超实数是“数”吗？

在我们国内，普遍认为：超实数不是“数”，请见科普中国以及全部高等数学教科书。

在国外，超实数是“数”,请见本文附件。

注：超实数是非标准书。数，因而是“数”。

袁萌  陈启清  7月14日

附件：

A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth.[1] A written symbol like "5" that represents a number is called a numeral. A numeral system is an organized way to write and manipulate this type of symbol, for example the Hindu–Arabic numeral system allows combinations of numerical digits like "5" and "0" to represent larger numbers like 50.[2] A numeral in linguistics can refer to a symbol like 5, the words or phrase that names a number, like "five hundred", or other words that mean a specific number, like "dozen". In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, number may refer to a symbol, a word or phrase, or the mathematical object.

In mathematics, the notion of number has been extended over the centuries to include 0,[3] negative numbers,[4] rational numbers such as

1

/

2

and −

2

/

3

, real numbers[5] such as √2 and π, and complex numbers,[6] which extend the real numbers with a square root of −1 (and its combinations with real numbers by addition and multiplication).[4] Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic. The same term may also refer to number theory, the study of the properties of numbers.

Besides their practical uses, numbers have cultural significance throughout the world.[7][8] For example, in Western society, the number 13 is regarded as unlucky, and "a million" may signify "a lot."[7] Though it is now regarded as pseudoscience, belief in a mystical significance of numbers, known as numerology, permeated ancient and medieval thought.[9] Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today.[9]

During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Today, number systems are considered important special examples of much more general categories such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.[10]

Contents

1 History

1.1 Numerals

1.2 First use of numbers

1.3 Zero

1.4 Negative numbers

1.5 Rational numbers

1.6 Irrational numbers

1.7 Transcendental numbers and reals

1.8 Infinity and infinitesimals

1.9 Complex numbers

1.10 Prime numbers

2 Main classification

2.1 Natural numbers

2.2 Integers

2.3 Rational numbers

2.4 Real numbers

2.5 Complex numbers

3 Subclasses of the integers

3.1 Even and odd numbers

3.2 Prime numbers

3.3 Other classes of integers

4 Subclasses of the complex numbers

4.1 Algebraic, irrational and transcendental numbers

4.2 Constructible numbers

4.3 Computable numbers

5 Extensions of the concept

5.1 p-adic numbers

5.2 Hypercomplex numbers

5.3 Transfinite numbers

5.4 Nonstandard numbers

6 See also

7 Notes

8 References

9 External links

History（全文请见“无穷小微积分”网站。）