de Gruyter Textbook Learning and Teaching Mathematics using Simulations Plus 2000 Examples from Physics Bearbeitet von Dieter Röss 1. Auflage 2011. Taschenbuch. XVII, 255 S. Paperback ISBN 978 3 11 025005 3 Format (B x L): 17 x 24 cm Gewicht: 449 g Weitere Fachgebiete > Mathematik > Numerik und Wissenschaftliches Rechnen > Computeranwendungen in der Mathematik Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 3 Numbers We first want to remind the reader of the different kinds of numbers that are used in arithmetic and to visualize their relationships with the elementary arithmetic op- erations. Here the number is the operand on which a certain operation is applied (arithmetic operations such as C, ,*,=,ˆ,D, >, <, and logical operations such as and,or,not,if-then,otherwise,...). The definitions of numbers and arithmetic operations are, up to the complex num- bers, synchronized in such a way that for any number z, the following fundamental rules of arithmetic operations apply. Here ./means that the operation in brackets is executed first. z1 C z2 D z2 C z1 .Commutative law of addition) z1 z2 D z2 z1 .Commutative law of multiplication) .z1 C z2/ C z3 D z1 C .z2 C z3/ (Associative law of addition) .z1 z2/ z3 D z1 .z2 z3/ (Associative law of multiplication) .z1 C z2/ z3 D z1z3 C z2z3 (Distributive Law of multiplication) Consequence: It does not matter in which sequence the operations are executed. 2 3 2 Shorthand notation in the text: z1z2 z1 z2I z D zzI z D z z.D zˆ3/I :::: The following sections will introduce different families of numbers in consecution of their historical development. The requirement to apply certain operations, which had been introduced for a certain kind of number, without restrictions, led to successive extensions of the usual concepts of numbers. 3.1 Natural numbers The natural numbers are 1; 2; 3; 4; 5; : : : in the set of natural numbers, which in math- Numbers ematics is referred to as N.10 Additions can be executed without limit as well as multiplications, which are to be understood as multiple additions: 3 4 D 4 C 4 C 4. In using number notation, one differentiates between ordinal numbers (the third –in an imagined sequence) and cardinal numbers (three pieces). Toddlers of 3–4 years of- ten know the ordinal numbers up to 10 and they can also execute simple additions via counting. The more abstract notion of the cardinal number children mostly understand only when they start school; in addition, even for the adult, the number of units that can 10 Historically, special symbols have been introduced for the number sets (see link in the margin). 14 3 Numbers Figure 3.1. Simulation. Spontaneous grasping of the number of elements in a set (cardinal numbers) A random number generator produces red points, whose number lies between 1 and the maximum number in the number field (in the figure the maximum number is 5, 5 are shown). The sets change with a frequency that can be adjusted with the slider from 1 to 10 per second. be grasped at a glance is quite limited (to around 5–7, which is also what intelligent animals are capable of); for fast calculations with cardinal numbers, the relationship is memorized or simplified in our thoughts (5C7 D 5C5C2 D 10C2 D 12). If one realizes this fact, one gains a deeper understanding of the difficulty that children have with learning the elementary rules of arithmetic. Simply assuming the memorized routines, which are present in an educated adult, leads to severely underestimating the natural hurdles of understanding that the children have to overcome when they learn arithmetic. The simulation in Figure 3.1 visualizes the sharp threshold that nature imposes for spontaneously grasping the number of elements of a set. In this simulation, points 3.2 Whole numbers 15 are shown in a random arrangement that can be spontaneously grasped as a group. The number changes with a frequency that can be specified between 1 and a max- imum number. You can establish experimentally where your own grasping thresh- old lies. The description pages of the simulation contain further details and hints for experiments. Even numbers are a multiple of the number 2; a prime number cannot be decom- posed into a product of natural numbers, excluding 1. The lower limit of the natural numbers is the unity 1. This number had a close to mystical meaning for number theoreticians of antiquity, as the symbol for the unity of the computable and the cosmos. It also has a special meaning in modern