Any line of the form x^ax (a sloping line) converges on 1 when x = 0, without dispute.

Edging along the line of y=0, and x ->0, is a kind of division by zero.

]]>where is a natural number and is an arbitrary complex number. This function is used to define polynomials and power series. With this function we must have for the function to be well-behaved.

where is an integer and is a nonzero complex number.

where is a complex number is the exponential function. The at the bottom is just a symbol, though by abuse of notation some people define the number , which makes this power compatible with other power functions.

where is a positive real number and is a complex number. We can also extend this to be defined for any in a set where can be defined consistently, or even to all nonzero if we make this a multi-valued function. We can also extend the function continuously so that for all positive real numbers . However, there is no way to extend this function continuously to (0,0), so with this function is undefined.

These functions are frequently confused because in any point where two of them are defined, they give the same value. However, it is crucial to distinguish them when analyzing the expression : According to the first definition it is , but according to the rest it is undefined. Finally, applying this to your riddle, I must object to your solution as well: There is no single power function such that looks like that.

]]>Correction. Last line should be: The only value that works for is .

]]>Consider multiplying repeatedly by . We can agree that . Then . The trouble is, this equation will work for any value of . You don’t run into this problem with repeated adding ‘s. The only value that works for $0+0$ is .

]]>That does work. And it looks like Matt Harden wrote it that way on G+.

If you ever need to write a URL with characters other than letters, numbers, dash, underscore, period or tilde (the “unreserved characters” in the table here), and they aren’t playing a role in the syntax of the URL (like a forward slash, a colon, or any of the other “reserved characters” listed in the same article), then the most robust way to write those characters is with a percent sign followed by two hexadecimal digits giving the character code.

You can even do this for characters with codes beyond the normal 8-bit range, but then you need to convert the code into UTF-8 and express it as multiple bytes. This is a bit of a pain in the neck, and it can make some links to Wikipedia articles tedious to write, because they like using em-dashes and accented characters that can only be expressed this way.

]]>I wonder if WordPress will allow https://www.google.com/search?q=0%5E0

That’s “escaping” the symbol ^ as a hexadecimal version of its character code.

]]>Knuth writes: “Some textbooks leave the quantity 0^0 undefined, because the functions x^0 and 0^x have different limiting values when x decreases to 0. But this is a mistake. We must define x^0 = 1, for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = -y. The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant.”

Matt Harden replied:

Google must have listened to Knuth: https://www.google.com/search?q=0^0

If you do a Google search on “0^0”, you’ll see makes a definitive claim about what 0^{0} equals. This will annoy the ‘undefined’ camp.

However, if you click on the above link you don’t get to the same page. Apparently WordPress gags when it sees the symbol ^ in a URL. Or maybe it just doesn’t like 0^{0}.

@David: that’s the correct answer, but a much more conceptual derivation is possible. The determinant of a linear transformation from a finite-dimensional vector space to itself is the scalar by which acts on the top exterior power of , which is -dimensional. If then the top exterior power is the zeroth one, which is the unit (that is, the underlying field, as a -dimensional vector space), with acting by multiplication by .

A related but more concrete argument is that is the unique answer consistent with cofactor expansion for a matrix.

]]>This is the sense in which the integers have to be considered on their own in their fully discrete glory instead of as a subset of the continuum.

]]>The horizontal arrows demarcate the first quadrant, that is, the region where and You can see that the function being graphed, namely equals zero along the axis, since for . And if you look harder you can see that it takes a constant positive value along the axis, consistent with the fact that for

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