Why is 0^0=?

A

AvgStudent

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Hi,
If [imath]0^a=0[/imath], and [imath]a^0=1[/imath], then what is [imath]0^0[/imath] equal to?
I've read multiple sources, but they all give different answers. Some say 1, and some say it depends on the context of the problem. I do not understand the ambiguity of the answer. Can someone provide me with further insight?
 
It's been discussed here just recently:

What is a quadratic polynomial?

The definition is mostly good, for a polynomial. However, they need to include a statement about the exponents. As far as defining the function part, their definition is completely lacking. By the way, does the word 'quadratic' appear in that book's index? :) No, it doesn't. It was the first...
www.freemathhelp.com

In calculus, it's called an indeterminate form; but in other contexts, it can make sense to assign a specific value, 1. If you always did that, though, calculus wouldn't work!
 
AvgStudent said:
Hi,
If [imath]0^a=0[/imath], and [imath]a^0=1[/imath], then what is [imath]0^0[/imath] equal to?
I've read multiple sources, but they all give different answers. Some say 1, and some say it depends on the context of the problem. I do not understand the ambiguity of the answer. Can someone provide me with further insight?
Click to expand...
There are reasons to say 0^0 = 1 and reasons to say 0^0 = 0. that is why it is an indeterminate form. (Though I once met a String Theorist who lambasted a student for saying that. She had decided that 0^0 = 1 and that was that.)

-Dan
 
Many students and most mathematicians treat mathematics as dealing with Plato’s reality. If you have not read Plato, you may not understand Platonic reality: in that reality, that you do not even exist; you are merely a shadow of a hermaphrodite combining male and female attributes.

Now personally, I find Platonism to be silliness compounded with confusion. The meaning of [imath]x^0[/imath] depends on how we define the term. In some cases, it is convenient to define it so that [imath]0^0[/imath] is equal to 1; in other cases, it is convenient to say that [imath]0^0[/imath] is indeterminate, just like 0/0.

In other words, we determine what we mean by [imath]0^0[/imath]. Sensible people (meaning those who have not had their brains scrambled by Plato) choose the definition that is appropriate to what they are discussing. So, sometimes those sensible people use one definition and sometimes another. For example, if I am talking about horses, “lead” describes an activity, but if I am talking about materials, “lead” describes a metal.
 
JeffM said:
Many students and most mathematicians treat mathematics as dealing with Plato’s reality. If you have not read Plato, you may not understand Platonic reality: in that reality, that you do not even exist; you are merely a shadow of a hermaphrodite combining male and female attributes.

Now personally, I find Platonism to be silliness compounded with confusion. The meaning of [imath]x^0[/imath] depends on how we define the term. In some cases, it is convenient to define it so that [imath]0^0[/imath] is equal to 1; in other cases, it is convenient to say that [imath]0^0[/imath] is indeterminate, just like 0/0.

In other words, we determine what we mean by [imath]0^0[/imath]. Sensible people (meaning those who have not had their brains scrambled by Plato) choose the definition that is appropriate to what they are discussing. So, sometimes those sensible people use one definition and sometimes another. For example, if I am talking about horses, “lead” describes an activity, but if I am talking about materials, “lead” describes a metal.
Click to expand...
The example of the word "lead" is perfect. I didn't know that there can be ambiguity in math...
 
AvgStudent said:
The example of the word "lead" is perfect. I didn't know that there can be ambiguity in math...
Click to expand...
It is not so much ambiguity as it is that consequences flow from definitions. Definitions are crucial. Sometimes, it happens that one definition is useful in some contexts and a different definition is useful in other contexts. When that happens, it is silly to say “this definition is right, and the other is wrong.” Instead, you need to ask which definition is being used and whether it is the more useful in this specific context.

Fortunately, there are very few cases in mathematics where more than one definition is useful. This allows people to apply Platonism to mathematics. I am not a mathematician. I was a banker: circumstances alter cases. I rejected every loan presented on Platonist principles.
 
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Hello, this is an interesting question.
Multiply 0 as many times as you like, you get 0.

0 ^ 3 = 0

0 ^ 2 = 0

0 ^ 1 = 0

0 ^ 0 = 0

However, any number raise to the power zero is equal to 1

3 ^ 0 = 1

2 ^ 0 = 1

1 ^ 0 = 1

0 ^ 0 = 1

That's why there is a dispute about the value of 0^0

The mathematical phrase 00 (zero to the power of zero) has no agreed-upon value. The most common options are 1 or leaving the phrase undefined, both of which have reasons based on the context. The commonly accepted value in algebra and combinatorics is 0 ^ 0 = 1, whereas in mathematical analysis, the expression is sometimes left undefined. This expression is handled differently in different programming languages and software.
0 ^ 0 is an "indeterminate form," according to certain Calculus textbooks. When assessing a limit of the form 0 ^ 0, you must be aware that such limits are known as "indeterminate forms," and that they must be evaluated using a particular approach such as L'Hopital's rule. Otherwise, 00 = 1 appears to be the most practical option for 0 ^ 0. This rule allows us to broaden concepts in areas of mathematics where treating 0 as a special case would otherwise be necessary. The value 0 ^ 0 denotes a break in the XY function. More significantly, keep in mind that a function's value and limit do not have to be the same, and functions do not have to be continuous if that serves a purpose.
 
AvgStudent said:
I didn't know that there can be ambiguity in math
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The sooner you accept it, the better off you'll be.

There's no single set of definitions compatible with human nature.

?

[imath]\;[/imath]
 
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