Is 0^0 Equal to 1? An Explanation of Mathematical Concepts and Terminology

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Discussion Overview

The discussion centers around the mathematical expression 0^0 and whether it should be defined as equal to 1. Participants explore various definitions, conventions, and implications of this expression within different mathematical contexts, including algebra, set theory, and limits.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that 0^0 should be defined as 1 for convenience in mathematical contexts such as the binomial theorem and set theory.
  • Others argue that defining 0^0 as 1 leads to contradictions, particularly concerning division by zero and the rules of exponents.
  • A few participants highlight that while defining 0^0 as 1 can simplify expressions, it may also introduce complications and should be approached with caution.
  • Some contributions emphasize that 0^0 is an indeterminate form and cannot be consistently defined without leading to issues in certain mathematical frameworks.
  • There are mentions of the historical context of mathematical definitions and how they evolve, with some participants questioning the validity of changing definitions to suit convenience.
  • Several participants express that the disagreement over the definition of 0^0 reflects broader debates in mathematics about notation versus established rules.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether 0^0 should be defined as 1. There are multiple competing views, with some advocating for the definition as 1 and others rejecting it due to potential contradictions and the indeterminate nature of the expression.

Contextual Notes

Limitations in the discussion include the reliance on specific definitions and the potential for confusion when applying rules of exponents to 0^0. The implications of defining 0^0 as 1 are not universally accepted and vary depending on the mathematical context.

  • #91
micromass said:
OK, what do you consider to be "a number"? Do you think -1 is a number? What about pi or i?

I think that 0 is just a "landmark" on the field of numbers. All numbers are calculated with respect to 0. E.g. -1 is defined as a number with difference 1 from 0 in a certain direction (the negative direction).
 
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  • #92
dimension10 said:
Isn't a transcendental number defined as an irrational number that is not the multiple of any 2 or more numbers?

No, a transcendental number is defined as a number which is not the root of a polynomial

a_nX^n+...+a_1X+a_0

with an,...,a0 rational.

That said, it is true that q=0 is the only rational number such that q*pi is not transcendental. But that's hardly a good reason to exclude 0 from being a number!

Without 0, we wouldn't have integers (because what would 1+(-1) be?), and without integers, we would have very ugly things. And mathematics would be pretty useless.
 
  • #93
dimension10 said:
I think that 0 is just a "landmark" on the field of numbers. All numbers are calculated with respect to 0. E.g. -1 is defined as a number with difference 1 from 0 in a certain direction (the negative direction).

I really don't understand why you consider -1 a number, but 0 not. You can't have -1 without having 0!

And 0 is a much more natural concept than -1. Say that I have 8 apples and I give 8 apples away, then I have 0 apples left. What's the problem with this?
 
  • #94
micromass said:
I really don't understand why you consider -1 a number, but 0 not. You can't have -1 without having 0!

And 0 is a much more natural concept than -1. Say that I have 8 apples and I give 8 apples away, then I have 0 apples left. What's the problem with this?
language constraints thought. :frown: Negative numbers and zero have different linguistic hurdles to overcome.

With negative numbers, we have to overcome our language's preference to express everything in terms of "magnitude and direction". e.g. to reject the notion of "a surplus of -5" because one would simply say "a deficit of 5".

With zero, we have to overcome our language's preference for using negation. e.g. to reject the notion of "traveled a distance of 0 meters" because one would simply say "didn't travel".

(disclaimer: I am not a psychologist, nor have I actually studied the phenomenon. However, the observation does appear to fit observed fact, and the underlying cause is plausible)
 
  • #95
micromass said:
Without 0, we wouldn't have integers (because what would 1+(-1) be?), and without integers, we would have very ugly things. And mathematics would be pretty useless.

I am not saying that 0 does not exist. I am saying that 0 is not a number.
 
  • #96
For example, if you say that a=b+0, we do not say that a and be are unequal do we?
 
  • #97
2 is not a prime number. Every other prime number is odd, only 2 stands out. 2 ought to be odd. But it isn't.

So I would say that 2 is more of a "curiosity" than a prime number.
 
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  • #98
dimension10 said:
I am not saying that 0 does not exist. I am saying that 0 is not a number.

dimension10 said:
For example, if you say that a=b+0, we do not say that a and be are unequal do we?

Replace "b plus 0" with "b times 1". Ergo 1 is not a number!
 
  • #99
pwsnafu said:
Replace "b plus 0" with "b times 1". Ergo 1 is not a number!

b multiplied by 1 means 1 times of b. Means if you put b 1 time you get 1, right? So, 1 is a number.
 
  • #100
disregardthat said:
2 is not a prime number. Every other prime number is odd, only 2 stands out. 2 ought to be odd. But it isn't.

So I would say that 2 is more of a "curiosity" than a prime number.

But the definition of a prime number, is that, it has only 2 factors. 2 has 2 factors and is thus a prime numbers.
 
  • #101
dimension10 said:
But the definition of a prime number, is that, it has only 2 factors. 2 has 2 factors and is thus a prime numbers.

I have provided a sound argument as compared to yours, that "0 is not a number because it does not share a certain property with all other rationals". Believe me, every rational has some property not shared with all other rationals.

2 does similarly not share a property with all other primes, but this provides no basis for denying that 2 is a prime number.

What I'm most interested in is what you consider "a number" is. It is essentially a conventional label (which leaves it up to personal, or more correctly, collective opinion), and no argument could prove that 0 is not a number. This applies to the primes as well. The definition "a natural number which is divisible only by 1 and itself is a prime" corresponds to "an element of Q is a number". We don't leave out 2 as a prime by the same type of reason we don't leave out 0 as a number.
 
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  • #102
dimension10, what exactly is your definition of a number?
 
  • #103
dimension10 said:
Means if you put b 1 time you get 1, right?

:rolleyes: I think my brain melted...

disregardthat said:
What I'm most interested in is what you consider "a number" is.

micromass said:
dimension10, what exactly is your definition of a number?

Thirded. I too would like to know.
 
  • #104
dimension10 said:
b multiplied by 1 means 1 times of b. Means if you put b 1 time you get 1, right? So, 1 is a number.
No, you get b.
 
  • #105
disregardthat said:
2 is not a prime number. Every other prime number is odd, only 2 stands out.

Doesn't that make 2 rather odd?
 
  • #106
dimension10: you have an opinion on the meaning of the word "number". Your opinion differs from the established usage of the word. That fact will not change no matter how many rationalizations you give.


(moderator hat on) If you wish to continue to try and tell people that they ought to redefine number to match what you want things to be, then do one of the following two things:
  1. Go to another forum
  2. Find a really, strongly compelling reason why such a convention is actually useful

(hint: some minor technical condition you have arbitrarily decided should define "number" does not count as a "compelling reason", especially when lacking motivation)[/color]
 
  • #107
dimension 10, what you must realize is that calling something a number doesn't mean that this something suddenly is something magical. It's just another name. That's all it is.

If I would call the integers bazalbieba's, then I could, and everything would still work the same way. But mathematicians have not decided to use the word bazalbieba's, but to use the word number. It's just a name..

I agree that 0 is just a concept, but so are 1,2 and 3. These are all just concepts, which we happen to call "number". Like I said, you can call them something else if you want to, but mathematicians still use the word "number"...

When I call 6 a perfect number, it just means that the sum of it's proper divisors is 6. It means nothing more. It doesn't mean that 6 is suddenly perfection or something. It means exactly what the definition says it means, nothing more and nothing less.

Sorry, Hurk, if a reply wasn't allowed anymore, but I think this could end some confusion.
 

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