Question

Graph each complex number. $$2+5 i$$

Short answer

The complex number \(2+5i\) is plotted at the point (2,5) on the complex plane.

Step-by-step solution

- Plot the Real Part

On a complex plane, start by plotting the real component of the number. For the complex number \(2+5i\), the real part is 2. This is plotted on the horizontal axis (the real axis) at the point (2,0).

- Plot the Imaginary Part

Next, plot the imaginary component. The imaginary part of \(2+5i\) is 5. From the point you marked on the real axis, move vertically (parallel to the imaginary axis) 5 units upwards, since the imaginary component is positive.

- Draw the Point

The point corresponding to the complex number \(2+5i\) is at the coordinates (2,5) on the complex plane. Mark this point clearly to indicate the position of the complex number.

Key concepts

Complex Plane

In mathematics, the complex plane is a fundamental concept used to visually represent complex numbers. It functions much like a two-dimensional graph where the horizontal axis, commonly referred to as the x-axis in standard graphing, is known as the 'real axis' in the complex plane. Similarly, the vertical axis, equivalent to the y-axis, is called the 'imaginary axis.'

Every point on this plane corresponds to one and only one complex number, with the location along the real axis indicating the real part of the number, and the position along the imaginary axis showing the imaginary part. Graphing complex numbers such as the one given in the example, \(2 + 5i\), allows for a visual understanding of their properties and how they relate to each other. The complex plane is integral in various fields, like engineering, physics, and of course, complex analysis in mathematics.

Real Part of Complex Number

The real part of a complex number is essentially the 'anchor' that relates the number to the concept of normal, everyday quantities. If you consider a complex number to be like a destination, the real part tells you how far to travel along the familiar road of the horizontal axis on the complex plane. For the complex number \(2 + 5i\), the number '2' is the real part. It's plotted as a simple point on the horizontal or real axis at (2, 0).

This is no different than plotting a point on a typical number line, which is why the real part is often perceived as more intuitive to grasp. It represents numbers we're used to dealing with in everyday math. When graphing, it's helpful to start with the real part to establish a foundation for where the complex number will sit on the complex plane.

Imaginary Part of Complex Number

While the real part of a complex number is anchored in the familiar, the imaginary part is a little more abstract, often causing confusion among students. However, it's not as complex as it seems! The imaginary part is represented by the number that is multiplied by the imaginary unit \(i\), which is the square root of -1. In our example \(2 + 5i\), the '5' is the imaginary part.

It indicates how far to move along the vertical or imaginary axis. A positive value, as in this example, means moving upward from the origin, while a negative value would mean moving downward. This imaginary journey you take upwards or downwards on the graph adds a second dimension to the concept of the number, providing a full picture of what a complex number represents in geometry. In our case, from point (2, 0) on the real axis, you'd go straight up five units to plot the imaginary part, and the full complex number is realized at (2, 5).