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The crowd the natural numbers |N|={1,2,3,4,...} arose from the Verschmelzung the crowd all possible "Zählzahlen" (aka 'Kardinalzahlen') with the crowd the Platznummern (aka 'Ordnungszahlen', 'Ordinalzahlen'). Later watts too The from the Arabern dazuerfundene zero '0' as 'Natürliche Zahl' hinzugenommen (The roman were z.B. not yet so far), sodaß one |N|_0={0,1,2,3,4,...}+Inf knew. so could releases the addition and Multiplikation positiver Ganzzahlen always resolved go.

long Time could Kaufleute to owe of Credit only discern, in the tappt im dunkeln the numbers on unterschiedlichen Buchseiten anschrieben, namely either in the SOLL (left) or in the HABEN (right, with Bankbilanzen inverse). first sometime in the 15. centenary (so circa 1450 around) was is normally, The Wörter „plus“ and „minus“ auszuschreiben. loudly Wikipedia watts in italian and french Schriften thereby often only the character „m“ for less ausgeschrieben and one waagerechten Strich slip, around the symbol rather To mark. this Querstrich over the „m“ is a Statement for coming into being the present Minuszeichens.

so was The crowd the Negativen Ganzzahlen born, The - mating with the natural numbers - The crowd all whole numbers inklusive zero |Z|_0 yielded. so could too releases as Subtraktion of/ one more number a smaller resolved go.

The treatment of Brüchen (Division whole numbers) by the middle-aged Griechen listed a new achievement: On The crowd the broken (aka 'Rationalen') numbers |R|. These can in the principle always as Gleitkomma-numbers with eingetragener (ggf. very langer) Periodizität exakt displayed go. but not, How in the maths normally, whom periodic part by a "Überstrich" (frz.: Macron) To mark, becomes this part in the nachstehenden statement between _ and _ stehend described.

P.s.: Umso erstaunter was one as one notice had to, that sizes How the Relationship of Kreisumfang To Durchmesser or the Relationship the Diagonalen in the Quadrat To which Seitenlänge not entire through fractures representable was, these means 'Irrationale Zahlen' were. The pursuit with the question, what really from Negativen numbers becomes, if one from them a geradzahlige root wander wants, led Carl Friedrich Gauss (1831) then on The crowd the complex numbers |K|. further Mathematiker created others regulate for Mengenelemente; the led u.a. To 'Topologischen Halbordnungen', Abelschen and not-Abelschen groups, on Verbände, rings and Zahlenkörper...

Compile Mark Separation

everything exakte embodiments:   period ab_Dezimale   Vorkomma.Nachkommalänge
1     1                                0
1/2   0.5                              0
1/3   0._3_                            1.1
1/4   0.25                             0
1/5   0.2                              0
1/6   0.1_6_                           2.1
1/7   0._142857_                       1.6
1/8   0.125                            0
1/9   0._1_                            1.1
1/10  0.1                              0
1/11  0._09_                           1.2
1/12  0.08_3_                          3.1
1/13  0._076923_                       1.6
1/14  0.0_714285_                      2.6
1/15  0.0_6_                           2.1
1/16  0.0625                           0
1/17  0._0598235294117647_             1.16
1/18  0.0_5_                           2.1
1/19  0._052631578947368421_           1.18
1/20  0.05                             0
1/21  0._047619_                       1.6
1/22  0.0_45_                          2.2
1/23  0._0434782608695652173913_       1.22
1/24  0.041_6_                         4.1
1/25  0.04                             0
1/26  0.0_384615_                      2.6
1/27  0._037_                          1.3
1/28  0.03_571428_                     3.6
1/29  0._0344827586206896551724137931_ 1.28
1/30  0.0_3_                           2.1
1/31  0._032258064516129_              1.15
1/32  0.03125                          0
1/33  0._03_                           1.2
1/34  0.0_2941176470588235_            2.16
1/35  0.0_285714_                      2.6
1/36  0.02_7_                          3.1
1/37  0._027_                          1.3
1/38  0.0_263157894736842105_          2.18
1/39  0._025641_                       1.6
1/40  0.025                            0
1/41  0._02439_                        1.5
1/42  0.0_238095_                      2.6
1/43  0._023255813953488372093_        1.21
1/44  0.022_72_                        4.2
1/45  0.0_2_                           2.1
1/46  0.0_2173913043478260869565_      2.22
Microsoft-Desktop-Taschenrechner:
1/47    0.02127659574468085106382978723404?...  NO PERIODIZITÄT ???
Maxima:  0.0212765957446808505471036454537170357070863246917724609375  NANU ?
BigNum: 0._0212765957446808510638297872340425531914893617_  DOCH ! 1.46
1/48  0.0208_3_                        5.1
1/49    0.02040816326530612244897959183673?...  NO PERIODIZITÄT ???
Maxima:   0.0204081632653061208204636756136096664704382419586181640625  ÄHM???
BigNum: 0._020408163265306122448979591836734693877551_  DOCH !     1.42
1/50  0.02                             0
1/51  0._0196078431372549_             1.16
1/52  0.01_923076_                     3.6
1/53  0._0188679245283_                1.13
1/54  0.0_185_                         2.3
1/55  0.0_18_                          2.2
1/56  0.017_857142_                    4.6
1/57  0._017543859649122807_           1.18
1/58   0.0_1724137931034482758620689655_  2.28
1/59   BigNum:0._0169491525423728813559322033898305084745762711864406779661_
1/60  0.01_6_
1/61  0._016393442622950819672131147540983606557377049180327868852459_ 1.60
1/62  0.0_161290322580645_             2.15
...