Is zero included in the "natural" numbers, or not?

[soroban]

From this thread:

Calculate how many groups of 4 natural numbes are able to solve this \(\displaystyle x_1+x_2+x_3+x_4\:=\:7\)

Solution is 120 but I don't understand the steps to solve it. . . I don 't agree.

We have a 7-inch board, marked off with six inch-marks.

.
  [SIZE=4][SIZE=3][SIZE=3]     * - - - - - - - - - - - - - *
       |   :   :   :   :   :   :   |
  [/SIZE][/SIZE][/SIZE][SIZE=4][SIZE=3][SIZE=3][SIZE=4][SIZE=3][SIZE=3]     |   :   :   :   :   :   :   |
  [/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][SIZE=4][SIZE=3][SIZE=3][SIZE=4][SIZE=3][SIZE=3]     |   :   :   :   :   :   :   |
  [/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][SIZE=4][SIZE=3][SIZE=3][SIZE=4][SIZE=3][SIZE=3][SIZE=4][SIZE=3][SIZE=3]     |   :   :   :   :   :   :   |[/SIZE][/SIZE][/SIZE]
    [/SIZE][/SIZE][/SIZE][/SIZE][SIZE=3][SIZE=3]   * - - - - - - - - - - - - - *
[/SIZE][/SIZE][/SIZE][/SIZE]

Choose three of the six inch-marks and cut along those lines.

Hence, there are: \(\displaystyle \displaystyle{6\choose3} \,=\,20\) outcomes.

Here they are: . \(\displaystyle \begin{array}{c}1114 \\ 1141\\ 1411\\4111\end{array}\quad
\begin{array}{c}1222 \\ 2122 \\ 2212 \\ 2221 \end{array}\quad
\begin{array}{c} 1123 \\ 1213 \\ 2113\\1231 \\2131\\2311 \end{array} \quad
\begin{array}{c} 1132\\1312\\3112\\1321\\3121\\3211 \end{array}\)

 

[pka]

Calculate how many group of 4 digit natural number are able to solve this x1+x2+x3+x4=7

Solution is 120 but I don't understand the steps to solve it. I don't agree.

We have a 7-inch board, marked off with six inch-marks.

  [SIZE=4][SIZE=3][SIZE=3]     * - - - - - - - - - - - - - *
       |   :   :   :   :   :   :   |
  [/SIZE][/SIZE][/SIZE][SIZE=4][SIZE=3][SIZE=3][SIZE=4][SIZE=3][SIZE=3]     |   :   :   :   :   :   :   |
  [/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][/SIZE]

[SIZE=4][SIZE=3][SIZE=3][SIZE=4][SIZE=3][SIZE=3]     |   :   :   :   :   :   :   |
  [/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][SIZE=4][SIZE=3][SIZE=3][SIZE=4][SIZE=3][SIZE=3][SIZE=4][SIZE=3][SIZE=3]     |   :   :   :   :   :   :   |[/SIZE][/SIZE][/SIZE]
    [/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][/SIZE][SIZE=3][SIZE=3]   * - - - - - - - - - - - - - *[/SIZE][/SIZE]

Choose three of the six inch-marks and cut along those lines.
Hence, there are: \(\displaystyle \displaystyle{6\choose3} \,=\,20\) outcomes.
Here they are: . \(\displaystyle \begin{array}{c}1114 \\ 1141\\ 1411\\4111\end{array}\quad
\begin{array}{c}1222 \\ 2122 \\ 2212 \\ 2221 \end{array}\quad
\begin{array}{c} 1123 \\ 1213 \\ 2113\\1231 \\2131\\2311 \end{array} \quad
\begin{array}{c} 1132\\1312\\3112\\1321\\3121\\3211 \end{array}\)

No, 120 is correct. You fell into an old mathematics educator's trap of not thinking of zero as a natural number.

 

[soroban]

No, 120 is correct. You fell into an old mathematics educator's trap of not thinking of zero as a natural number.

I'm not familiar with that "old mathematical educator's trap".

For me, Natural Numbers have never included zero.
These are the "counting numbers", which begin with "one".

 

[pka]

For me, Natural Numbers have never included zero.
These are the "counting numbers", which begin with "one".

You have made my point for me. mathematics educators make distinction among: natural numbers, whole numbers, & counting numbers.

A mathematician will not have such an artificial partition . Here is the standard construction of \(\displaystyle \mathbb{N}\) in academic mathematics:
\(\displaystyle \begin{align*}0&=\emptyset \\1&=0\cup \{0\}\\2&=1\cup\{1\}\\&~~\vdots \end{align*}\)

 

[stapel]

...Hence, there are: \(\displaystyle \displaystyle{6\choose3} \,=\,20\) outcomes.

