Mathematicians modern rigorous definition of number?

In summary, the modern rigorous definition of a number is an element of a number system, which is a set associated with operations such as addition and multiplication. This allows for the definition of different types of numbers, such as natural numbers, integers, and complex numbers. However, not all numbers represent quantities, as some are purely qualitative. Additionally, mathematicians have the ability to define operations and create number systems for a wide range of "things", showing that anything can be defined as a number in a specific context.
  • #1
roger
318
0
What is the mathematicians modern rigorous definition of number ?


thanks

Roger
 
Mathematics news on Phys.org
  • #2
There isn't one!
 
  • #3
If there is one, a "number" is an element of a "number system"
A "number system" is some set, associated typically by some "operations" that you can use upon the elements of the set, for example "adding" two of the "numbers" together.

This is my very informal view of this, however..
 
  • #4
roger said:
What is the mathematicians modern rigorous definition of number ?


thanks

Roger
What type of number? A natural number? An integer? A quotient? A real number? A complex number? A hyper-real number? A hyper-complex number? A trans-finite number? A surreal number?...

All of these have different definitions.
 
  • #5
Apparently it seems that a number is defined as being an element of some defined set.
It is quite funny that "element" and "number" mean the same thing. So in fact we can define anything we want as a number !
 
  • #6
hello3719 said:
Apparently it seems that a number is defined as being an element of some defined set.
It is quite funny that "element" and "number" mean the same thing. So in fact we can define anything we want as a number !

Some collection of things whose members we often refer to as numbers are not sets, to give you an example the 'surreal numbers' form a proper class (i.e. they do not form a set).

Element and number are not synonyms; it certainly is not common to call every member of a set (or a class) a number.
 
  • #7
I didn't state which kind of number because, I didn't feel that it would ultimately make any difference to the question.

the last comment made is true I guess in the sense that the two words element and number, are equivalent in meaning.

is it wrong to define it as a quantity of things eg apples ?
 
  • #8
roger said:
is it wrong to define it as a quantity of things eg apples ?
That definition is misleading. You end up having to twist and distort it to an unrecognizable lump after encountering various number systems. Considering just the negative integers, you then have to modify it by "also an absence of quantity" or some other interpretation. It only goes downhill from there. What quantity does sqrt(-1) measure ? Then you start to redefine quantity until the original statement is meaningless. While all quantities may be described by numbers, not all numbers represent quantities. Some are quite qualitative.
 
Last edited:
  • #9
hello3719 said:
Apparently it seems that a number is defined as being an element of some defined set.
It is quite funny that "element" and "number" mean the same thing. So in fact we can define anything we want as a number !

No, no one said that- a "number" is an element of some specifically defined sets, not just any set! In order to be a "number system" the set must have other things associated with it- primarily operations such as addition or multiplication. Of course,mathematicians do, regularly, define such operations for all kinds of "things" so we could in a very specific way "define" anything we want as a number!
 
  • #10
HallsofIvy said:
No, no one said that- a "number" is an element of some specifically defined sets, not just any set! In order to be a "number system" the set must have other things associated with it- primarily operations such as addition or multiplication. Of course,mathematicians do, regularly, define such operations for all kinds of "things" so we could in a very specific way "define" anything we want as a number!
In fact to demonstrate such a thing one of my first lectures last year for a course started off by creating a set of cutlery and using them as numbers after defining addition and multiplication on them.
 

1. What is the modern rigorous definition of number?

The modern rigorous definition of number is based on the concept of sets, and defines numbers as elements of sets. This definition was developed by mathematicians in the late 19th and early 20th centuries, and is the foundation of modern number theory and algebra.

2. How is this definition different from previous definitions of number?

Previous definitions of number, such as the natural numbers or the real numbers, were based on intuitive concepts and did not have a rigorous mathematical foundation. The modern rigorous definition is based on set theory, which provides a rigorous and consistent framework for defining numbers.

3. What are the key principles of the modern rigorous definition of number?

The key principles of the modern rigorous definition of number are: numbers are defined as elements of sets, sets can be manipulated using operations such as union and intersection, and numbers can be compared and ordered using set inclusion.

4. How does the modern rigorous definition of number apply to different types of numbers?

The modern rigorous definition can be applied to all types of numbers, including natural numbers, integers, rational numbers, real numbers, and complex numbers. Each type of number is defined as a set with specific properties and operations.

5. What are the implications of the modern rigorous definition of number for other areas of mathematics?

The modern rigorous definition has had a significant impact on other areas of mathematics, such as algebra, geometry, and analysis. It has allowed for a more rigorous and consistent approach to studying numbers and their properties, and has led to the development of new mathematical concepts and theories.

Similar threads

Replies
9
Views
1K
Replies
7
Views
741
Replies
8
Views
1K
  • General Math
2
Replies
47
Views
2K
Replies
4
Views
1K
Replies
56
Views
4K
  • Set Theory, Logic, Probability, Statistics
Replies
13
Views
2K
  • General Math
Replies
5
Views
1K
  • General Math
Replies
10
Views
3K
Replies
11
Views
4K
Back
Top