Natural numbers The sequence \(0, 1,2,3,\ldots\) is the sequence of the natural numbers. These are the numbers to count with or to index. The set of natural numbers is symbolized by \(\mathbb{N}\). so \[\mathbb{N}=\{0,1,2,3,\ldots\}\]

Integers The arithmetic of natural numbers goes well if you add such numbers. But subtraction of natural numbers sometimes causes problems. What, for example, would the outcome of \(1-2\) or \(2-3\) be and do we mean the same number with that? In other words, what is the solution of the equation \(x+2=1\) or the equation \(x+3=2\)? The wish to be able to solve such equations gives rise to the introduction of the set of integers, symbolized by \(\mathbb{ℤ}\).

The integers can be placed on a number line: beginn with a first point on a straight line and give it the label 0 and choose a second point that get the label 1. We agree that the distance between these two points is equal to 1. Now continue with steps of length 1 and place the next integer on the number line with its label. Each number has a successor. Do the same now in the other direction and you get the following number line:

So \[\mathbb{ℤ}=\{\ldots, -2, -1, 0, 1, 2, \ldots\}\] The numbers from this collection can be added, subtracted and multiplied with each other.

Rational numbers In \(\mathbb{ℤ}\) we can add, subtract, multiply without any problems. But division of two integers not always produces an integral number. What, for example, would the outcome of \(1\div 2\) or \(2\div 4\) be and do we mean the same number with that? In other words, what is the solution of the equation \(2x=1\) or the equation \(4x=2\)? The wish to be able to solve such equations gives rise to the introduction of the set of rational numbers, symbolized \(\mathbb{Q}\).

Also, the rational numbers, i.e., the numbers that can be written as a fraction, can be placed on the number line. The construction of a pair of these points is shown in the picture below.

The notation for a fraction consists of two integers, the numerator and denominator separated by a horizontal line or slash. In the above picture, there is a rational number drawn on the number line that as fraction \(\tfrac{2}{3}\) is labeled with numerator \(2\) and denominator \(3\). But we also could have connected the point with label 6 on the vertical line through 0 on the horizontal number line to the point with label1 on the horizontal line and have drawn parallel lines. Then we could have given the point with the previous label \(\tfrac{2}{3}\) also the label \(\tfrac{4}{6}\).

In other words, the representation of a rational number as a fraction is not unique: for example, \(\tfrac{2}{3}\) and \(\tfrac{4}{6}\) represent the same rational number and we write \(\tfrac{2}{3}=\tfrac{4}{6}\). In general, if you multiply or divide the numerator and denominator of a fraction by the same whole number (non-zero), then the value of the fraction does not change. Division of numerator and denominator by the same factor greater than 1 is called simplification. For example, \(\tfrac{4}{6}\) can be simplified to \(\tfrac{2}{3}\) by dividing the numerator and denominator by 2.

We speak of a irreducible fraction if

• the greatest common divisor (GCD) of numerator and denominator equals 1
• the denominator is positive, and
• the denominator is 1 in case the numerator equsld 0.

So \[\mathbb{Q}=\Bigl\{\,\frac{t}{n} \Bigm| t, n \in \mathbb{ℤ}, n\neq 0\,\Bigr\}\] The numbers from this collection can be added, subtracted, multiplied and divided with each other.

Real numbers On the number line other numbers than the rational numbers can be constructed: for example, with ruler and compass, the number \(\sqrt{2}\). Consider the square with edge length 1.

From the Pythagorean theorem it follows that the square of the length of the diagonal line segment is equal to the sum of the squares of two sides of the square. Thus, the length of the diagonal line segment is equal to \(\sqrt{2}\). With a compass you can construct this length segment on the number line.

All numbers on the number line form the set of real numbers, symbolized by \(\mathbb{R}\). A real number that is not rational is called an irrational number. Known irrational numbers are \(\pi\approx 3.141592\ldots\) and the base of the natural logarithm \(e\approx 2.71828\ldots\).

Note that real numbers can be described by approximating them with decimal numbers. In case such a decimal numbers contains a sequence of digits that repeats itself over and over again, then the represented number is a rational number; otherwise it is an irrational number.