Everything about Mathematical Problem totally explained
A
mathematical problem is a problem that's amenable to being analyzed, and possibly solved, with the methods of
mathematics. This can be a real-world problem, such as computing the
orbits of the planets in the solar system, or a problem of a more abstract nature, such as
Hilbert's problems. It can also be a problem referring to the
nature of mathematics itself, such as
Russell's Paradox.
Real-world problems
Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam has five apples and gives John three. How many has he left?". Such questions are usually more difficult to solve than regular mathematical exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known as
word problems, they're used in
mathematics education to teach students to connect real-world situations to the abstract language of mathematics.
In general, to use mathematics for solving a real-world problem, the first step is to construct a
mathematical model of the problem. This involves abstraction from the details of the problem, and the modeller has to be careful not to lose essential aspects in translating the original problem into a mathematical one. After the problem has been solved in the world of mathematics, the solution must be translated back into the context of the original problem.
Abstract problems
Abstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so results may obtained that find application outside the realm of mathematics.
Theoretical physics has historically been, and remains, a rich source of inspiration.
Some abstract problems have been rigorously proved to be unsolvable, such as
squaring the circle and
trisecting the angle using only the
compass and straightedge constructions of classical geometry, and solving the general
quintic equation algebraically. Also provably unsolvable are so-called
undecidable problems, such as the
halting problem for
Turing machines.
Many abstract problems can be solved routinely, others have been solved with great effort, for some significant inroads have been made without having led yet to a full solution, and yet others have withstood all attempts, such as
Goldbach's conjecture and the
Collatz conjecture. Some well-known difficult abstract problems that have been solved relatively recently are the
four-colour theorem,
Fermat's Last Theorem, and the
Poincaré conjecture.
Further Information
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