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Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero. In 830, Mahāvīra unsuccessfully tried to correct the mistake Brahmagupta made in his book Ganita Sara Samgraha: "A number remains unchanged when divided by zero ...
A non-negative integer is a square number when its square root is again an integer. For example, =, so 9 is a square number. A positive integer that has no square divisors except 1 is called square-free. For a non-negative integer n, the n th square number is n 2, with 0 2 = 0 being the zeroth one. The concept of square can be extended to some ...
2. Orthogonal subspace in the dual space: If W is a linear subspace (or a submodule) of a vector space (or of a module) V, then may denote the orthogonal subspace of W, that is, the set of all linear forms that map W to zero. 3. For inline uses of the symbol, see ⊥.
Zero was not treated as a number at that time, but as a "vacant position". [39] Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol ‘〇’ for zero. [40] The origin of this symbol is unknown; it may have been produced by modifying a square symbol. [41]
radical symbol (for square root) 1525 (without the vinculum above the radicand) ... division slash (a.k.a. solidus) 1718 (deriving from horizontal fraction bar, ...
A square-free integer is an integer that is not divisible by any square other than 1. square number A square number is an integer that is the square of an integer. For example, 4 and 9 are squares, but 10 is not a square. Szpiro Szpiro's conjecture is, in a modified form, equivalent to the abc conjecture.
The concept of zero and the introduction of a notation for it are important developments in early mathematics, which predates for centuries the concept of zero as a number. It was used as a placeholder by the Babylonians and Greek Egyptians, and then as an integer by the Mayans, Indians and Arabs (see the history of zero).
For division to always yield one number rather than an integer quotient plus a remainder, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c / b means a × b = c, as long as b is not zero.