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This problem has over 2 million comments on Facebook and is getting coverage in mainstream media and blogs. Can you figure out this Facebook fruit algebra puzzle involving apples, bananas, and coconuts? The video presents what many people consider to be the correct answer.
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What is the car’s parking spot number? Hong Kong elementary students were asked to solve this problem in 20 seconds. The problem stumped a lot of adults and went viral in 2014.
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder.
The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert‘s axioms for geometry, and the theories of ordered groups, ordered fields, and local fields.
An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, a linearly ordered group that is Archimedean is an Archimedean group.
This can be made precise in various contexts with slightly different ways of formulation. For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.
Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation. Commonly studied types of magmas include
- From magma to group, via two alternative paths. Key:
- M = magma, d = divisibility, a = associativity,
- Q = quasigroup, S = semigroup, e = identity.
- L = loop, i = invertibility, N = monoid, G = group
- Note that both divisibility and invertibility imply
- the existence of the cancellation property.