"Number" Theory vs. The term Number vs. Diophantine equation - context meaning

[shahar]

"Number" Theory vs. The term Number vs. Diophantine equation - context meaning

Let me understand the terminology of math thing that I used.
OK.
Write for me true or false. And tell me why (if you want to)
(1) Number theory deal with concepts of Natural Number.
(2) Natural Number is positive number (x > 0).
(3) You can also define positive number as x belong to N and 0 belong to N or as can be define in terminology: non-negative integer number or: x >=0 N).
(4) In some "books" (1) and (2) statement can be use to answer by context but each book choose for you what you have to use.
(5) Equations that the results is Natural called Diophantine equation.
(6) There is not such terminology: Diophantine number. So the only terminology is Natural number.
(a) Is there a special field in math that deal only in the topic: real number,
If (a) is true so Why?
My "feeling is" because:
(1) historical reasons.
(2) you can't elaborate the terms of Natural number to Real Number? Because it a doing a "paradox"?
(b) Why there aren't tools to deal to with real numbers like Natural Number?
That all for now.

 

[HallsofIvy]

Let me understand the terminology of math thing that I used.
OK.
Write for me true or false. And tell me why (if you want to)
(1) Number theory deal with concepts of Natural Number.

Yes, this is true.

(2) Natural Number is positive number (x > 0).

No, this is false. A "natural number" is a positive integer. "1.5" is a positive number but not an integer so not a "natural number".

(3) You can also define positive number as x belong to N and 0 belong to N or as can be define in terminology: non-negative integer number or: x >=0 N).

You could but that would be a very peculiar and restrictive use of the word "number". The main problem is that the natural numbers do not include 0.

(4) In some "books" (1) and (2) statement can be use to answer by context but each book choose for you what you have to use.

If you mean that some text books include "0" as a natural number, I don't recall having seen such. The natural numbers, together with 0, are the "whole numbers".

(5) Equations that the results is Natural called Diophantine equation.

Diophantine equation are equations such that only natural numbers are accepted as solutions. Is that what you mean by "results"?

(6) There is not such terminology: Diophantine number. So the only terminology is Natural number.
(a) Is there a special field in math that deal only in the topic: real number,
If (a) is true so Why?

"Real analysis" deals with real numbers and functions of real numbers.

My "feeling is" because:
(1) historical reasons.
(2) you can't elaborate the terms of Natural number to Real Number? Because it a doing a "paradox"?
(b) Why there aren't tools to deal to with real numbers like Natural Number?
That all for now.

 

[Dr.Peterson]

Let me understand the terminology of math thing that I used.
OK.
Write for me true or false. And tell me why (if you want to)
(1) Number theory deal with concepts of Natural Number.
(2) Natural Number is positive number (x > 0).
(3) You can also define positive number as x belong to N and 0 belong to N or as can be define in terminology: non-negative integer number or: x >=0 N).
(4) In some "books" (1) and (2) statement can be use to answer by context but each book choose for you what you have to use.
(5) Equations that the results is Natural called Diophantine equation.
(6) There is not such terminology: Diophantine number. So the only terminology is Natural number.
(a) Is there a special field in math that deal only in the topic: real number,
If (a) is true so Why?
My "feeling is" because:
(1) historical reasons.
(2) you can't elaborate the terms of Natural number to Real Number? Because it a doing a "paradox"?
(b) Why there aren't tools to deal to with real numbers like Natural Number?
That all for now.

Terms vary somewhat in usage, so some of these questions have more than one answer, or need qualifications.

Some authors do define natural number in different ways; see http://mathworld.wolfram.com/NaturalNumber.html .

Diophantine equations in general have integer solutions, not necessarily natural numbers: https://en.wikipedia.org/wiki/Diophantine_equation .

The primary difference between integers and real numbers is that the integers are a discrete set, so that some things that can be done with real numbers (such as limits of a sequence) are impossible in general. Also, the integers lack multiplicative inverses. On the other hand, some things that can be done with integers (such as greatest common divisor) can't be defined for the real numbers. Their properties determine the kinds of math that can be done.

 

[JeffM]

i disagree with Halls of Ivy that there is a universal agreement on the definition of the natural numbers.

See for example

https://math.stackexchange.com/questions/283/is-0-a-natural-number

https://en.m.wikipedia.org/wiki/Natural_number

See in particular the reference to the International Standards Organization, the standard of which includes zero in the natural numbers.

http://mathworld.wolfram.com/NaturalNumber.html

There clearly is disagreement about the definition. (See the wolfram article about suggested notation to make clear which definition is being used.) Halls of Ivy may have strong views on which definition is preferable, but that does not warrant saying that contrary views do not exist. (I am perhaps being unfair because Halls said that he has never heard an opposing viewpoint.)

But definitions are a matter of convention and convenience. For many proofs in mathematics, it makes things more convenient to start N with 1. There are apparently fields where it makes things more convenient to start N with 0. But this is an argument about names, which ultimately are free creations of the human mind.

God dropped by the Garden of Eden one day and asked Adam to tell him how Adam had gone about naming the animals.

God pointed to one of the animals and asked Adam for its name. Adam said, "Monkey." God said, "Yes, nice choice."

God indicated another animal and posed the same question. Adam answered, "Mongoose." God said, "Excellent. Well done."

God gestured at a third animal and made the same query. Adam replied, "Hippopotamus."

God said, "Hippopotamus? But why?"

"Oh," announced Adam, "it looked like a hippopotamus to me."