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When calculus books state that 00 is an indeterminate form, they mean that there are functions f(x) và g(x) such that f(x) approaches 0 & g(x) approaches 0 as x approaches 0, & 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!


When calculus books state that 00 is an indeterminate khung, 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, & 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 what many texts say. In fact, 00 = 1!


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

We reGọi the rule for dividing powers with like bases:

xa/xb = xa-b (x not equal to lớn 0)(1)

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

x0 = 1 (x not equal lớn 0)(4)

Since the definition x0 = 1 is based upon division, & division by 0 is not possible, we have sầu stated that x is not equal to 0. Actually, the expression 00 (0 to the zero power) is one of several indeterminate expressions in mathematics. It is not possible lớn 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) and (lim_x ightarrow a g(x)=0.) When attempting to lớn 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 và that the limit can have sầu various values of fg. This begs the question: are these the same? Can we distinguish 00 as an indeterminate size & 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 to lớn the exponents, we immediately conclude, that the term which precedes the first must be a0; and 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, & even when a is nothing; that is to say, a0 is equal lớn 1.

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

Let the exponential to be considered be az where a is a constant & the exponent z is a variable .... If z = 0, then we have 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 lớn a possible problem with 00.