Numbers: mathematical units used for counting and comparing things. Practically all primitive peoples were familiar with numbers, though at first they were used only as a means to express quantities or volumes of objects: two apples, three pears, and so on. Isolating the numbers two and three from the apples and pears and treating them as objects in their own right requires abstraction, patience, leisure, and a certain distance from the things of everyday life. Professional priests and magicians were presumably the first who used numbers in isolation from things. Initially only a limited amount of numbers were recognized: one, two, three, infinity. More numbers were added in the course of time, but infinity remained the largest number. The Greeks, who were already able to count as far as to 100,000,000, were the first to realize that infinity is not an object but a concept. This view conforms to contemporary mathematics, which does not recognize infinity as a number object — not even in the sense of the result of dividing by zero — but as a property of a ►set. Large and Small Numbers Before the invention of ►numbering systems it was difficult to write down very small or very large numbers. To save paper and for reasons of clarity we now use powers of ten: for example, 1012 (pronounced "ten to the power of twelve") corresponds to 1,000,000,000,000, that is, to a 1 with 12 zeros. 10-9 corresponds to 0.000000001. Some numbers are so large that they cannot be reasonably represented even as powers of ten. These numbers are represented instead as double powers of ten. In the article on ►parallel worlds, for example, you encounter the number 10 to the power of 1080, that is, a 1 with 1080 zeros.* To be better able to pronounce large numerical amounts we use Greek prefixes or number words. However, the fact that numerical words — like so many other things — are clearly larger in the ►USA than in the rest of the world adds to the confusion (especially when reading internet articles):
More gigantic prefixes, such as Zeta (Z) = 1021 or Yota (Y) = 1024, are used rarely, and then only by number fanatics in certain online dictionaries. Not All Numbers are Alike Apart from their sometimes enormous sizes, numbers at first glance look like simple structures. Their potentially complex and even astonishing properties were discovered only gradually. Even contemporary mathematics is still far from knowing all the properties of numbers. Thinking of numbers, the first thing that comes to mind is that they can be classified into various types:
In addition to these normal numbers there are exotic numbers that are no longer subject to representation by means of numericals:
The various sets of numbers have different kinds of "infinity". Integers can become infinitely large. Rational numbers not only can become infinitely large, but can also be divided infinitely. Real numbers can become infinitely large, can be divided infinitely, and in addition may have infinitely many digits after the decimal point. Finally, the transfinite, infinitesimal, surreal, and hyperreal numbers are not just capable of becoming infinitely large or small as a limit value; they already are these things. For while the other numbers possess only potential infinity, these numbers are actually infinite. Integers, rational and algebraic numbers are ►countable while real numbers are not. For this the transcendental numbers are to blame. While in reality we know only a handful of transcendental numbers, among them the number ►Pi and Euler's number, they actually constitute the vast majority of all numbers. To complete this excursus on numbers, here are some examples of small, large, or otherwise remarkable numbers:
* If we printed this number out in a book, the latter would weigh 1070 tons, even if it were printed with minuscule letters on extremely thin paper. This would be far heavier than the entire observable part of the ►universe. Since the book would immediately collapse under its own weight, the energy released in this process would trigger a thermonuclear fusion reaction (12C+12C => 24Mg). As a result, the exterior book cover would heat up to around 600 million degrees Celsius, explode, and thereby create new galaxies. After only a few million years the further contracting book core would fall below the Schwarzschild radius and become a gigantic ►black hole devouring the Milky Way and all nearby galaxies. Thus, the advantages of double powers of ten should not be underestimated. ** Flops is a unit of measurement for the efficacy of microprocessors (Floating Point Operations per Second). *** Here is the proof. A number with finitely many digits after the decimal point (such as 0.123) is obviously rational, since it can be easily represented as a fraction (123/1000). A number with infinitely many digits after the decimal point such as z = 0.123123123..., also is a fraction, for 999∙z = 1000∙z - z = 123.123123123... - 0.1231231213... = 123, and therefore z = 123/999. † Here is the proof that †† An equation of the form a0 + a1x + a2x2 + ... +anxn = 0, where n is finite and all a are integers. ††† If, for example, we add the two periodic decadic numbers ...444444 and ...555556 in accordance with the standard rules of addition then we obtain 0. Thus, ...555556 is not the least bit larger but instead is the negative value of ...444444. †††† A class is an ordered set of elements such as numbers together with arithmetic operations such as +, -, ∙, /, that can be applied to these elements. Links Related to the Topic ■ Wikipedia:
List of Special Numbers
|