A mathematical problem is a problem that can be represented, 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 problemsInformal "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 are 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 problemsAbstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so, results may be obtained that find application outside the realm of mathematics. Theoretical physics has historically been 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. 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. Computers do not need to have a sense of the motivations of mathematicians in order to do what they do.[1] Formal definitions and computer-checkable deductions are absolutely central to mathematical science. Degradation of problems to exercisesMathematics educators using problem solving for evaluation have an issue phrased by Alan H. Schoenfeld: How can one compare test scores from year to year, when very different problems are used? (If similar problems are used year after year, teachers and students will learn what they are, students will practice them: problems become exercises, and the test no longer assesses problem solving).[2]The same issue was faced by Sylvestre Lacroix almost two centuries earlier: ... it is necessary to vary the questions that students might communicate with each other. Though they may fail the exam, they might pass later. Thus distribution of questions, the variety of topics, or the answers, risks losing the opportunity to compare, with precision, the candidates one-to-another.[3]Such degradation of problems into exercises is characteristic of mathematics in history. For example, describing the preparations for the Cambridge Mathematical Tripos in the 19th century, Andrew Warwick wrote: ... many families of the then standard problems had originally taxed the abilities of the greatest mathematicians of the 18th century.[4]See also
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In mathematics, symbols that have certain meanings in the English language can mean very specialized and different things. For example, consider the following expression: 3! No, we did not use the exclamation point to show that we’re excited about three, and we shouldn’t read the last sentence with emphasis. In mathematics, the expression 3! is read as "three factorial" and is really a shorthand way to denote the multiplication of several consecutive whole numbers. Since there are many places throughout mathematics and statistics where we need to multiply numbers together, the factorial is quite useful. Some of the main places where it shows up are combinatorics and probability calculus. The definition of the factorial is that for any positive whole number n, the factorial: n! = n x (n -1) x (n - 2) x . . . x 2 x 1 First we will look at a few examples of the factorial with small values of n:
As we can see the factorial gets very large very quickly. Something that may seem small, such as 20! actually has 19 digits. Factorials are easy to compute, but they can be somewhat tedious to calculate. Fortunately, many calculators have a factorial key (look for the ! symbol). This function of the calculator will automate the multiplications. One other value of the factorial and one for which the standard definition above does not hold is that of zero factorial. If we follow the formula, then we would not arrive at any value for 0!. There are no positive whole numbers less than 0. For several reasons, it is appropriate to define 0! = 1. The factorial for this value shows up particularly in the formulas for combinations and permutations. When dealing with calculations, it is important to think before we press the factorial key on our calculator. To calculate an expression such as 100!/98! there are a couple of different ways of going about this. One way is to use a calculator to find both 100! and 98!, then divide one by the other. Although this is a direct way to calculate, it has some difficulties associated with it. Some calculators cannot handle expressions as large as 100! = 9.33262154 x 10157. (The expression 10157 is a scientific notation that means that we multiply by 1 followed by 157 zeros.) Not only is this number massive, but it is also only an estimate to the real value of 100! Another way to simplify an expression with factorials like the one seen here does not require a calculator at all. The way to approach this problem is to recognize that we can rewrite 100! not as 100 x 99 x 98 x 97 x . . . x 2 x 1, but instead as 100 x 99 x 98! The expression 100!/98! now becomes (100 x 99 x 98!)/98! = 100 x 99 = 9900. |