The sixth number system

Question

In high school, I learned there are 5 number systems, namely:

• Natural numbers ($\Bbb N$)

• Integers ($\Bbb Z$)

• Rational numbers ($\Bbb Q$)

• Real numbers ($\Bbb R$)

• Complex numbers ($\Bbb C$)

I remember one time our teacher told us that there is a sixth number system denoted by $\Bbb H$ and that is used by video games developers. Is it right that there is such number system?

If so, are there other number systems? And why haven't we studied them at university?

Answer 1

The five systems you have listed are the most commonly seen systems of numbers, but there are indeed others. The $\Bbb{H}$ your teacher mentioned is most likely the quaternions, which is a way of extending the complex numbers: rather than elements that look like $a + bi$, you have things of the form $a + bi + cj + dk$, where $i^2 = j^2 = k^2 = -1$, and there are specific rules for multiplying any two of these "imaginary units." (You do lose commutativity of multiplication in the process of creating $\Bbb H$, though: that is, $ab$ might not be the same as $ba$ for $a,b\in\Bbb H$). Another example of a number system you might not have seen before is the $p$-adic numbers ($\Bbb{Q}_p$), which is important in number theory. These numbers are created by completing the rational numbers ($\Bbb Q$) with respect to a different absolute value that has to do with how many times a prime $p$ divides the numerator and denominator of your rational number. Many of these number systems are studied at university, but you have to take the right courses! Number theory will introduce you to $\Bbb H$ and $\Bbb{Q}_p$, and abstract algebra will also give you some insight into $\Bbb{H}$. Other systems of numbers add infinities and infinitesimals to the real numbers $\Bbb R$, and those are encountered in non-standard analysis.

Answer 2

Your teacher was probably talking about Quaternions, but there are many, many more number systems. They are studied in Modern Algebra, or more specifically Ring Theory.

Answer 3

The quaternions are a special collection of numbers that generate a number system called the quaternion algebra and denoted $\mathbb{H}$.

If you take a course on abstract algebra, then you will surely encounter the quaternions. My first encounter with them was in the study of groups, although you may also see them in a physics course where they are denoted $\hat{i},\hat{j},\hat{k}$ and not typically called by name.

There are many more number systems than those one encounters in high school or beginning university mathematics. For example, there are finite number systems referred to as modular arithmetic, many examples between $\mathbb{Q}$ and $\mathbb{C}$ called number fields, and for each prime number $p$ there are the $p$-adic numbers $\mathbb{Q}_p$. If one broadens their definition of number, then there is a vast supply of examples called fields, rings, and groups (descending in order of abstraction.)

Answer 4

Undoubtedly your teacher was referring to the quaternions. The use of $\mathbb{H}$ to represent them is due to Rowan Hamilton being the original discoverer. The following anecdote from Wikipedia's article on the history of quaternions is associated with Hamilton's discovery:

On October 16, 1843, Hamilton and his wife took a walk along the Royal Canal in Dublin. While they walked across Brougham Bridge (now Broom Bridge), a solution suddenly occurred to him. While he could not "multiply triples", he saw a way to do so for quadruples. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge:

        i^2 = j^2 = k^2 = ijk = -1

Answer 5

There are also quaternions and octonions. See the book by Conway and Smith on this subject. The former corresponds to four-dimensional numbers, the latter to eight-dimensional numbers.