In this chapter we became acquainted with natural numbers, integers, rational numbers, and real numbers.

We started by giving an idea of ​​the structure of, successively,

• natural numbers through the counting of objects
• integers through the subtraction of a natural number from another natural number
• rational numbers through the division of an integer on another integer unequal to #0#
• real numbers through the number line, on which numbers are placed that can be approximated systematically by special rational numbers: decimal numbers

If a real number is not rational, then it is difficult to indicate exactly how big it is. A precise definition of \(\pi\) for example, is "half the circumference of a circle with radius #1#". But if we want to work with this, we often use an approximation like #3.142#. In a formula we indicate this by use of the symbol #\approx# for approximately equal: \(\pi \approx 3.142\). This number is the decimal number #\frac{3142}{1000}# where we do not worry about the fact that it can be simplified. The approximation we choose is in fact a number that consists of a number of digits before and a number of digits after the decimal point.

Every real number can be described as accurate as you prefer. But the amount of work increases with increasing accuracy. Accuracy is defined as the number of digits (decimal places) behind the point. Exact determination of the number by means of this so-called decimal development for most numbers is only theoretically possible: you would have to write down an infinite number of digits.

In order to understand decimal numbers better, it is necessary to find out more about Sequences and series. In particular, rational numbers can be recognized as the repetitive tail with the aid of geometric series. But numbers like #\pi# are better understood by considering approximations as terms of a series.

Real numbers can be extended to complex numbers. Each real number is a complex number, but there is also exists a complex number #i# satisfying #i^2 = -1#. That number does not occur on the number line, hence not real. Every complex number can be written as #a+bi#, where #a# and #b# are real numbers. An introduction into this world is given later. If you are interested in this, it is a good idea to first read the chapter 2-Dimensional Geometry: points and lines.

Besides the introduction of numbers we have dealt with operations on numbers. Obviously we dealt with addition, subtraction, multiplying, dividing and exponentiation. But also paid a lot of attention to roots. We learned how to compare two roots of fractions by writing denominators without fractions. For this, a standard notation has been discussed. For an expression like #\frac{\sqrt{3}}{\sqrt{2}}# we have indicated that the standard notation is #\frac{1}{3}\sqrt{6}#. But a standard notation for #\frac{1}{2+\sqrt{3}}# is not discussed, let alone an expression such as #\frac{1}{2+\sqrt[3]{5}}#. In the chapter Algebra we will show how the root can be removed from the denominators of both expressions. Nevertheless it is far from trivial to find a standard notation for expressions containing multiple roots. By means of an example, we discussed that \[\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}\] is equal to #1#. The chapter Polynomials and rational functions contains a proof of this equality.

This difficulty is all the more reason to approach real numbers by decimal developments. We have shown that this can determine if one number is bigger than another. A shortcoming of the decimal development is that, in the case two numbers have the same head with a lot of decimal digits, it cannot be determined if these two numbers are exactly the same.

We recommend Algebra as the next chapter for study; here, the emphasis is on the manipulation of variables.