2.2 Numbers
2.2.4 Real Numbers
5 3
= 5!
3!(5−3)! = 1·2·3·4·5 1·2·3·1·2 = 10
or find them in the Pascal Triangle. This triangle is constructed in the following way:
n = 0 : 1
n = 1 : 1 1
n = 2 : 1 2 1
n = 3 : 1 3 3 1
n = 4 : 1 4 6 4 1
n = 5 : 1 5 10 10 5 1
n = 6 : 1 6 15 20 15 6 1
We start with the number 1 in the linen= 0. In the next line (n = 1) we write two ones, one on each side. Then for n = 2 we add two ones to the left and right side once again, and in the gap between them a 2 = 1 + 1 as the sum of the left and right “front man”
(in each case a 1). In the framed box, we once again recognize the formation rule. The required binomial coefficient 53
is then found in line n= 5 on position 3.
Exercise 2.2
a) Determine the length of the space diagonal in a cube with side length a.
b) Calculate (a4−b4)/(a−b).
c) Calculate n0
and nn . d) Calculate 74
and 83 . e) Show that n−kn
= nk
holds true.
f ) Prove the formation rule for the Pascal Triangle: k−1n + nk
= n+1k .
Insert: History: Already in antiquity some mathematicians knew that there are numbers which cannot be represented as fractions. They showed this with a so-called indirect proof:
If e.g. the diagonal of a square with side length 1 were a rational number, like
√2 = b/a, two natural numbers b, a∈ N would exist with b2 = 2a2. Think now of the prime factor decompositions of b and a. On the left hand side of the equation there stands an even number of these factors, because of the square each factor appears twice. On the right hand side, however, an odd number of factors shows up, because in addition the factor 2 appears. Since the prime factor decomposition is unique, the equation cannot be right.
With this it is shown that the assumption,√
2can be represented as a fraction, leads to a contradiction and thus must be wrong.
With the real numbers, which have the same calculation rules of a field as the rational numbers, both solutions of the general
quadratic equation: x2+ax+b= 0, x1,2 =−a 2 ±
ra2 4 −b
will then be real numbers, as long as the discriminant under the root is not negative:
a2 ≥4b.
Insert: Preview: complex numbers: Later in Chapter 8 we will go one step further by introducing the complex numbers C for which e.g. also x2 = a for a <0 is always solvable and, amazingly enough, many other beautiful laws hold.
Chapter 3
SEQUENCES AND SERIES and Their Limits
Direct mathematical study of sequences and series are, for natural scientists, less im- portant than the fact that they greatly help us to understand and perform the limiting procedures which are of fundamental importance in physics. For this reason, we have combined in this chapter the most important facts of this part of mathematics. Later you will deal in greater detail with these things in your future mathematics lectures.
3.1 Sequences
The first important mathematical concept we have to inspect is that of a sequence.
With this physicists think for instance of the sequence of the bounce heights of a steel ball on a plate, which due to the inevitable dissipation of energy decrease with time and tend more or less quickly to zero. After a while, the ball remains still. The resulting physical sequence of the jump heights has only a finite number of non-vanishing members in contrast to the ones that are of interest to mathematicians: Mathematically, a sequence is an infinite set of numbers which can be numbered consecutively, i.e. labelled by the set of the natural numbers: (an)n∈N. Because it is impossible to list all infinite many members (a1, a2, a3, a4, a5, a6, . . .), a sequence is mostly defined by the “general member”
an, which is a law stating how to calculate the individual members of the sequence. Let us look at the following typical examples which already enable us to display all important concepts:
(F1) 1,2,3,4,5,6,7, . . . = (n)n∈N the natural numbers themselves (F2) 1,−1,1,−1,1,−1, . . .= ((−1)n+1)n∈
N a simple “alternating” sequence, (F3) 1,12,13,14,15, . . .= n1
n∈N the inverse natural numbers, the so-called “harmonic” sequence, (F4) 1,12,16,241, . . . = n!1
n∈N the inverse factorials, (F5) 12,23,34,45, . . . = n+1n
n∈N a sequence of proper fractions and (F6) q, q2, q3, q4, q5. . .= (qn)n∈
N, q∈R the “geometric” sequence.
Insert: Compound interest: Many of you know the geometrical sequence from school because it causes a capitalK0 atp% compound interest afternyears to increase to Kn=K0qn with q= 1 +100p .
In order to give us a first clear idea of these sample sequences, we have plotted the sequence membersan (in the 2-direction) over the equidistant natural numbers n (in the 1-direction) in the following Cartesian coordinate system in a plane:
Figure 3.1: Visualization of our sample sequences over the natural numbers, in case of the geometrical sequence (F6) forq = 2 andq= 12.
Also the sum, the difference or the product of two sequences are again a sequence. For example, the sample sequence (F5) with an = n+1n = n+1−1n+1 = 1− n+11 is the difference of the trivial sequence (1)n∈N = 1,1,1, . . ., consisting purely of ones, and the harmonic sequence (F3) except for the first member.
The termwise product of the sample sequences (F2) and (F3) makes up a new sequence:
(F7) 1,−12,13,−14, ...=(−1)n+1
n
n∈N
the “alternating” harmonic sequence.
Similarly the termwise product of the harmonic sequence (F3) with itself is once again a sequence:
(F8) 1,14,19,161, ...= n12
n∈N the sequence of the inverse natural squares.
The termwise product of the sample sequences (F1) and (F6), too, gives a new sequence:
(F9) q,2q2,3q3,4q4,5q5. . .= (nqn)n∈N, q ∈R a modified geometric sequence.
An other more complicated combined sequence will attract our attention later:
(F10) 2,(32)2,(43)3, . . .= (1 + n1)n
n∈N the so-called exponential sequence.
Exercise 3.1 Illustrate these additional sample sequences graphically. Project the points on the 2-axis.
There arethree characteristicsthat are of special interest to us as far as sequences are concerned: boundedness, monotony and convergence: