**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**.

**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 **x ^{n} - q = 0**, i.e. it's a root of an

**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**.