## RolyPoly one year ago "The numbers 1, 1+1=2, 2+1=3, etc. are said to be natural; it is assumed that none of these numbers is zero". Why do we have/need such an assumption?

1. kc_kennylau

Because it makes that $$\forall A,B\in\mathbb N, \frac AB\in\mathbb N$$. @ikram002p am I right?

2. RolyPoly

We can also have A = 1, B = 3, but 1/3 $$\notin \mathbb N$$?

3. kc_kennylau

then i'm not right lol

4. kc_kennylau

oh, i see why 0's not natural, coz you can't have 0 things.

5. kc_kennylau

NATURal numbers are numbers that occur in the NATURE.

6. RolyPoly

-_- You can have nothing, right?

7. RolyPoly

There's something you CAN'T find in the nature though!

8. kc_kennylau

How do you count?

9. RolyPoly

From 0

10. kc_kennylau

cool

11. kc_kennylau

"There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term designates the non-negative integers {0, 1, 2, 3, ...}. The former definition is the traditional one, with the latter definition having first appeared in the 19th century." https://en.wikipedia.org/wiki/Natural_number

12. Spacelimbus

Not sure if I understand the concept of this question. But I remember the following words from my Professor "First thing, you always do when you buy a boot about Mathematics, check on their definition of $$\mathbb{N}$$" What he meant by that, was that whether or not $$0 \in \mathbb{N}$$ or not is still a big discussion. For references check for example Zorich Analysis 1, Blatter Analysis and so on. Also consider the notation $$\mathbb{N}_0$$

13. ikram002p

cuz 0 is the identity of any groupe on the binary operation of addition , so it must be UNIQE its thm from (elementary properties of groups)

14. RolyPoly

The footnote of this quotation was as follows: "Given two elements N and E, say, we can construct a field by the rules N+N=N, N+E=E, E+E=N, N$$\cdot$$N=N, N$$\cdot$$E=N, E$$\cdot$$E=E. Then, in keeping with our notation, we should write N=0, E=1 and hence 2=1+1=0. To exclude such number systems, we require that all natural field elements be nonzero" Basically, I don't know what's going on here.

15. kc_kennylau

The problem is that the question itself is already wrong. It isn't universal to exclude 0 from the list of natural numbers.

16. RolyPoly

But it also isn't universal to include 0 as well?

17. RolyPoly

Warning: Please ignore the following. ---------------------------------------------------------------- Should 0 be included in the set of natural numbers? ---------------------------------------------------------------- <Saying I> No, $$0\notin \mathbb N$$ The numbers 1, 1+1=2, 2+1=3, ... are said to be natural. Given two elements N and E, we can struct a field by the rules (i) N+N=N (ii) N+E=E (iii) E+E=N (iv) N$$\cdot$$N = N (v) N$$\cdot$$E = N (vi) E$$\cdot$$E = E Then in keeping with our notation, we should be N=0, E=1. From (iii), E+E = N, so, we have 1+1 = 0 However from the first statement, we have 1+1=2 $$\ne$$ 0, so we need to exclude 0. Problems: 1) Why would we construct such a field with rules (i) to (vi), particularly with rule (iii)? 2) What would happen if we picked some other values for N and E?