This chapter is about arithmetic with numbers. We will introduce all kinds of numbers: natural numbers, integers, rational and irrational numbers. All of these numbers are real. Every real number is rational or irrational. Natural numbers are integers. Integers are rational numbers. We focus on the systematic calculations with such numbers.

The row \(1,2,3,\ldots\) is the sequence of natural numbers. These are the numbers with which we count a finite number of objects. The set of natural numbers is denoted by \(\mathbb{N}\). Often #0# is also seen as natural number, but that is not the case in this course. The number #0# expresses the number of objects in the case that there are no objects to be counted. The numbers \(0,1,2,3,\ldots\) will be described as the non-negative integers. This will become clear below.

The arithmetic of natural numbers will work if you add or multiply numbers. But subtracting numbers sometimes causes problems. For example, what is the result of \(1-2\) or \(2-3\) ? This gives rise to the introduction of the set of integers, denoted by \(\mathbb{ℤ}\).

Integers can be put on a number line: start with a first point on a straight line and give it the name card #0#; choose a second point that gets the name card #1#. We agree that the distance between these two points is equal to #1#. Now continue in the same manner with steps of length #1# and placing name cards for every next integer. Do the same in the other direction and you get the following number line:

In \(\mathbb{ℤ}\) there are no problems with adding subtraction and multiplication. But dividing two integers does not always result in an integer. For example, #{3}:{2}# is not an integer. This gives rise to the introduction of the set of rational numbers, denoted by \(\mathbb{Q}\).

Also, the rational numbers, i.e., the numbers that can be written as a fraction of two integers, can be placed on the number line. In a fraction there are two integers, the numerator and the denominator separated by a horizontal or slash. The notation of a rational number as a fraction is not unique: \(\tfrac{2}{3}\) and \(\tfrac{4}{6}\) for example both indicate the same rational number, and we can write \(\tfrac{2}{3}=\tfrac{4}{6}\). However, it is possible to make the notation uniquely by the following two requirements:

• the denominator must be positive;
• the numerator and denominator may have no common divisors.

All the numbers on the number line form the set of real numbers, denoted by \(\mathbb{R}\). A real number which is not rational is called an irrational number. Some well known irrational numbers are \(\pi\approx 3.141592\ldots\), the surface of a circle with radius #1#, and \(\e\approx 2.71828\ldots\), Euler's constant, or the base of the natural logarithm (the details are discussed later on).

The numbers are ordered on the number line: for every two real numbers we can determine which one is most on the right of the number line: this is the larger of the two. With #2\lt 20# we indicate that #2# is smaller than #20#. A number is called negative if it is smaller than #0# and positive if it is greater than #0#. A number is non-negative if it is positive or equal to #0#.

The result of repeatedly multiplying by the same number is a power. #2^5# (pronounced two to the power of five) is #2\times 2\times 2\times 2\times 2#, so #32#. Roots are associated with it. The root of #4# is #2#, a rational number, but the roots of #2# and #3# are irrational. We pay a lot of attention to roots of rational numbers.

Not all real numbers are rational numbers or roots. At the end of this chapter we discuss real numbers in general: how to perform arithmetic with them without having to write them completely. This part is more difficult than the rest.

In this chapter we sometimes calculate with variables, an issue that will be discussed in more detail in the chapter Algebra. To help you become familiar with the notion and to prepare for its use in this chapter, a short introduction to Variables is included.