Question

Graph each complex number. $$-1-3 i$$

Short answer

The complex number \(-1 - 3i\) is graphed as the point \((-1, -3)\) on the complex plane.

Step-by-step solution

Understanding the Complex Number

A complex number in the form of \( a + bi \) consists of a real part \( a \) and an imaginary part \( bi \). In the complex number \( -1 - 3i \), the real part is \( -1 \) and the imaginary part is \( -3i \), where \( i \) is the imaginary unit.

Identify the Coordinates

To graph the complex number, translate the real part to the x-coordinate and the imaginary part to the y-coordinate of a point in the complex plane. Here, the x-coordinate is \( -1 \) and the y-coordinate is \( -3 \) (note that the coefficient of \( i \) is the imaginary part).

Plotting the Point

On the complex plane, plot the point \( (-1, -3) \) which corresponds to the complex number \( -1 - 3i \). The x-axis represents the real axis and the y-axis represents the imaginary axis.

Key concepts

Complex Plane

The complex plane is a fundamental concept when dealing with complex numbers. It's essentially a two-dimensional plane for representing these numbers visually. Think of it like a regular Cartesian coordinate system, but instead of plotting just x and y values, we’re dealing with the real and imaginary components of complex numbers.

The 'x-axis' now becomes the real axis, while the 'y-axis' stands in as the imaginary axis. So, whenever you have a complex number like \( -1 - 3i \), the real part (\( -1 \)) sits on the horizontal real axis, while the imaginary part (the coefficient of \( i \), which is \( -3 \) in this case) lines up along the vertical imaginary axis.

This system allows complex numbers to be easily plotted, and relationships between them to be visualized – such as addition or subtraction – which is done by vector summation.

Imaginary Unit

At the heart of complex numbers lies the imaginary unit, denoted by \( i \). It's defined by the property that \( i^2 = -1 \). This might seem a bit odd at first since no real number squared gives a negative result. That's exactly why we need this new kind of number!

It's important to not let the term 'imaginary' mislead you into thinking these numbers aren't 'real' or useful. They're a crucial part of mathematics and physics, helping to solve equations that would otherwise have no solution. When graphing complex numbers, we often encounter terms like \( 3i \) or \( -3i \). It’s \( i \) that allows us to plot these on the vertical axis of the complex plane, giving us a complete picture of the complex number.

Real and Imaginary Parts of Complex Numbers

Every complex number is built from two components: a real part and an imaginary part. These are usually expressed as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( b \) being a real number and \( i \) representing the square root of \( -1 \).

The real part is what you would normally think of in terms of traditional numbers. It's the portion that lies along the real axis on the complex plane. The imaginary part, on the other hand, is a multiple of \( i \), and its position is along the imaginary axis.

In our example \( -1 - 3i \), the real part \( -1 \) tells us how far to move left or right from the origin of the complex plane, while the imaginary part \( -3i \), or just \( -3 \) when plotting on the plane, tells us how far to move up or down. Together, they pinpoint the exact location of the complex number in this two-dimensional space.