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When calculus books state that 00 is an indeterminate form, they mean that there are functions f(x) and g(x) such that f(x) approaches 0 & g(x) approaches 0 as x approaches 0, and that one must evaluate the limit of g(x) as x approaches 0. But what if 0 is just a number? Then, we argue, the value is perfectly well-defined, contrary to lớn what many texts say. In fact, 00 = 1!

When calculus books state that 00 is an indeterminate form, they mean that there are functions f(x) and g(x) such that f(x) approaches 0 & g(x) approaches 0 as x approaches 0, and that one must evaluate the limit of g(x) as x approaches 0. But what if 0 is just a number? Then, we argue, the value is perfectly well-defined, contrary khổng lồ what many texts say. In fact, 00 = 1!

Pick up a high school mathematics textbook today và you will see that 00 is treated as an indeterminate form. For example, the following is taken from a current Thủ đô New York Regents text <6>:

We reCall the rule for dividing powers with like bases:

xa/xb = xa-b (x not equal to lớn 0)

(1)

Therefore, in order for x0 lớn be meaningful, we must make the following definition:

x0 = 1 (x not equal to lớn 0)

(4)

Since the definition x0 = 1 is based upon division, & division by 0 is not possible, we have stated that x is not equal to 0. Actually, the expression 00 (0 lớn the zero power) is one of several indeterminate expressions in mathematics. It is not possible khổng lồ assign a value khổng lồ an indeterminate expression.

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Calculus textbooks also discuss the problem, usually in a section dealing with L"Hospital"s Rule. Suppose we are given two functions, f(x) & g(x), with the properties that (lim_x ightarrow a f(x)=0) và (lim_x ightarrow a g(x)=0.) When attempting khổng lồ evaluate <f(x)>g(x) in the limit as x approaches a, we are told rightly that this is an indeterminate form of type 00 and that the limit can have sầu various values of f and g. This begs the question: are these the same? Can we distinguish 00 as an indeterminate form and 00 as a number? The treatment of 00 has been discussed for several hundred years. Donald Knuth <7> points out that an Italian count by the name of Guglielmo Libri published several papers in the 1830s on the subject of 00 & its properties. However, in his Elements of Algebra, (1770) <4>, which was published years before Libri, Euler wrote,

As in this series of powers each term is found by multiplying the preceding term by a, which increases the exponent by 1; so when any term is given, we may also find the preceding term, if we divide by a, because this diminishes the exponent by 1. This shews that the term which precedes the first term a1 must necessarily be a/a or 1; và, if we proceed according khổng lồ the exponents, we immediately conclude, that the term which precedes the first must be a0; & hence we deduce this remarkable property, that a0 is always equal khổng lồ 1, however great or small the value of the number a may be, và even when a is nothing; that is lớn say, a0 is equal to 1.

More from Euler: In his Introduction khổng lồ Analysis of the Infinite (1748) <5>, he writes :

Let the exponential lớn be considered be az where a is a constant và the exponent z is a variable .... If z = 0, then we have sầu a0 = 1. If a = 0, we take a huge jump in the values of az. As long as the value of z remains positive sầu, or greater than zero, then we always have sầu az = 0. If z = 0, then a0 = 1.

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Euler defines the logarithm of y as the value of the function z, such that az = y. He writes that it is understood that the base a of the logarithm should be a number greater than 1, thus avoiding his earlier reference to a possible problem with 00.