Foundations of Pure Mathematics

Syllabus

This is an introductory undergraduate course aimed at introducing the student to fundamental mathematical concepts such as sets, number systems and proofs.

The following topics will be covered: Sets. Integers, rational numbers, real numbers. Decimal expansions for rationals and reals. Inequalities, complex numbers. Induction; examples and applications.

The practice and language of mathematics

Throughout pre-university level education, mathematics focuses on arithmetic and on manual computation. This leads to the common misconception that mathematics is fundamentally about numbers and calculation. In reality, mathematics is a much vaster field which explores many concepts such as uncertainty, change, space and information, and whose boundaries blur with other subject areas such as philosophy, physics, and economics.

Mathematics is not characterised by the phenomenon (e.g. the stock market) or structure (e.g. a number system) we choose to analyse, but rather by the method with which we analyse them. Central to this method is the concept of a proof.

Proofs

In mathematics a proof is a logical argument used to demonstrate to the reader the validity of a given statement. For example, suppose I wanted to prove to you that there is an infinite number of primes. One approach would be to create a long list of primes: 2, 3, 5, 7, 11, 13, ... and show you that I can keep going. This would not be considered a mathematical proof. Perhaps I've persuaded you that there are a lot of primes, maybe even an infinite number of them, but it's not certain - after a few hours of listing prime numbers, I might run out! That would serve me right for making long lists of prime numbers.

The other approach would be to come up with a cunning argument that would eliminate any possibility that there were not an infinite number of primes - such as this one, invented by Euclid:

Suppose there are only 100 prime numbers, which we'll list as $p_{1}$, $p_{2}$, $p_{3}$, $p_{4}$, ..., $p_{100}$ to indicate the first, the second, the third, ..., up to the 100th prime number. Now consider the number $N=p_{1}\times p_{2}\times \cdots \times p_{100}+1$. We cannot divide $N$ by $p_{1}$, because then we'd have a remainder of 1. The same applies to $p_{2}$ and $p_{3}$ and so on. Therefore, we have two possibilities:

• $N$ is a prime number which is not in our list (since it's bigger than all the primes in our list).
• $N$ is not a prime number, but in this case it must be divisible by a prime number. $N$ isn't divisible by any of the prime numbers in our list, therefore there must be a prime number which isn't in our list.

In either case, we have found a prime number which is not in our list of 100 prime numbers. This implies that there are at least 101 prime numbers.

Now suppose there are only 101 prime numbers... but then we can repeat the same argument with a list of 101 prime numbers! But suppose there are only 102 prime numbers? And so on.

Hence we conclude there is an infinite number of primes.

Don't worry if at this stage you don't understand every step in the proof - the important thing is to understand that mathematics is not about being 'fairly sure' that something is true (by gathering evidence, like we do in other subjects), it's about coming up with an argument which shows that it must be true. That is the essence of a mathematical proof.

Sets

What is a set?

The concept of a set is central to university-level mathematics. Intuitively sets can be thought of as 'collections' of mathematical objects, which we call elements of the set. For example, the set

$X=\{1,2,3\}$

contains the elements 1, 2 and 3. To say that 1 is an element of X (i.e. that X contains 1) we write

$1\in X$

Similarly, we can write

$1\not \in \{4,5\}$

to express that 1 is not an element of the set $\{4,5\}$.

Often it is not possible to write every element of a set - for example, the set may be infinite or even just very large. In this case we can write

$\{1,2,3,...\}$

to describe the set containing all whole numbers from 1 onwards (this is an infinite set). Alternatively, we can write

$\{1,2,3,...,n\}$

to describe the set of all whole numbers between 1 and $n$ inclusive. At other times, it may be desirable to describe a set by the properties of its elements.

$\{x:10\leq x\leq 100\}$

denotes the set containing the numbers 10 to 100. We can read this as 'the set of $x$ such that $x$ is between 10 and 100'.

