{"id":361943,"date":"2024-05-21T01:24:17","date_gmt":"2024-05-21T01:24:17","guid":{"rendered":"http:\/\/savepearlharbor.com\/?p=361943"},"modified":"-0001-11-30T00:00:00","modified_gmt":"2024-05-21T01:24:17","slug":"","status":"publish","type":"post","link":"https:\/\/savepearlharbor.com\/?p=361943","title":{"rendered":"Why x^0 = 1 visually<\/span>"},"content":{"rendered":"
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The traditional definition for the operation of exponentiation to a natural power (or a positive integer) had introduced approximately as follows:<\/p>\n

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Exponentiation<\/strong><\/a> is an arithmetic operation originally defined as the result of multiple multiplications a number by itself<\/em>.<\/p>\n<\/blockquote>\n

At first glance, this definition is difficult to generalize for integer powers less than one <\/p>\n

    ... x^3  = x*x*x x^2  = x*x x^1  = x x^0  = ????? x^-1 = ?????     ...<\/code><\/pre>\n
but if we remember that the operation of multiplication has an inverse — division, then an extension arises for negative powers<\/p>\n
    ... x^3  = x*x*x x^-3 = 1\/(x*x*x) = 1\/x\/x\/x     ...<\/code><\/pre>\n
therefore<\/p>\n
    ... x^3  = x*x*x x^2  = x*x x^1  = x x^0  = ? x^-1 = 1\/x x^-2 = 1\/x\/x x^-3 = 1\/x\/x\/x     ...<\/code><\/pre>\n
But the question still remains open for 0<\/code>: on the one hand, in ?<\/code> place 1<\/code> fits, but on the other hand, this definition remains not entirely symmetrical for positive and negative values…<\/p>\n
Therefore, upon careful examination, it is quite natural to want to slightly modify and improve the original<\/p>\n
    ... x^3  = 1*x*x*x x^2  = 1*x*x x^1  = 1*x x^0  = 1 x^-1 = 1\/x x^-2 = 1\/x\/x x^-3 = 1\/x\/x\/x     ...<\/code><\/pre>\n
In this representation, the definition seems to acquire harmony coupled with completeness:<\/p>\n
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Raising a number X to an integer power N<\/strong> is an arithmetic operation defined as the result of multiple [<\/em>N<\/em><\/strong> by mod times] multiplications or divisions <\/em>one<\/em><\/strong> by number <\/em>X<\/em><\/strong>.<\/p>\n<\/blockquote>\n
\"\u0425\u043e\u0434<\/p>\n
The way of thinking<\/figcaption><\/div>\n<\/figure>\n
A slightly more general case*<\/a> in programming language C#<\/p>\n
static double Pow(double x, int pow, double seed = 1d) {     var value = seed;          if (pow < 0)         for (var i = 0; i > pow; i--) value \/= x;     else         for (var i = 0; i < pow; i++) value *= x;          return value; }<\/code><\/pre>\n
 * in a general case the <\/em>seed<\/code> may be different from <\/em>1<\/code><\/p>\n
In this form, the definition can easily and naturally be extended to the case of 0^0<\/code>, which has long been the subject of heated debate in the mathematical community.<\/p>\n
It naturally follows from this that 0^0 = 1<\/code>.<\/p>\n
Indeed, if we never multiply one by zero, then the result will be the original 1<\/code>, but if we multiply it even once, we will already get 0<\/code>.<\/p>\n\n
If you think about it carefully, it is impossible to strictly prove the identity X^0 = 1<\/code>, and the reason is that we ourselves define functions in such a way that they have properties convenient for applied calculations.<\/p>\n
That is, when trying to prove the identity X^0 = 1<\/code> using various methods, in fact, we are only demonstrating that we have defined the exponentiation function in precisely such a way as to have those specified useful properties through which the \u201cproof\u201d itself is carried out. That is, an implicit vicious circle arises.<\/p>\n
To the question:<\/p>\n