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Eryk Longin Hinc: Number systems beyond the reals

Bachelor thesis presentation

Time: Wed 2022-06-08 10.30 - 11.30

Location: Kräftriket, house 6, room 306

Respondent: Eryk Longin Hinc

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From childhood, people are subconsciously exposed to number systems; from counting the number of objects of something to understanding the concept of having parts of a whole or owing someone something. However, while this natural exposure to number systems helps people comprehend number systems up to the reals rather intuitively, number systems beyond the reals are more troublesome for our intuition. This paper explores complex numbers and quaternions in order to further develop understanding of how one can operate with them as well as to create some sort of an intuition for these numbers. In this paper, it is shown that complex numbers can be expressed and represented in different forms, most of which can be compared to geometrical objects; for instance, complex numbers can be represented as pairs of real numbers as well as rotation matrices. Similarly, albeit different in certain aspects, quaternions can be compared to complex numbers to find that, like the complex numbers, they can be represented as tuples of real numbers and through rotation matrices. However, they differ in that complex numbers can be represented as 2-tuples of real numbers that can describe rotations in \(\mathbb{R}^2\) , while quaternions can be represented as 4-tuples of real numbers that can describe rotations in \(\mathbb{R}^3\) . Subsequently, in this text, the history of complex numbers and quaternions is presented as it is discussed and explained in other sources, this in order to facilitate the understanding of how these number systems originated and thus strengthening the intuition. Finally, the text touches on an approach to teaching number systems; the approach in question mostly relies on teaching complex numbers and quaternions based on the pupils’ prior knowledge, this can be achieved by for instance comparing complex numbers to by the pupils’ previously known strategies for operating on expressions with unknown variables. One can also use the similarities that the complex numbers and the quaternions share in order to facilitate the teaching of quaternions as a mathematical object