I described the many types of numbers in my last article: natural, integers, rational, real, and complex.
To recap, the natural numbers N are whole positive numbers, the integers Z are whole negative and positive numbers, the rational numbers Q are fractions, the real numbers R are irrational numbers (numbers that cannot be expressed in fraction form), and the complex numbers C are numbers that cannot be expressed in fraction form. The numbers acquired from real are added to the number I which has the property that its square is -1.

Except for the complex, all of the numbers described above can be represented as points on a line. We know which one is bigger and which one is smaller because we know their order. We have the number zero in the centre of the line when working with integers and larger groupings. We also know that as you move to the left, the numbers get smaller and as you move to the right, they get bigger (as shown on the picture below). We can think of numbers as a one-dimensional entity since we only need one measure to specify each number, which is the number itself. This is also in keeping with their representation on a line, which is a one-dimensional space.

Image courtesy of mathisfun.com

When it comes to complex numbers, we’re in a slightly different situation. As we can see from the way they’re stated, we’ll need two measures — the two real numbers a and b — to express the number a+bi. When two measurements are required to specify a mathematical entity, we refer to it as 2-dimensional. More crucially, we can’t utilise the number line to depict a complex number on a picture because the line only allows us to supply one of the parameters.

Academic Master is a US based writing company that provides thousands of free essays to the students all over the World. If you want your essay written by a highly professional writers, then you are in a right place. We have hundreds of highly skilled writers working 24/7 to provide quality  essay writing services  to the students all over the World

Let’s take a closer look at the complicated numbers. We mentioned that they’re commonly written as a+bi, where a and b are real numbers. As a result, we can observe that there are two parts: a and bi. The actual part (a) is called the real part, while the imaginary part (bi) is called the imaginary part (since it is the one that involves the imaginary number i). To visualise this, we’ll need two number lines: one for the real part and another for the imagined part. We have all the data we need to depict the complex number in issue if we draw a horizontal line to map our real part and a vertical line to plot the imaginary part. The real axis is the horizontal line, and the imaginary axis is the vertical line. They meet at 0 degrees. This graph exists on a plane, which is a flat two-dimensional surface that extends in all directions indefinitely.

The graphic below depicts the real and imaginary axes, as well as an example of how to plot a complex number, in this case 3+4i, on the plane.

Image courtesy of mathisfun.com

A complex number is a point on the plane rather than a point on the line, as seen in the diagram above. As a result, we consider complex numbers to be a two-dimensional number system.

Simple movements on the plane can be simply represented using complex numbers. We would be more interested in tracing motions in our reality because we live in a three-dimensional cosmos. The obvious assumption is that we’ll need to use 3-dimensional integers to accomplish so. For years, scientists believed this until they realised it was not the case.

Let’s take a step back. Rotation is one movement that we’d like to numerically depict. If we’re on a plane, or in a flat space, we’ll require the following information to encode a rotation:

The fixed point we are rotating around is called the centre of rotation.

Whether to rotate in a clockwise or anticlockwise direction;

Angle of rotation – this is the measurement in degrees of how much we rotate about the centre in the direction we want to go.

The dot in the image below represents the centre, the arrow indicates the direction, and the angle of the turn is described beneath the image.

Collins GCSE Maths 2-tier foundation image

We may think of complex numbers in a more geometric approach since they are points on a plane. I won’t go into depth here, but the notion is that each complex number may be assigned an angle and orientation. So, if we wish to describe a rotation algebraically (i.e. using numbers), all we have to do is choose the shape’s initial state (one complex number), then the angle and direction we want to rotate in (a second complex number). In mathematics, there is a neat theorem that says that rotating the two integers equates to multiplying them. We may use the same technique with translations, enlargements, and other things.

Even if the facts above were not entirely clear, we can deduce that the complex numbers’ 2-dimensional feature allows us to express transformations in 2-dimensional space.

Let us now return to our three-dimensional reality. Let’s imagine we wish to rotate a solid item around a pen. How can we mathematically represent this movement?

To begin, Euclid’s Theorem asserts that in order to define a rotation in 3-space, we must have a fixed axis (called the Euler axis and denoted by ê below) travelling through a fixed point, as well as a measurement of the turn we will make — i.e., a specified angle of rotation (see the diagram below).

The three spatial parameters x,y,z, as well as the rotation axis and angle Wikipedia’s image.

It’s worth noting that a three-dimensional body is encoded by three measures — three integers. They can be thought of as height, width, and depth. So, with only three parameters, how are we going to convey all of the information needed in Euclid’s Theorem?

When we switched from the real numbers to the complex numbers (our “1-dimensional” number system), we added a new abstract number I and an extra line to the visual depiction. To make a three-dimensional number system, we’d need to include a second number, say j, as well as a third axis.

It turns out that this isn’t mathematically sound. However, it wasn’t until 1843 that Sir William Rowan Hamilton realised that a three-dimensional number system isn’t the best way to express movements in three dimensions. According to legend, he came to this realisation while walking with his wife to the Royal Academy in Dublin. The perfect setting occurred to him while they were crossing the Brougham Bridge, and he was so enthused that he scrawled it on the bridge so he wouldn’t forget what was needed for the definition of this new number system. In fact, a plaque honouring this discovery may still be found on the bridge today.

offline data entry services are the secret sauce many organizations have used to improve customer experiences, innovate products, and disrupt entire industries. At CloudFactory, we’ve been providing data entry services for more than a decade for more than 360 organizations. We’ve developed the people, processes, and technology it takes to scale data entry without compromising quality

The plaque on Dublin’s Brougham (Broom) Bridge.

To express movements in our three-dimensional reality, Hamilton discovered that a four-dimensional number system is required. As a result, we’ll need to add three new abstract integers in total — I j, and k — to build this. These aren’t just any symbols, either. They must meet certain criteria. The main point of adding the symbol I to the complex numbers was to create a new abstract number that squares to something negative. As a result, the condition was enforced that if I was multiplied by itself, the result would be -1. The two new symbols j and k are subjected to the same criteria. In this way, we can see how the new number system is a development of complex numbers. There are a few more conditions that we won’t go into now, but the reason for them is that they made the new number system the best framework for defining the intended movements. The quaternions were given to the new numbers.