Exploring the Role of Zero in Math and Science

[{~}]

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?

 

[Dr. Courtney]

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

 

[Svein]

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?

• 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.

 

[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.

 

[symbolipoint]

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?

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.

 

[PeroK]

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?

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 $f(x) = 362$?

 

[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?

 

[{~}]

What good is the additive identity in symbolic math? Is it actually necessary? Why can't 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?

 

[Samy_A]

What good is the additive identity in symbolic math? Is it actually necessary? Why can't 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?

Rings, fields, vector spaces need aan additive identity.

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

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

 

[PeroK]

Why can't 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?

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.

 

[Svein]

Why can't 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?

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!

 

[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.

 

[symbolipoint]

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

 

[WWGD]

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

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

 

[Mark44]

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

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

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

 

[WWGD]

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

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

 

[symbolipoint]

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

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,...

 

[pwsnafu]

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,...

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

 

[pbuk]

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

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

 

[WWGD]

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

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

 

[micromass]

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

Yes. And Kronecker still remains a brilliant mathematician.

 

[micromass]

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

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

 

[pbuk]

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

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?

 

[WWGD]

Yes. And Kronecker still remains a brilliant mathematician.

Just not too decent of a person, it seems.

 

[micromass]

Just not too decent of a person, it seems.

Many mathematicians weren't for some reason.

 

[Physicalchemist]

No zero? Means no difference quotient, which means no differentiation, which means no integration, which means an extremely difficult way to calculate area between the x-axis and a curve/line, which means no calculus, which means I have nothing to learn in seventh grade in math.
also, zero is both a real number and an imaginary number because it can be written as 0 and 0i. so it is also a complex number 0 + 0i. There would also be basically no definition for e (for the Taylor series) which is the sum from ZERO to infinity (or sometimes thought of as x/ZERO where x is positive) x^n/n! with n being the variable and x being the exponent in e^x. Plus, it would be impossible to define the x and y-axis on the coordinate plane if a point was on those axes.
All in all, without zero everything would fall apart.

 

[{~}]

No zero? Means no difference quotient, which means no differentiation, which means no integration, which means an extremely difficult way to calculate area between the x-axis and a curve/line, which means no calculus, which means I have nothing to learn in seventh grade in math.
also, zero is both a real number and an imaginary number because it can be written as 0 and 0i. so it is also a complex number 0 + 0i. There would also be basically no definition for e (for the Taylor series) which is the sum from ZERO to infinity (or sometimes thought of as x/ZERO where x is positive) x^n/n! with n being the variable and x being the exponent in e^x. Plus, it would be impossible to define the x and y-axis on the coordinate plane if a point was on those axes.
All in all, without zero everything would fall apart.

It seems to me slope is computed as a ratio. I don't see how this explicitly requires zero. In modern calculus the rise and run both approach zero but zero itself is a no go for the denominator at least. I don't see why we can't keep the numerator and denominator largish but increasingly more precise, perhaps by multiplying both by some common factor related to how precise an increment we are talking about. Probably I am mad but I don't see specifically why it wouldn't work.

 

[WWGD]

In a structure where you allow for subtraction ( as , maybe, the operational inverse of addition, as a group), how would you define, for any element x, the expression : x-x , or x+(-x) ?

 

[Mark44]

It seems to me slope is computed as a ratio. I don't see how this explicitly requires zero. In modern calculus the rise and run both approach zero but zero itself is a no go for the denominator at least. I don't see why we can't keep the numerator and denominator largish but increasingly more precise, perhaps by multiplying both by some common factor related to how precise an increment we are talking about. Probably I am mad but I don't see specifically why it wouldn't work.

Consider f(x) = 1, a function whose graph is a horizontal line.

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)} h = \lim_{h \to 0} \frac{1 - 1} h = \lim_{h \to 0} \frac 0 h = 0$
The line is horizontal, thus we would expect its slope (i.e., f'(x)) to be zero.