Question

In the complex plane, the \(x\) -axis is referred to as the _________ axis, and the \(y\) -axis is called the _______ axis.

Short answer

In the complex plane, the x-axis is the real axis, and the y-axis is the imaginary axis.

Step-by-step solution

- Identify the Axes in the Complex Plane

The complex plane is a two-dimensional plane where each point represents a complex number. The plane consists of a horizontal axis and a vertical axis.

- Name the Horizontal Axis

In the complex plane, the horizontal axis (corresponding to real numbers) is called the 'real axis'.

- Name the Vertical Axis

In the complex plane, the vertical axis (corresponding to imaginary numbers) is called the 'imaginary axis'.

Key concepts

Real Axis

The real axis is a crucial part of the complex plane. It runs horizontally and represents the real part of a complex number. For example, in the complex number \(3 + 4i\), the \(3\) is located on the real axis. It is similar to the \(x\)-axis you are familiar with from basic coordinate systems.

Understanding the real axis:

• Every real number in a complex number is placed along this axis.
• Any point on the real axis has an imaginary part of \(0\). For example, the number \(5\) is simply represented as \(5 + 0i\).
• The real axis forms the backbone for plotting complex numbers on the complex plane.

By plotting just the real part of a complex number, you can see how it contributes to the overall position of the point in two-dimensional space. This axis is essential because it gives a visual representation of the real components.

Imaginary Axis

The imaginary axis is the vertical counterpart to the real axis in the complex plane. It represents the imaginary component of complex numbers, running up and down. For example, in the complex number \(3 + 4i\), the \(4i\) is located on the imaginary axis.

Key points to know about the imaginary axis:

• It is similar to the \(y\)-axis found in standard Cartesian coordinates.
• Points on this axis have real parts equal to \(0\). For instance, \(6i\) is represented as \(0 + 6i\) on the complex plane.
• The imaginary axis helps in distinguishing the imaginary component of complex numbers from the real component.
• When combined with the real axis, it forms a complete plane for mapping all complex numbers.

This axis allows you to properly visualize how imaginary numbers contribute to complex numbers, giving you a complete understanding of their behavior in the complex plane.

Complex Number

A complex number is a number that comprises both a real part and an imaginary part. It is usually expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Main attributes of complex numbers:

• The real part \(a\) is plotted along the real axis.
• The imaginary part \(b\) is multiplied by the imaginary unit \(i\), and plotted along the imaginary axis.
• The combination of both parts creates a point in the two-dimensional complex plane.

For example, the complex number \(3 + 4i\) consists of:

• A real part \(3\) that lies on the real axis.
• An imaginary part \(4i\) that lies on the imaginary axis.

Together, they determine a point \((3, 4)\) in the complex plane. Understanding complex numbers is crucial as they extend the concept of one-dimensional real numbers to two dimensions, offering a comprehensive mathematical framework for diverse applications in engineering, physics, and beyond.