Notes on Mathematics, part I - Number Systems

Description and some basic properties of the familiar sets of numbers. They are listed, in terms of number of elements contained in each set, from the smallest to the largest. Each set presented is always a subset of all sets listed after it. So an element in a set is also in all subsequent sets, e.g. any natural number is also an integer, therefore a rational number, therefore an algebraic number, and so forth. Such inclusion relation is not reciprocal - while, say, a real number might be an integer, it certainly does not have to be an integer.

Natural Numbers: The most basic set of numbers, natural numbers are just the set of all positive integers (the number zero is also considered as a natural number by many). They are the numbers used to count objects. Not surprisingly, natural numbers is by far the earliest number set in existence. The set of natural numbers is usually denoted by the capital letter N.

Integers: Take the set of natural numbers then add to it the negatives of all natural numbers. What we get is of course the set of integers, usually denoted by the capital letter Z. Note that while addition/subtraction and multiplication are defined on integers, division operation is not. Because, in most cases, dividing one integer by another will not yield a third integer as the result. (Even though you might not be aware of it, when you divide an integer by another, you are really performing the operation over the set of rational numbers or real numbers, rather than over integers.)

Rational Numbers: Expand the set of integers by adding to it all the quotients (ratios) of integers. The result is the rational numbers, the set of numbers of the form a/b, where a and b are integers (and b not equals to zero, of course). Division is defined on rational numbers - any rational number divided by any nonzero rational yields another rational number. Thus, in a sense, the rational numbers has "nicer" arithmetic properties than that of the integers or natural numbers. Because of this, the rational numbers is an algebraic structure known as a field. As well, all sets listed below are fields, i.e. division is a legal operation on those sets. The set of all rational numbers is usually denoted by the capital letter Q (stands for quotient).

Algebraic Numbers: By far the least known set on the list, it is included here for completeness. The set of algebraic numbers consists of all the real numbers that are also the roots of at least one polynomial (of any degree) with rational numbers as its coefficients. Therefore, the n-th root of any rational number q is always an algebraic number, for any positive integer n. Because the n-th root of a rational number q is a solution of the equation xn - q = 0, i.e. it's a root of an n-th degree polynomial with rational coefficients. Those real numbers that are not algebraic are called transcendental numbers. Numbers such as pi and e are transcendental, i.e. neither is a root of any polynomial with rational number coefficients. Transcendental numbers do not form a field.

Real Numbers: Other than perhaps integers/natural numbers, you are most familiar with the real numbers. Most of the calculations you have done so far are done over the field of real numbers. While it might not be intuitive, you should look at the real numbers as a subset of the complex numbers. That is, a real number r is just a complex number without the imaginary part, r + 0i. The set of all real numbers is usually denoted by the capital letter R.

Complex Numbers: The next step above the real numbers is the set of complex numbers. It is the largest number system that you will normally encounter. For those of you familiar with linear algebra, the complex numbers form a 2-dimensional vector space over the real numbers, with the set {1, i} as its standard basis. While, for those of us used to do calculations in real numbers, complex numbers arithmetic might seemed cumbersome. The arithmetic nevertheless obeys exactly the same rules as that of real numbers arithmetic. In addition, the complex numbers has the following nice property which real numbers lack. The field of complex number is algebraically closed, meaning that every n-th degree polynomial with complex number coefficients has n complex roots. Equivalently, this means that every n-th degree polynomial with complex number coefficients can be factored into n linear (degree 1) polynomials, with complex coefficients. (This is known as the Fundamental Theorem of Algebra.) This result might not be surprising to you for polynomials with real coefficients; it can thusly be extended to all polynomials with complex coefficients a very remarkable property of complex numbers. The set of all complex numbers is usually denoted by the capital letter C.