The number zero is essential in mathematics and science, serving as the additive identity in various mathematical systems, including group theory, rings, and fields. Without zero, fundamental concepts such as differentiation, integration, and the definition of equality become problematic, as equality is often defined by the condition that the difference equals zero. The absence of zero complicates number representation and calculations, as demonstrated by the limitations of systems like Roman numerals. Overall, zero is crucial for the completeness and functionality of mathematical frameworks.
PREREQUISITESMathematicians, educators, students in advanced mathematics, and anyone interested in the foundational concepts of mathematics and their implications in scientific disciplines.
{~} said: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.{~} said: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.{~} said: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?
Rings, fields, vector spaces need aan additive identity.{~} said: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?
While anecdotical, this example shows that without 0, you have to devise tricks for the simplest of things,{~} said:What does it mater which number the football game starts at? If it is the same number isn't it still a fair match?
{~} said: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?
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!{~} said: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?
symbolipoint said:...And God created zero and permitted man to discover it, and this pleased God that it was good.
symbolipoint said:...And God created zero and permitted man to discover it, and this pleased God that it was good.
I'm pretty sure symbolipoint is speaking tongue in cheek...WWGD said:Would you please leave the religious references for...a religious forum?
Ah, good deal, my non-religious refuge here at PF is safe, phew.Mark44 said:I'm pretty sure symbolipoint is speaking tongue in cheek...
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,...WWGD said:Ah, good deal, my non-religious refuge here at PF is safe, phew.
"God made the integers, all else is the work of man."symbolipoint said: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,...
The integers I use were made by Peano just a couple of years after Kronecker wrote that.pwsnafu said:"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?pwsnafu said:"God made the integers, all else is the work of man."
Leopard Kronecker, 1886
WWGD said:Isn't Kronecker one of the Mathematicians that pushed Cantor to a breakdown because of his insults and accusations?
MrAnchovy said:The integers I use were made by Peano just a couple of years after Kronecker wrote that.
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?micromass said:Made? Peano just defined a very useful set of axioms. He did not construct or make anything however.
Just not too decent of a person, it seems.micromass said:Yes. And Kronecker still remains a brilliant mathematician.
WWGD said:Just not too decent of a person, it seems.
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.Physicalchemist said: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.
Consider f(x) = 1, a function whose graph is a horizontal line.{~} said: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.