arithmetic as that number which, when multiplied with another number, produces the same number again. There is, however, no upper limit of the natural numbers: for each number there exists an even larger number. As a token for this boundlessness, the notion of infinity developed, with the symbol 1, which does not represent a number in the usual sense. Already, the preplatonic natural philosophers (Plato himself lived from 427–347 BC) worked on the question of the infinite divisibility of matter (If one divides a sand grain infinitely often, is it then still sand?) and time (if one adds to a given time interval infinitely often half of itself, will that take infinitely long?) Zenon of Elea (490–430 BC) showed in his astute paradoxes, Achilles and the tor- toise and the arrows,11 that the ideas of movement and number theory at the time were in contradiction to each other. Subtraction is the logical inversion of addition: for natural numbers it is only permissible if the number to subtract is smaller than the original number by at least 1. Division is the natural inversion of multiplication. For natural numbers it is permis- sible if the dividend is an integer multiple of the divisor – 6 W 2 D 3. 3.2 Whole numbers In order for the operation of subtraction to be always possible, we have the extend the natural numbers by zero (the “neutral” element of addition) and the negative numbers to the set Z of whole numbers. The introduction of the zero as a number was, historically, not a trivial step. Zero is connected with the notion of nothing, and for the pre-socratic natural philosophers 11 Achilles and the tortoise: during a footrace, Achilles allows the tortoise a head start. When he has reached its starting point, the tortoise has already crawled a certain distance further. When he reaches this point, the tortoise has again had a head start, and so on. Thus Achilles cannot reach the tortoise. (Solution: Convergence of the geometric series, which had not been recognized at the time). The flying arrow: at every instant, the arrow is situated at a certain point, which is fixed in space; therefore, the arrow is at rest in every instant. Therefore the arrow cannot move. (Solution: In every instant, the arrow does not have only a location but also a velocity; this differential concept was not recognized.) 16 3 Numbers it was a fundamental question, whether nothing (the emptiness, something that is not) can exist or not. Parmenides of Elea (around 600 BC) taught that nothing cannot exist, but that every space has to filled by something, which leads to the paradoxical logical consequence that movement is impossible and everything is unchangeable. The atomicists Leukipp (5th century BC) and Democrit (460–371 BC) taught, however, that the world consists mostly of nothing (today we would call it a vacuum), in which objects that consist of material atoms can move. In the 3rd century BC, the zero was imported from the east in connection with the campaigns of Alexander the Great as a sign indicating the position in the decimal number system (as today in 10,100). The concept of zero as a whole number was only reached in the 17th century. The set of whole numbers contains the natural numbers as a subset Whole numbers: :::;3; 2; 1; 0; 1; 2; 3; : : : : Multiplication is admissible without exception if one defines: .1/ 1 D1; .1/ .1/ D 1 and .0/ .1/ D 0. In the domain of whole numbers, the symbol for positive infinity must be necessar- ily supplemented by negative infinity 1; both numbers are not numbers in the usual sense. Division can be applied for whole numbers, as for natural numbers, if the divi- sor is contained as a factor in the dividend, i.e. if the division works out, as for 30 W 5 D6. Division by zero is not a well defined inversion of multiplication: for integers a; b; c b b D c uniquely leads to a D a c 0 for D 0acan be any number. a 1 0 and is therefore excluded. The expressions 0 1, 1 and 0 are not defined. Whole numbers are visualized as a discrete ladder on the number line (see Fig- ure 3.2). Arithmetic operations amount to jumping back and forth on this ladder – in the same way that toddlers indeed make calculations with natural numbers by counting. Figure 3.2. Number line with whole numbers. 3.3 Rational numbers 17 3.3 Rational numbers In order for division to be admissible in all cases, except for division by zero,the whole numbers have to supplemented by the “broken numbers” to form the set of rational numbers Q.