Real Numbers

Click on an arrow to get a description of the connection!

Click on an arrow to get a description of the connection!

Everybody has got to know at school, the rational numbers, the real, and basic arithmetics. There are certain rules that we can apply, and usually we do not think about them.

In our course, we will get to know other objects than real numbers (vectors, matrices), where some of these laws do not apply any more. So try to have a fresh look at those well known laws:

We can add (\(a+b\)) and multiply (\(ab\) or \(a\cdot b\)) real numbers and use parentheses \((\),\()\) to describe the order of the computations. We have the notational convention that multiplication binds stronger than addition: (\(ab+c\) means \((ab)+c\) and not \(ab+c=a(b+c)\))

Some laws apply: \[\begin{aligned} a+(b+c)&=(a+b)+c, \qquad a(bc)=(ab)c & \mbox{ associative law }\\ a+b &=b+a \qquad \qquad \; ab = ba & \mbox{ commutative law }\\ a(b+c) &= ab+ac & \mbox{ distributive law } \end{aligned}\] Furthermore, we are used to have the neutral numbers \(0\) and \(1\) with special properties: \[\begin{aligned} a+0 = a \qquad a \cdot 1 = a \end{aligned}\] and additive inverse element \(-a\) and also the multiplicative inverse \(a^{-1}=1/a\) for \(a \neq 0\). They fulfil \(a+(-a)=0\) and \(a a^{-1}=1\).

A set with such properties is called a field. Here we have the field of real numbers \(\mathbb{R}\).

It is also well known that the real numbers can be ordered, i.e., the relation \(a < b\) makes sense. It has turned out, that the following rules are sufficient to derive all known rules concerning ordering of numbers.

• For any \(a \in \mathbb{R}\) exactly one of the three relations hold \[a < 0, \mbox{ or } a > 0 \mbox{ or } a = 0\]

• For all \(a,b\in \mathbb{R}\) with \(a>0\) and \(b>0\) one has \(a+b > 0\) and \(a b > 0\).

Then, as a definition we write: \[a < b \quad : \Leftrightarrow \quad a-b < 0\] and \[a \leq b \quad : \Leftrightarrow \quad a-b < 0 \text{ or } a = b \,.\]

In particular, we have for \(a \neq 0\) that always \(a^2 > 0\), because \(a^2=(-a)^2 > 0\) by the last rule applied to one of these terms.

The order relations are the reason, why we can think of the real numbers as a line, the ”real line“.

For describing subsets of the real numbers, we will use intervals. Let \(a,b\in\mathbb{R}\). Then we define \[\begin{aligned} \left[a,b\right] &:= \{x\in\mathbb{R} \mid a\le x\le b\}\\ (a,b] &:= \{x\in\mathbb{R} \mid a< x\le b\} \\ [a,b) &:= \{x\in\mathbb{R} \mid a\le x< b\}\\ (a,b) &:= \{x\in\mathbb{R} \mid a< x< b\}. \end{aligned}\]

Obviously, in the case \(a>b\), all the sets above are empty. We also can define unbounded intervals: \[\begin{aligned} \left[a,\infty\right) := \{x\in\mathbb{R} \mid a\le x\}&,\qquad (a,\infty) := \{x\in\mathbb{R} \mid a< x\} \\ \left(-\infty,b\right] := \{x\in\mathbb{R}\mid x\le b\}&,\qquad (-\infty,b) := \{x\in\mathbb{R}\mid x< b\}. \end{aligned}\]

Rational versus real numbers

For most practical purposes the rational numbers (all fractions) \[\mathbb{Q} = \left\{ x : x = \frac{n}{d} \text{ with } n \in \mathbb{Z}, ~ d \in \mathbb{N} \right\}\] are enough. All numbers that can somehow be stored sensibly on a computer are rational.

Mathematicians say: \(\mathbb{R}\) is complete, \(\mathbb{Q}\) is dense in \(\mathbb{R}\), \(\mathbb{R}\) is the completion of \(\mathbb{Q}\).

Real numbers are like public transport. While quite easy to use, it is not easy to understand in detail why and how it works. (The only difference: real numbers can be used reliably).