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What is 0^0? - George Baron

Author(s): 
Michael Huber and V. Frederick Rickey

Defining powers is often carelessly done. Almost thirty years before Libri's first paper, George Baron published "A short Disquisition, concerning the Definition, of the word Power, in Arithmetic and Algebra" in The Mathematical Correspondent (1804). In this paper [1], Baron begins the discussion with the following definition:

The powers of any number, are the successive products, arising from unity, continually multiplied, by that number.

As an example, he writes that 1 × 5 = 5, which is the first power of 5, and 1 × 5 × 5 = 25, which is the second power of 5, etc. The first, second, etc., powers are then conveniently expressed as 51, 52, etc. In the same manner, the powers of any number x might be represented as x1, x2, etc., in which x1 = 1 × x, x2 = x1 × x, etc. After stating a few corollaries, Baron writes:

Let us, therefore, next inquire, whether the same definition, will not lead us to a clear and intelligible solution, of the mysterious paradoxes, resulting from the common definition, when applied, to what is denominated, the nothingth power of numbers.

Baron then addresses the rules for dividing powers (look back to the argument from the high school text), but he develops a different conclusion:

If the multiplication by x, be abstracted from the first power of x, by means of division; the power will become nothing but the unit will remain: for \(\frac{x^1}{x} = \frac{1\times x}{x} =1,\) and hence it is plain that x0 = 1, when x represents any number whatever. But since the number x, is here unlimited with regard to greatness, it follows, that, the nothingth power of an infinite number is equal to a unit.

Baron gives credit to both William Emerson (1780) [3] and Jared Mansfield (1802) [9] who wrote on the subject of "nothing." Baron takes their arguments one step further and postulates that the number x can be any number, great or small:

To pursue the application of our definition, to quantity in the ultimate extremity of smallness, let us suppose x to represent any fractional quantity; or in other words, let x denote any magnitude, expressed in numbers, by means of some part of its measuring unit: then by the definition x1 = 1 × x. Let now this multiplication by x, be abstracted; and for the reasons heretofore advanced, we have x0 = 1. Now since x here represents a fractional quantity, independent of any limitation, in respect to smallness; we may therefore suppose x, by means of continual diminution, or decrease, to pass from its present value, through every degree of smallness, until it become nothing; then it will be evident, that, during this diminution or decrease of x, x0 will continue equal to an invariable unit; and that precisely at the instant, when x becomes nothing, x0, or 00 = 1.

Baron never mentions the term indeterminate form, and he in fact ends his treatise with the following:

Also, since x0 = 1, whatever be the value of x; of consequence; in every system of logarithms, the logarithm of 1 = 0.

Michael Huber and V. Frederick Rickey, "What is 0^0? - George Baron," Loci (July 2012)

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