NTIC The Group of Integers Modulo n

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Chapter 8 The Group of Integers Modulo $n$

This chapter does not do any number theory, per se. Yet it is at the heart of the text. We introduce two powerful methods to deal with integers modulo $n$ – visualizing them graphically, and the language of group theory.

There is no prerequisite in either case; do not feel worried if you have not encountered algebraic structures like groups before. We will only take and introduce what we need, and refer back to fundamental properties often.

Summary: The Group of Integers Modulo $n$

In this chapter, it is high time to introduce a few algebraic innovations that allow a unified presentation of our ideas about modular arithmetic.

Most importantly, we officially define Integers Modulo $n$ and reconfigure what an inverse is in Fact 8.1.5. We not only make tables of operations, but in Subsection 8.1.2 we start visualizing them!
We will see later that the visualization of powers in Figure 8.2.1 is extremely powerful.
In the final section, we build our way up to the definition of a group in Definition 8.3.3, so that in the future we can use the important ideas of the Order of an element of a group and Lagrange’s Theorem on Group Order.

The Exercises give a chance to try some algebraic theory we otherwise avoid in this course.

Number Theory: In Context and Interactive

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Chapter 8 The Group of Integers Modulo \(n\)

Summary: The Group of Integers Modulo \(n\)