Beginning Mathematics/Number Theory

Following on from the introduction of sets and their manipulation, this section introduces formal number theory.

Numbers

The use of numbers was probably the first application of mathematical theory. As we have seen, many many objects can be defined using mathematical logic, but the study of quantities of these objects is based in number theory. As a human, it is natural to have an implicit understanding of quantities, but defining numbers formally will require definitions from the previous section. Again, let us start with nothing; in particular the number zero (0), representing the quantity of no objects. Formally it can be stated that a set with zero objects is empty. If an object, any at all, is introduced into this set, the set is no longer empty, and the quantity of objects within it is no longer zero. As long as the introduced object is solitary, we can state in English that the set now contains one object. Progressing onwards from this results in the concept of numbers that we know and love.

• Definition: A number is representative of the quantity of arbitrary objects in an arbitrary set.
• Definition: $n(A)$ is an operation performed on a set A that gives the quantity of objects contained within.

These definitions lead to the progression below

$0=n(\{\}),\;1=n(\{a\}),\;2=n(\{a,b\}),...$

giving us the set of symbols {0,1,2,3,4,5,6,7,8,9} to represent quantities.

When progressing beyond the value of 9 we could try to represent each value with a different symbol to infinity, but this would quickly become tedious, as the symbols would either have to differ by infinitesimally small amounts or grow to very large sizes to represent the changes in value, and in practice they would have to grow anyway due to limitations in resolution and the particular nature of the universe. So we do not do this; here the concept of a positional numeral system needs introduction.

Within a positional numeral system, the actual value of a numeral depends (as the name implies) on its position within the collection of numerals that make up the number. Following on from the progression before, the sequence continues with the number 10. Here the numeral 1 holds the value of $n(A)$ where A is a set holding one more object than the number represented by the symbol 9. The numeral 0 is a placeholder, important in that it increases the value of the numeral to the left by its presence. A numeral holding the position of the 1 in this, the number ten, will have its value similarly scaled. For example, jumping forward in the progression, the number 83. The numeral 3 holds its base value, and the numeral 8 has its base value scaled by the value the 1 represents in the number 10. In this, the commonly used Hindu-Arabic system, the value of a numeral is ten times the value it would hold were it placed one position to the right, down to the base value of the numeral at the rightmost position. The result of this is a powerful system that allows the concept of the value of large and diverse numbers to be communicated in concise written form.