# Real Numbers

In a real analysis, the real numbers are the largest number set we need. They satisfy axioms that represent the idea of a number line.

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Concept | Content |
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Sets | Sets are the basic building blocks for a lot of mathematics. In order to rigorously define numbers and doing real analysis, we need to know how to work with sets. |

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Concept | Content |
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Supremum and Infimum of Sets | Bounded sets always have an supremum and infimum which are generalizations of maximum and minimum. |

Sequences | These object are needed to define limits later on. |

Convergence | Convergent sequences have a well-defined limit. |

Completeness | Completeness says that Cauchy sequences must converge. |

Partitions and Step Functions | Splitting up an interval in subintervals and defining functions that are constant on them. |

## Study *Real Numbers* #

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: \[\begin{aligned} \left[a,\infty\right) := \{x\in\mathbb{R} \mid a\le x\}&,\qquad (a,\infty) := \{x\in\mathbb{R} \mid a< x\} \\ \left(-\infty,b\right] := \{x\in\mathbb{R}\mid x\le b\}&,\qquad (-\infty,b) := \{x\in\mathbb{R}\mid x< b\}. \end{aligned}\]

**Definition 1** (Absolute value for real numbers).
*The absolute value of a number \(x\in\mathbb{R}\) is defined by \[|x|:=\begin{cases}
~~x & \text{ if } x\ge 0,\\
-x & \text{ if } x< 0.
\end{cases}\]*

**Question 2**. *Which of the following claims are
true? \[|-3.14|=3.14, \quad |3|=3 , \quad
|-\tfrac75|=\tfrac75,
\quad {-|-\tfrac35|=\tfrac35},
\quad {|0| \text{ is not well-defined}}.\]*

For any two real numbers \(x,y\in\mathbb{R}\), one has

\(|x\cdot y| = |x| \cdot |y|\), (\(|\cdot|\) is multiplicative),

\(|x+y| \le |x| + |y|\), (\(|\cdot|\) fulfils the triangle inequality).

The real numbers are a non-empty set \(\mathbb{R}\) together with the operations \(+ : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) and \(\cdot : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) and an ordering relation \(<: \mathbb{R} \times \mathbb{R} \rightarrow \{\text{True}, \text{False}\}\) that fulfil the following rules

Addition

associative: \(x + (y + z) = (x + y) + z\)

neutral element: There is a (unique) element \(0\) with \(x + 0 = x\) for all \(x\).

inverse element: For all \(x\) there is a (unique) \(y\) with \(x + y = 0\). We write for this element simply \(-x\).

commutative: \(x + y = y + x\)

Multiplication

associative: \(x \cdot (y \cdot z) = (x \cdot y) \cdot z\)

neutral element: There is a (unique) element \(1 \! \neq \! 0\) with \(x \! \cdot \! 1 = x\) for all \(x\).

inverse element: For all \(x \neq 0\) there is a (unique) \(y\) with \(x \cdot y = 1\). We write for this element simply \(x^{-1}\).

commutative: \(x \cdot y = y \cdot x\)

Distributivity: \(x \cdot (y + z) = x \cdot y + x \cdot z\).

Ordering

for given \(x,y\) exactly one of the following three assertions is true: \(x<y\), \(y<x\), \(x=y\).

transitive: \(x<y\) and \(y<z\) imply \(x<z\).

\(x < y\) implies \(x + z < y + z\) for all \(z\).

\(x < y\) implies \(x \cdot z < y \cdot z\) for all \(z>0\).

\(x>0\) and \(\varepsilon>0\) implies \(x < \varepsilon + \cdots + \varepsilon\) for sufficiently many summands.

Completeness: Every sequence \((a_n)_{n\in \mathbb{N}}\) with the property

*For all \(\varepsilon > 0\) there is an \(N \in \mathbb{N}\) with \(|a_n - a_m| < \varepsilon\) for all \(n,m > N\)*has a limit.

**Definition 3** (Field). *Every set \(M\) together with two the operations \(+ : M \times M \rightarrow M\) and \(\cdot : M \times M \rightarrow M\) that
fulfil (A), (M) and (D) is called a field.*

**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).

Courtesy of Marcus Waurick. *Well-defined & Wonderful podcast*, marcus-waurick.de.

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