Numbers
| A History of Numbers
| Propositional Logic
| Logical Completeness
| The Liar's Paradox

Logical Consistency
| Basic Methods of Mathematical Proof
| Integers and Natural Numbers

Rational Numbers
| Irrational Numbers
| Imaginary Numbers
| The Euler Equation

Numbers have had a fascinating history over the past three thousand years. The counting systems used have been added to many times, and they have been tweaked to allow solutions to be found to problems that would otherwise be insoluble.

**Why Numbers?**

Most people would probably ask this question as 'Why Maths^{1}?', but that isn't really the same thing. Maths isn't about counting or arithmetic; maths isn't about being a human calculator. The theory of numbers, likewise, isn't about adding 2 and 2 and getting 4, it's about the logic behind the way numbers work. It's about the *reason* that numbers follow certain rules. The answer to the above question has echoes of the climber's mantra 'because it's there'; only maths, though, has the logical qualifiers needed to justify that answer.

Over the millennia of mathematical thought the number space^{2} has been extended and generalised, and the logical grounds affirmed.

**The Development of Numbers**

When people are asked what the number system looks like, they will usually 'um' and 'ah' before saying that it looks like a line with the whole numbers strung out like pearls along a necklace. This point of view goes back a very long way. The natural numbers (those integers which are positive and non-zero) were used by people in the Stone Age for counting things like property and enemies. Zero was used as a number representing the state of *not* having something^{3}. Negatives came along much later to allow debt collectors to work out how much they could extract^{4} from those suffering from an overload of zeroes.

Next, we can imagine the intervening space on the number line to be completely filled by a continuum of rational fractions. These are the ratio of two whole numbers and allow us to solve division problems.

Unfortunately this picture isn't adequate. Even considering the relatively easy concept of a number line, the image breaks down. Where do irrational numbers like *pi* and *e* fit? Neither of these can be expressed as a ratio of two integers; neither can be ascribed an exact value; and yet they pop up all the time. They must be able to fit somewhere. To allow a class of problems to be solved, mathematicians formalised the irrational numbers and interwove them between the rationals. The answer space grew.

Since then many other classes of problems have been found that have no answer using the normal numbers we are used to, so new number spaces are invented to solve these problems, such as imaginary numbers and hyperreals. The sole reason for adding these new systems is to try to achieve completeness - a system is defined as being complete when every problem can be solved using that system. Recently, Gödel's Incompleteness Theorem put a stop to that by proving that there is *no* system that can solve every problem, but for most problems, the principle holds.

**Elementary, my Dear Watson!**

People often think that this way of getting around awkward problems is cheating, and that it defines the answer in terms of itself. Mathematicians shrug these off by answering back: 'How do you define anything?' It's a simple trick but a necessary one. How can you define red without a red object? How can you define '1' without using '1 of something' to do so? It's not cheating; it's just how things are.

Behind numbers is a foundation of logic. This logic is rigorously proved and is known with absolute certainty to be true. You can't argue against it because there are no arguments. This is the basis of mathematical knowledge and it is what makes it the most secure subject possible.

Through the years, this subject has picked up and proved the rigorousness of many different methods of proof. These methods have allowed mathematicians to take shortcuts that don't invalidate the absolute nature of the proof. It is these shortcuts which have allowed mathematics to evolve so quickly after the Renaissance.

**A Timeline of Numbers**

3000 BC: The Egyptians and Babylonians were first recorded as using the natural numbers and rationals. While Egyptians use base 10, Babylonians use base 60, probably because of the large number of factors^{5} that 60 has.

500 BC: The Greek philosopher sect called the Pythagoreans supresses evidence that irrationals exist, by murdering the mathematician responsible for the proof.

300 BC: Euclid releases his proof of the existence of the irrationals.

600 AD: The Arabic number system is started. This is a place value system^{6} in base 10, including the number zero.

15th Century: Arabic number system becomes prevalent in Europe. Negative numbers are defined.

16th Century: Imaginary numbers are first used to give notional solutions.

17th Century: Complex arithmetic is defined as part of mathematics (arithmetic that uses the imaginary number *i*). This leads to the development of the Argand Diagram and much, much more.

18th Century: The Euler Equation is used to define the natural logarithm of negative numbers.

19th Century: Cantor and his diagonal argument define the real numbers absolutely and defend them rigorously. The argument also establishes countable and uncountable infinities.

20th Century: Gödel's Incompleteness Theorem (as expressed by the Liar's Paradox) rules out total completeness of mathematics. Consistency is not affected.