Problem 51 - Algebra 2 and Trigonometry

Question

In \(44-51,\) locate the point that corresponds to each of the given complex numbers. $$ 0+0 i $$

Step-by-Step Solution

Answer

The complex number \(0 + 0i\) is located at the origin \((0, 0)\) on the complex plane.

Step 1: Understanding the Complex Number

The given complex number is \(0 + 0i\). This is the simplest form of a complex number, where both the real part and the imaginary part are zero.

Step 2: Identifying the Real Part

The real part of the complex number \(0 + 0i\) is \(0\). This is the horizontal component when plotting on the complex plane.

Step 3: Identifying the Imaginary Part

The imaginary part of the complex number \(0 + 0i\) is \(0\). This is the vertical component when plotting on the complex plane.

Step 4: Plotting on the Complex Plane

On the complex plane, the x-axis represents the real part, and the y-axis represents the imaginary part. With both parts being zero, the point is located at the origin, \((0, 0)\).

Key Concepts

Complex Plane

The complex plane is a two-dimensional space used to visually represent complex numbers. In this plane, each complex number is represented as a unique point.
On the complex plane:

• The horizontal axis is called the real axis.
• The vertical axis is called the imaginary axis.

Each complex number, in the form of \(a + bi\), can be plotted by finding the real part on the real axis and the imaginary part on the imaginary axis.
This approach allows us to visually interpret complex numbers, making them easier to understand and work with in mathematical problems.

Real and Imaginary Parts

Complex numbers are composed of two key components: the real part and the imaginary part. The complex number \(a + bi\) consists of:

• Real Part: \(a\), which determines the position horizontally on the complex plane.
• Imaginary Part: \(bi\), where \(b\) is the coefficient of the imaginary unit \(i\), determining the position vertically.

When arranging a complex number, it is essential to identify which part is real and which is imaginary, as it influences how the number is represented on the complex plane.
Understanding these parts helps in grasping more advanced concepts involving complex numbers, such as addition, subtraction, or even finding magnitudes and phase angles.

Origin on Complex Plane

The origin on the complex plane is the point where both the real and imaginary parts are zero. For the complex number \(0 + 0i\), both components fall at zero.
This means the number is located exactly at the origin, represented by the point \((0, 0)\) on the complex plane.

• Since there is no horizontal movement, the real part is zero.
• Similarly, with no vertical movement, the imaginary part is also zero.

The origin is a crucial reference point in the complex plane, often serving as a starting point for calculating distances to other complex numbers and for understanding transformations.
Recognizing the location of the origin helps when solving higher-level mathematical problems involving complex numbers.