are . In the set of natural numbers, which in mathematics is referred to as
Numbers1. additions can be executed without limit as well as multiplications that are to be understood as multiple additions:
In using the notion of numbers one differentiates between ordinal numbers (the third – in a imagined sequence ) and cardinal numbers (three pieces). Toddlers of 3-4 years often 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; also for the adult the number of units that can be grasped at a glance is quite limited (to around 5 -7, which also intelligent animals are capable of); for fast calculations with cardinal numbers the relationship is memorized or simplified in our thoughts (5 + 7 = 5 + 5 + 2 = 10 + 2 = 12). If one realizes this fact, one gains a deeper understanding of the difficulty that children have with learning the elementary rules of arithmetics. Simply assuming the memorized routines which are present with an educated adult leads to severely underestimating the natural hurdles of understanding that the children have to overcome when they learn arithmetics.
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 are shown in a disordered arrangement, that can be spontaneously grasped as a group. The number changes with a frequency that can be specified between 1 and a maximum number of 10. You can establish experimentally , where your own grasping threshold lies The description pages of the simulation contain further details and hints for experiments.
Even numbers are a multiple of the number 2; Prime number cannot be decomposed 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 symbol for the unity of the computable and the cosmos. It also has a special meaning in modern arithmetics 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 token for being unlimited the notion of infinity developed, with the symbol , which does not represent a real number in the usual sense.
Already the preplatonic natural philosophers (Platon himself lived from 427 - 347 b.C.) 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 b.C.) showed in his astute paradoxes (Achilles and the tortoise and the arrows 2 .) , 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 subtracted it at least by 1 smaller than the original number.
Division is the natural inversion of multiplication. For natural numbers it is permissible, if the dividend is an integer multiple of the divisor.