{"id":672,"date":"2010-02-25T14:56:16","date_gmt":"2010-02-25T14:56:16","guid":{"rendered":"http:\/\/mathscitech.org\/articles\/?p=672"},"modified":"2024-01-01T12:02:44","modified_gmt":"2024-01-01T12:02:44","slug":"zero-to-zero-power","status":"publish","type":"post","link":"https:\/\/mathscitech.org\/articles\/zero-to-zero-power","title":{"rendered":"Why Zero Raised to the Zero Power is defined to be One"},"content":{"rendered":"

Updated! February 5, 2017<\/font><\/p>\n

The value of zero raised to the zero power, (0^0), has been discussed since the time of Euler in the 18th century (1700s). There are three reasonable choices: 1,0, or “indeterminate”. Despite consensus amongst mathematicians that the correct answer is one, computing platforms seem to have reached a variety of conclusions: Google, R, Octave, Ruby, and Microsoft Calculator choose 1; Hexelon Max and TI-36 calculator choose 0; and Maxima and Excel throw an error (indeterminate). In this article, I’ll explain why, for discrete mathematics, the correct answer cannot be anything other than 0^0=1, for reasons that go beyond consistency with the Binomial Theorem (Knuth’s argument).
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Context of the Debate: Continuous Mathematics<\/h4>\n

The three choices for the value of 0^0 appear because x^y, as a function of two continuous variables, is discontinuous at (0,0) and takes three different values depending on the direction of approach to the discontinuity:<\/p>\n

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  1. Fixing y=0, we have x^0=1 for all x \\neq 0. (Proof: x^0 = x^{(1-1)} = x^1 x^{-1} = x\/x = 1, each statement holding for all x \\neq 0). Indeed, x^y \\rightarrow 1 as x \\rightarrow 0, approaching from left or right, with y=0. (This was Euler’s reason.)\n
  2. Fixing x=0, we have 0^y=0 for y >0. (When y < 0 we have division by zero which is undefined in the reals and +\\infty in the extended reals). Taking limits, x^y \\rightarrow 0 as y \\searrow 0, approaching from above only, with x=0.\n
  3. Fixing x=0, we have an undefined value when y < 0 due to division by zero.\n<\/ol>\n

    Notice that the discontinuity is not a simple (point) discontinuity, but rather a pole discontinuity due to the approach from below. (Exercise: what happens as the origin is approached from 45 degrees?)<\/p>\n

    Principles for a Decision in Mathematics: Extension and Consistency<\/h4>\n

    In mathematics, when there is more than one choice, a decision is typically made by extending an existing precedent to maintain consistency with the evidence that is already accumulated and accepted. <\/p>\n

    An elementary example is the way ordinary multiplication is extended from two positive numbers to a positive and a negative number, then to two negative numbers, i.e. (-1)(-1) = 1.<\/p>\n

    “Minus times minus is plus.
    The reason for this we need not discuss!”
    — W.H. Auden<\/p><\/blockquote>\n

    Empirically, multiplication of two positive numbers has a well-defined, tangible meaning as repeated addition. This meaning holds when one of the numbers is negative. But when both are negative, the empirical meaning fails.<\/p>\n

    For the mathematician, declaring something to be undefined (throwing an error) means a loss of efficiency because every instance now has to be checked for the undefined case, and this must be treated separately. If a definition could be found that remains consistent with all other empirically obtained rules, and if that definition means that calculation can proceed indifferent to the decision, then that is a big win. <\/p>\n

    The consistency in this particular case is the distributivity of multiplication over addition, a law which, for positive numbers, can be accepted on entirely empirical grounds. (See the footnote for the full argument.1<\/a><\/sup>.)<\/p>\n

    Turning to Discrete Mathematics – Consistency with the Binomial Theorem<\/h4>\n

    In discrete mathematics, there is no notion of “approaching” — one is either at<\/em> 0^0 or away from it, in which case 1^0 = 1 or 0^1 = 0.<\/p>\n

    The case of 0^0 can be decided on consistency grounds with respect to the binomial theorem, i.e. loss of computational efficiency to have to treat this case separately. This is the argument of Knuth (of The Art of Computer Programming<\/a>, and TeX<\/a> fame), based on maintaining consistency with the binomial theorem (a+b)^x when x=0, due to its fundamental place in both discrete and continuous mathematics:<\/p>\n

    “Some textbooks leave the quantity 0^0 undefined, because the functions 0^x and x^0 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.”
    \n – from Concrete Mathematics, p.162, R. Graham, D. Knuth, O. Patashnik, Addison-Wesley, 1988\n<\/p><\/blockquote>\n

    Different Conventions Among Mathematical Computing Platforms<\/h4>\n

    Given the universality of the 0^0=1 convention amongst mathematicians, it is surprising to find that various computing platforms have implemented different values:<\/p>\n