Finally, it's important to realise that a set need not only contain numbers, as you might have assumed from reading the above section. A set can contain points on a plane, or geometrical shapes, or people or animals or vegetables. For example we can imagine the following sets:

$\{{\mbox{points on the plane with a distance less than 1 from the origin}}\}$

$\{{\mbox{people with brown eyes}}\}$

In fact sets can even contain other sets:

$Y=\{\{1\},\{1,2\},\{1,2,3\}\}$

Note that in this last case it is not correct to write $1\in Y$, since in fact $Y$ contains a set containing 1. Therefore instead we should write $\{1\}\in Y$.

Subsets

We say that a set $A$ is a subset of another set $B$ if every element of $A$ is also an element of $B$. For example, the set of people with brown eyes is a subset of the set of all people. Also, the set $\{1,2\}$ is a subset of the set $\{1,2,3,...\}$. To express this in mathematical notation we write

$\{1,2\}\subset \{1,2,3,...\}$

Note that the subset does not need to be a finite set - it is correct to write

$\{2,4,6,...\}\subset \{1,2,3,...\}$

This is because every even number is a whole number.

Number systems

It's a common misconception that mathematics is only about numbers - it would be more accurate to say that mathematics is a method for analysing well-defined generalized structures. However, it's undeniable that 99% of the time these structures are of a numerical nature, so it makes sense to start studying mathematics by understanding the distinction between different types of numbers.

Natural numbers

Positive whole numbers (excluding zero) are known as natural numbers. The set of natural numbers is therefore denoted by the letter N.

$\mathbb {N} =\{1,2,3,4,...\}$

Integers

Integer is the correct mathematical term for 'whole number'. This includes all natural numbers, all negative integers and zero. We use the letter Z to denote the set of integers.

$\mathbb {Z} =\{...,-3,-2,-1,0,1,2,3,...\}$

Rational numbers

The integers are closed under the operations of addition, subtraction and multiplication. By this we mean that if $a$ and $b$ are both integers, then so is $a+b$, $a-b$ and $ab$ (mathematicians tend to leave out the $\times$ sign when writing the product of two numbers, so $a\times b$ becomes $ab$).

However, if we divide one integer by another, we are not guaranteed that the result is also an integer. For example, dividing 6 by 10 we get the fraction ${\frac {3}{5}}$. The fractions are known rational numbers and the set of rational numbers is denoted by the letter Q. Using set notation, we can write

$\mathbb {Q} =\left\{{\frac {a}{b}}:a\in \mathbb {Z} {\mbox{ and }}b\in \mathbb {Z} \right\}$

Note that a rational number $q\in \mathbb {Q}$ can have more than one representation as a fraction - for example ${\frac {4}{7}}$ and ${\frac {8}{14}}$ are the same number.

Real numbers

It is possible to construct numbers which are not rational numbers. For example, ${\sqrt {2}}\not \in \mathbb {Q}$. By this, we mean that it is impossible to represent ${\sqrt {2}}$ as a fraction - there does not exist a pair of integers $a,b$ such that ${\sqrt {2}}=a/b$. These numbers (in addition to rational numbers) are known as real numbers.

To come up with a rigorous description of the real numbers requires some higher level maths (in particular, the real numbers can be described as the completion of the rational numbers), however for now we will content ourselves to say that the real numbers are the numbers you can represent with a finite or infinite decimal representations. Some examples of real numbers are

• $1$
• $0$
• $0.25$
• $3.3333333...$
• ${\sqrt {3}}$
• $\pi$

and we denote the real numbers with the symbol $\mathbb {R}$

Proof that the square root of two is irrational

Assume that the contrary is true, that is, that ${\sqrt {2}}$ can be expressed as a rational number. If so, then $\exists p,q\in \mathbb {N}$ such that $q$ and $p$ have no common factors and ${\frac {p}{q}}\times {\frac {p}{q}}=2$. Then $p^{2}=2q^{2}$, so $p^{2}$ is even, so $p$ must be even, say $p=2r$ where $r\in \mathbb {N}$. Substituting $2r$ in the previous equation, we get $4r^{2}=2q^{2}$, or $2r^{2}=q^{2}$. Thus $q$ must be even. But if $p$ and $q$ are both even, then they have 2 as a common factor, which contradicts our assumption. Thus the assumption must be incorrect, and ${\sqrt {2}}$ cannot be expressed as a rational number.