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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."
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy.There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or ...
The report implied that Anderson had discovered the solution to division by zero, rather than simply attempting to formalize it. The report also suggested that Anderson was the first to solve this problem, when in fact the result of zero divided by zero has been expressed formally in a number of different ways (for example, NaN ).
Sometimes, the whole numbers are the natural numbers plus zero. In other cases, the whole numbers refer to all of the integers , including negative integers. [ 3 ] The counting numbers are another term for the natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1.
The theorem is frequently referred to as the division algorithm (although it is a theorem and not an algorithm), because its proof as given below lends itself to a simple division algorithm for computing q and r (see the section Proof for more). Division is not defined in the case where b = 0; see division by zero.
This is because if b were a negative number then dividing by a negative would change the ≥ relationship into a ≤ relationship. For example, although 2 is more than 1, –2 is less than –1. Also if b were zero then zero times anything is zero and cancelling out would mean dividing by zero in that case which cannot be done.
In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring . The term wheel is inspired by the topological picture ⊙ {\displaystyle \odot } of the real projective line together with an extra point ⊥ ( bottom element ) such that ⊥ = 0 / 0 {\displaystyle \bot =0/0} .
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. If b = 0, then this is a division by zero, which is not defined. [a] [4]: 246 In the 21-apples example, everyone would receive 5 apple and a quarter of an apple, thus avoiding any leftover.