NUMBER SYSTEMS AND ANALYSIS

Natural Numbers

The natural numbers

1, 2, 3, ...

are used for counting, and for labeling the terms of an infinite sequence. They obey the following laws.

Laws of Algebra

Associative law of addition:: (a + b) + c = a + (b + c).
Commutative law of addition:: a + b = b + a.
Cancellation law of addition:: If a + c = b + c, then a = b.

Distributive law with multiplication on the left:: a.(b + c) = a.b + a.c.
Distributive law with multiplication on the right:: (a + b).c = a.c + b.c.

Associative law of multiplication:: (a.b).c = a.(b.c).
Commutative law of multiplication:: a.b = b.a.
Cancellation law of multiplication:: If a.c = b.c, then a = b.

Laws of Order

Trichotomy law:: For two given natural numbers, a and b, exactly one of the following relations is true:
a < b, a = b, a > b.
Transitive law:: If a < b and b < c, then a < c.
Order and addition:: If a < b, then a + c < b + c.
Order and multiplication:: If a < b, then a.c < b.c.

Other Number Systems

It is not always possible to subtract one natural number from another. For example, there is no natural number n such that 8 + n = 5. In other words we cannot subtract 8 from 5. Therefore the natural numbers are not a group under addition.

In order to get a group under addition we introduce the number zero and negative whole numbers to make the system of integers:

... -3, -2, -1, 0, 1, 2, 3, ...

Also, it is not always possible to divide one natural number by another. For example, there is no natural number n such that 5 × n = 8. In other words we cannot divide 8 by 5. Therefore the natural numbers are not a group under multiplication.

In order to get a group under multiplication we introduce the fractions to make the system of positive rational numbers:

   1/1, 2/1, 3/1  ...
   
   1/2, 2/2, 3/2  ...

   1/3, 2/3, 3/3  ...

   ...  ...  ...  ...

In order to get a number system that is a group under addition and also a group under multiplication, we introduce the negative fractions to make the rational numbers:

   ... -2/1, -1/1, 0/1, 1/1, 2/1, ...

   ... -2/2, -1/2, 0/2, 1/2, 2/2, ...

   ... -2/3, -1/3, 0/3, 1/3, 2/3, ...

   ...  ...   ... ... ...  ...  ...

The system of rational numbers is sufficient for all practical arithmetical calculations. But it is not sufficient for mathematical analysis, because there are theoretical problems that cannot be solved using rational numbers. Here are two examples:

The equation x² = 2 cannot be solved in the rational number system.
The equation x² = -1 cannot be solved in the rational number system.

To solve these problems we need real numbers and complex numbers.

Home Page Real Numbers Real Variables Complex Numbers Complex Variables

By R. H. B. Exell, 1998. King Mongkut's University of Technology Thonburi.