# Why zero?

1. Dec 26, 2015

### {~}

Why do we have the number zero? What makes it important? Can we create math like system that don't use zero but are nonetheless useful. What limits or biases does having or not having zero create when we do science?

2. Dec 26, 2015

### Dr. Courtney

Calculus would be hard without the limit as stuff goes to zero.

3. Dec 26, 2015

### Svein

• Number representations would be hard (just try do do calculations in the roman number system which has no zero)
• Group theory (and everything based on it) would not exist
• Since equality is defined as "difference equals zero" in several mathematical systems, equality would be hard to define
... and so on.

4. Dec 27, 2015

### jambaugh

One way to define numbers in a pragmatic, constructive way is as actions. This especially gives you meaning for the negatives. For example the number +3 indicates incrementing in the positve direction 3 steps, -3 is the reverse action and 0 is the identity, the do-nothing action. The zero is necessary for completeness when numbers are used in this way (as elements of a group under addition). It is in fact referred to as the additive identity for this reason.

5. Dec 27, 2015

### symbolipoint

God gave us the Whole Numbers. He did not give us zero because he did not need to. Humans are smart enough to find what zero means, using our own efforts.

6. Dec 27, 2015

### PeroK

I'd say 0 is a lot more important than, say, 362. It would be easier to get rid of that one.

When did you last look for $f(x) = 362$?

7. Dec 27, 2015

### PeroK

If you didn't have 0, then I guess every football match would have to start at 1-1? Who would be credited with scoring those goals?

8. Dec 27, 2015

### {~}

What good is the additive identity in symbolic math? Is it actually necessary? Why cant our number system be based on multiplication rather than addition and use 1 as the identity to which transformations are made to get other numbers? I think I read somewhere that that would work under group theory. Why does group theory require zero?

What does it mater which number the football game starts at? If it is the same number isn't it still a fair match?

9. Dec 27, 2015

### Samy_A

Rings, fields, vector spaces need aan additive identity.

While anecdotical, this example shows that without 0, you have to devise tricks for the simplest of things,

10. Dec 27, 2015

### PeroK

What world are you living in that you want to replace addition with multiplication? They are different you know!

Back to football, if you start at 1 and multiply by 1 every time you score a goal, the score would always be 1-1. There's a certain fairness in that, I guess.

11. Dec 27, 2015

### Svein

It does not, You can create a group using multiplication as the operation - but you will soon run into problems. First - you cannot use the integers, since that would not give you an inverse, You have to use the rationals - and you will run into a lot of problems. But feel free to try!

12. Dec 27, 2015

### WWGD

No number system using addition would be closed under it : how would you define a-a in such system? It may be an interesting exercise to show that
0 (the right type of 0 object) is the only viable definition for the expression a-a.

13. Dec 27, 2015

### symbolipoint

....And God created zero and permitted man to discover it, and this pleased God that it was good.

14. Dec 27, 2015

### WWGD

Would you please leave the religious references for....a religious forum?

15. Dec 27, 2015

### Staff: Mentor

I'm pretty sure symbolipoint is speaking tongue in cheek...

16. Dec 27, 2015

### WWGD

Ah, good deal, my non-religious refuge here at PF is safe, phew.

17. Dec 27, 2015

### symbolipoint

My higher-level human feelings were active on this, so what I said was not any quotation from religious sources. The remark I made comes from the sometimes quoted statement that God gave us the integers,...

18. Dec 27, 2015

### pwsnafu

"God made the integers, all else is the work of man."
Leopard Kronecker, 1886

19. Dec 27, 2015

### MrAnchovy

The integers I use were made by Peano just a couple of years after Kronecker wrote that.

20. Dec 27, 2015

### WWGD

Isn't Kronecker one of the Mathematicians that pushed Cantor to a breakdown because of his insults and accusations?

21. Dec 28, 2015

### micromass

Yes. And Kronecker still remains a brilliant mathematician.

22. Dec 28, 2015

### micromass

Made? Peano just defined a very useful set of axioms. He did not construct or make anything however.

23. Dec 28, 2015

### MrAnchovy

How would you describe this part of his work? Or is it your point that the integers, or rather the natural numbers, can not be considered constructed by the Peano axioms without Dedekind's isomorphism proof?

24. Dec 28, 2015

### WWGD

Just not too decent of a person, it seems.

25. Dec 28, 2015

### micromass

Many mathematicians weren't for some reason.