary unit 'i'. For example, in the complex number -4 -2i, the real part is -4.

• Real Numbers: These are all the positive and negative whole numbers, fractions, and decimals we are used to.
• Solid Foundation: The real part provides a grounding or starting point for complex numbers, connecting the complex to the practical world of real numbers.

Including the real part in a complex number lets us maintain a link to the real-world calculations we often perform, making the incorporation of imaginary numbers less daunting.

Imaginary Part

At first glance, the term 'imaginary' might seem to suggest something fictional or not real, yet it widens the scope of what can be calculated or expressed in mathematics. The imaginary part of a complex number appears with the imaginary unit 'i,' which provides depth and dimension to these numbers. From our example, -4 - 2i, the imaginary part is -2.

• Imaginary Numbers: Imaginary numbers are multiples of 'i,' where 'i' is the square root of -1, and help in solving equations that cannot be solved using just real numbers.
• Coordinate Placement: On the complex plane, the imaginary part determines how far up or down a point is from the horizontal axis.

By pairing real and imaginary numbers, you can represent an even broader range of motion, heat transfer equations, and oscillations, making this concept vital in fields such as engineering and physics.