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In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system. [ 1 ] Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that system.
For a vertical line, this is 1 : 0, a kind of division by zero. In another interpretation, the quotient Q {\displaystyle Q} represents the ratio N : D . {\displaystyle N:D.} [ 6 ] For example, a cake recipe might call for ten cups of flour and two cups of sugar, a ratio of 10 : 2 {\displaystyle 10:2} or, proportionally, 5 : 1. {\displaystyle 5:1.}
The expressions 0 0, ∞ 0 and 1 ∞ are considered indeterminate forms when they occur as limits (just like ∞ × 0), and the question of whether zero to the zero power should be defined as 1 has divided opinion. If the output is considered as undefined when a parameter is undefined, then pow(1, qNaN) should produce a qNaN.
Zero-based numbering is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday non-mathematical or non-programming circumstances.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.
Since cos(x) ≤ 1 for all x and x 3 > 1 for x > 1, we know that our solution lies between 0 and 1. A starting value of 0 will lead to an undefined result which illustrates the importance of using a starting point close to the solution. For example, with an initial guess x 0 = 0.5, the sequence given by Newton's method is:
A critical point of a function of a single real variable, f (x), is a value x 0 in the domain of f where f is not differentiable or its derivative is 0 (i.e. ′ =). [2] A critical value is the image under f of a critical point.
An undefined value must not be confused with empty string, Boolean "false" or other "empty" (but defined) values. Depending on circumstances, evaluation to an undefined value may lead to exception or undefined behaviour, but in some programming languages undefined values can occur during a normal, predictable course of program execution.