The Restful Maths Podcast

The square numbers are generated by multiplying integers by themselves (aka “squaring”). Given that multiplying 2 numbers with the same sign always yields a positive product, it follows that square numbers are always positive, whether they are the result of squaring a negative or positive integer, so, for example, the squares of both fifteen and negative fifteen is two hundred and twenty five.

Since even the natural numbers form an infinite set, it follows that the squares also form an infinite set.

One way to define a square number is to consider its factors, and in particular its factor pairs. Since one of the factor pairs of any square number is a repeated factor, it follows that all square numbers have an odd number of distinct (positive) factors, whereas all non square  numbers have an even number of distinct factors.

As we progress through the square numbers they grow further and further apart, alternate between odd and even numbers and are progressively separated by the odd numbers.

Perhaps the most common application of the squares we meet in high school maths is their use in Pythagoras’ Theorem which gives the relationship between the three sides of right angle triangles. We shall explore this in episode 5.

Links:

nrich activities with square numbers 

What are prime numbers, and how did they move from the backwaters of abstract mathematical analysis to being vital for the way we live in the computer age? These are a couple of the questions that I look at in this episode of Restful Maths before reading a few out...ok before reading about 500 of them out.

Transcription:

The prime numbers form an infinite subset of the natural numbers; that is to say that they are all positive integers. It is the Greek mathematician Euclid who is credited with the proof that there are infinitely many prime numbers, back in the day, over 2000 years ago. Now although there are infinitely many of them, as we go along the number line they tend to get more difficult to identify, and currently the largest known prime is only about 23 and a quarter million digits long.

 A common explanation of what a prime number is is that it can only be divided by itself or one, but this does not stand up as a proper definition in itself. A more elegant definition of what a prime is is that it is a natural number with precisely 2 distinct factors; the only divisible by one and itself idea does not make explicit the distinct nature of the factors, so when asked to list the prime numbers people armed with this concept often start with the number 1, as it cannot be denied that it is divisible by one and itself. But one is not a prime number since it only has one factor.

Throughout the history of classical and modern mathematics from Euclid in 300 BC, through the medieval Islamic mathematicians, the enlightenment and on to today, one has largely been excluded from the primes. Indeed up until the renaissance, whether one was even a number at all for the purpose of analysis was a big can of worms in the world of mathematics.

You notice I say that one has largely been excluded from the list of primes. In the 18th and 19th centuries there were some mathematicians who posited that one is prime, but it was never generally accepted.  Apart from not conforming to our modern definition of what a prime is, acceptance of the primality of one would require a lot of established maths, such as the fundamental theorem of arithmetic, to be redefined to exclude it.

So, what is the first prime number? Well it is also the only even prime number, and that number is 2. Interestingly to some classical Greek mathematicians 2 was not prime as they considered primes to be a subset of the odd naturals. Euclid considered it to be prime, so I guess the ancient Greeks weren’t all barking mad.

With our reliance on computers and the internet the primes have been flung from the backwaters of abstract mathematical analysis and philosophy to one of the cornerstones of how we live today. Computer encryption is dependent on using  prime factors pairs of very large numbers – numbers so large that even a computer is unable to work out which 2 prime numbers were used to make them.

Links:

Prime numbers - the Wikipedia take

Prime numbers and encryption (ABC Australia)

Euclid and the infinity of primes

The largest known prime

Getting it wrong

In this episode I look at the Natural Numbers; also known as the counting numbers they form the countable infinite set of whole numbers beginning with one...or sometimes zero. I briefly discuss their origins and why they are so important before I tell you what some of them...ok quite a few of them...are.

Transcript:

Hello listeners welcome to the first episode of the restful maths podcast the, podcast that aims to make maths restful not stressful. Today I thought I’d start with the natural numbers. The what now? What are natural numbers? Well the natural numbers are the numbers which occur most commonly in nature; they are also known as the counting numbers: the whole positive numbers: 1, 2, 3, 4,…in other words the first numbers you learned about.

What’s so special about the natural numbers I hear you ask. Well they are the set of numbers which forms the foundation of modern civilization. At some points in our diverse histories as human kind, all civilizations needed the ability to count things in order to develop; so the concept of counting things and the creation of formal counting systems which evolved into what we now call the natural numbers is an important step in human evolution. There is modern day evidence that in less developed civilizations, the restricted need to count meant the existence of a very limited formal counting system; this is often known as the “one, two, many” phenomenon. I’ll post a link to an interesting article on this in the show notes, but even where there is evidence of the existence of one, two, many, the concept of tallying, one of the most basic ways to count things still exists. And this is important because the to us simple ability to tally – the concept of oneness, twoness, threeness and so on - is an intellectual and philosophical keystone in the development of humanity.

Now I said at the beginning that the natural numbers are also known as the counting numbers, so traditionally they started with one. After all zero tends not to exist in nature and it is challenging to say the least to count something which doesn’t exist. In fact the whole concept of the number zero is a human invention which reflected different developmental needs around the world, but it is generally agreed that the concept of zero as we understand and use it today has its origins in India. I’ll put a couple of links to books about zero which are worth reading in the show notes.

So traditionally zero is not included in the natural numbers, but in some fields of maths such as set theory, it makes sense to include zero in the counting system and so it is sometimes added. But for the purpose of this podcast I’m going to stick to the counting definition and start with the number...

Links:

Is one, two, many a myth? 

The Nothing That Is (Link to Amazon)

Zero: the Biography of a Dangerous Idea (Link to Amazon)

Title music: Music box melody by xanadou at Freesound

In this trailer I describe what the  point of the Restful Maths podcast is and what it will look like, to begin with at least. To many people mathematics is stressful. The idea of the Restful Maths podcast is that mathematics, and in particular number, can be an aid to a restful end of a stressful day.

Introductory music: Guitarholica by Pax11 at www.freeesound.org.

Birdsong: Summer by dobroide at www.freeesound.org.