Here they are: . \(\displaystyle \begin{array}{c}1114 \\ 1141\\ 1411\\4111\end{array}\quad
\begin{array}{c}1222 \\ 2122 \\ 2212 \\ 2221 \end{array}\quad
\begin{array}{c} 1123 \\ 1213 \\ 2113\\1231 \\2131\\2311 \end{array} \quad
\begin{array}{c} 1132\\1312\\3112\\1321\\3121\\3211 \end{array}\)

Please stop posting fully-worked solutions.

I'm not familiar with that "old mathematical educator's trap".

For me, Natural Numbers have never included zero.
These are the "counting numbers", which begin with "one".

Unfortunately, not all use the same definition. This is why the original poster was specifically asked for the book's definition:

Also, in your book, is zero included in the "natural" numbers?

It's just possible that the purpose of this forum is to encourage students to learn, not to enable others to boast of how smart they are.

:roll:

 

[mmm4444bot]

I agree with Stapel and Denis. It's handy to make distinctions between specific subsets of the Reals. People are free to name specific subsets whatever they like. There are no global definitions for many parameters in mathematics. We serve a very diverse membership. Makes little sense to interject such opinions in members' threads, but everyone is welcome to toss their two cents worth anywhere on the Odds & Ends board (in accordance with the site rules), as this board is an appropriate location to posit such arguments. :cool:

 

[pka]

Who gives a third anyway!
A bit like 1 not "being" a prime number:
if it was, does it mean the end of the world .

This is the night for training is mathematically correct writing.

To say that "a prime number is a integer having only one and itself as a divisor" makes one a prime number.
(Think about why that is the case)!

The correct definition is: "a prime number is a integer having exactly two divisors."
(How does that exclude one from the list of primes?)

 

[pka]

There are no global definitions for many parameters in mathematics. We serve a very diverse membership. Makes little sense to interject such opinions in members' threads, but everyone is welcome to toss their two cents worth anywhere on the Odds & Ends board (in accordance with the site rules), as this board is an appropriate location to posit such arguments.

You would be hard pressed to find a mathematician who would not agree that positive & non-positive are global definitions. So why not use those adjectives to replace the artificial terms: whole numbers, counting numbers, or natural numbers. The set of non-negative integers contains zero and positive integers do not. If you want to write questions used for mathematics contests, you must learn these standards.

Just because you may serve a diverse membership does not relieve you of the obligation to stamp out ignorance.

 

[Otis]

... mathematics educators make distinction among: natural numbers ... & counting numbers.

I haven't heard of that one.

What is the distinction?

 

[lookagain]

To say that "a prime number is a integer having only one and itself as a divisor" makes one a prime number.
(Think about why that is the case)!

The correct definition is: "a prime number is a integer having exactly two divisors."
(How does that exclude one from the list of primes?) \(\displaystyle \ \ \ \ \) <==== That would make one ** a prime number.

** The divisors of 1 are -1 and 1.

"Divisors can be negative as well as positive, although sometimes the term is restricted to
positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4,
but only the positive ones (1, 2, and 4) would usually be mentioned."


Source:

https://en.wikipedia.org/wiki/Divisor

A correct definition is "a prime number is a positive integer having exactly two positive divisors."

 

[mmm4444bot]

... why not use [positive & non-positive] to replace the artificial terms: whole numbers, counting numbers, or natural numbers.

Because we don't teach mathematics by introducing so much of the subject at once.

Counting numbers (and the concept of Whole numbers) come first; the introduction of negative numbers comes later. Arguments concerning merits of terminology come even later.

If you want to write questions used for mathematics contests, you must learn these standards

The thread in which you posted is not a contest. :?

[Serving] a diverse membership does not relieve [one] of the obligation to stamp out ignorance.

Differing opinions (i.e., contexts) are not synonymous with ignorance. Please refrain from such stamping in other people's threads. Thank you! :cool:

 

[stapel]

If you want to write questions used for mathematics contests, you must learn these standards.

If, on the other hand, you're wanting to help students complete their homework correctly, you must learn that these differences are common, and point the student to his/her particular book's particular definitions, pointing out that there are differences between books, so they're aware.

Just because you may serve a diverse membership does not relieve you of the obligation to stamp out ignorance.

You're welcome.

 

[Ishuda]

I haven't heard of that one.

What is the distinction?

Yes and no and maybe. I am positive that that might be the case and, if fact, if there are only two cases, yes and no, then either yes or no unless, of course the two cases are no and maybe in which case there can no yes unless a mistake has been made and t