We wrote last week on *modular arithmetic*, the arithmetic of remainders. Here we will examine a few other aspects of this huge subject. Modular arithmetic was advanced by Gauss in his *Disquisitiones Arithmeticae*. In this system, number wrap around when they reach a point known as the modulus. Numbers that differ by a multiple of the modulus are called congruent. Thus 4, 11 and 18 are all congruent modulo 7.

## Posts Tagged 'Arithmetic'

### More on Moduli

Published November 6, 2017 Occasional Leave a CommentTags: Arithmetic, Number Theory

### Modular Arithmetic: from Clock Time to High Tech

Published November 2, 2017 Irish Times Leave a CommentTags: Arithmetic, Time measurement

You may never have heard of *modular arithmetic*, but you use it every day without the slightest difficulty. In this system, numbers wrap around when they reach a certain size called the modulus; it is the arithmetic of remainders [TM126 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Modular Arithmetic: from Clock Time to High Tech’

Amongst the many joys of mathematics are the beautiful patterns that appear unexpectedly from time to time. In 1951, Alfred Moessner discovered a delightful and ingenious way of generating sequences of powers of natural numbers. It is surprisingly simple and offers great opportunities for development and generalization.

It is well-known that the sum of odd numbers yields a perfect square:

1 + 3 + 5 + … + (2*n* – 1) = *n *^{2}

This is easily demonstrated in a geometric way. We start with a unit square, and repeatedly add an additional row and column on the “east” and “north” sides and a unit square at the “north-east” corner. This amounts to adding the next odd number and, at each stage, a new square is produced.

### Fractions of Fractions of Fractions

Published August 10, 2017 Occasional 1 CommentTags: Arithmetic, Number Theory, Recreational Maths

Numbers can be expressed in several different ways. We are familiar with whole numbers, fractions and decimals. But there is a wide range of other forms, and we examine one of them in this article. Every rational number can be expanded as a continued fraction:

where all are integers, all positive except perhaps . If we add it to ; then the expansion is unique.

### Quadrivium: The Noble Fourfold Way

Published July 20, 2017 Irish Times Leave a CommentTags: Arithmetic, Astronomy, Geometry, History, Music

According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s *Republic*. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6^{th} century AD [see TM119 or search for “thatsmaths” at irishtimes.com].

It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.

### Patterns in Poetry, Music and Morse Code

Published June 29, 2017 Occasional Leave a CommentTags: Arithmetic, History, Recreational Maths

Suppose we have to ascent a flight of stairs and can take only one or two steps at a time. How many different patterns of ascent are there? We start with the simplest cases. With one step there is only one way; with two, there are two: take two single steps or one double step. With three steps, there are three possibilities. We can now proceed in an inductive manner.

### Brun’s Constant and the Pentium Bug

Published March 9, 2017 Occasional 1 CommentTags: Arithmetic, Euler, Number Theory

Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler’s conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges:

Obviously, this could not happen if there were only finitely many primes.