# Portal:Mathematics

**Mathematics** is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

**Euclidean geometry** is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text *Elements* was the first systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could fit together into a comprehensive deductive and logical system.

The *Elements* begin with plane geometry, still often taught in secondary school as the first axiomatic system and the first examples of formal proof. The *Elements* goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the *Elements* states results of what is now called number theory, proved using geometrical methods.

For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is only a good approximation to the properties of physical space if the gravitational field is not too strong.

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**Anscombe's quartet** is a collection of four sets of bivariate data (paired *x*–*y* observations) illustrating the importance of graphical displays of data when analyzing relationships among variables. The data sets were specially constructed in 1973 by English statistician Frank Anscombe to have the same (or nearly the same) values for many commonly computed descriptive statistics (values which summarize different aspects of the data) and yet to look very different when their scatter plots are compared. The four *x* variables share exactly the same mean (or "average value") of 9; the four *y* variables have approximately the same mean of 7.50, to 2 decimal places of precision. Similarly, the data sets share at least approximately the same standard deviations for *x* and *y*, and correlation between the two variables. When *y* is viewed as being dependent on *x* and a least-squares regression line is fit to each data set, almost the same slope and *y*-intercept are found in all cases, resulting in almost the same predicted values of *y* for any given *x* value, and approximately the same coefficient of determination or *R*² value (a measure of the fraction of variation in *y* that can be "explained" by *x*, or more intuitively "how well *y* can be predicted" from *x*). Many other commonly computed statistics are also almost the same for the four data sets, including the standard error of the regression equation and the *t* statistic and accompanying *p*-value for testing the significance of the slope. Clear differences between the data sets are apparent, however, when they are graphed using scatter plots. The plots even suggest particular reasons why *y* cannot be perfectly predicted from *x* using each regression line: (1) While the variables are roughly linearly related in the first data set, there is more variability in *y* than can be accounted for by *x*, as seen in the vertical spread of the points around the regression line; in this case, one or more additional independent variables may be needed to account for some of this "residual" variation in *y*. (2) The second scatter plot shows strong curvature, so a simple linear model is not even appropriate for the data; polynomial regression or some other model allowing for nonlinear relationships may be appropriate. (3) The third data set contains an outlier, which ruins the otherwise perfect linear relationship between the variables; this may indicate that an error was made in collecting or recording the data, or may reveal an aspect of the variation of *y* that has not been considered. (4) The fourth data set contains an influential point that is almost completely determining the slope of the regression line; the reliability of the line would be increased if more data were collected at the high *x* value, or at any other *x* values besides 8. Although some other common summary statistics such as quartiles could have revealed differences across the four data sets, the plots give additional information that would be difficult to glean from mere numerical summaries. The importance of visualizing data is magnified (and made more complicated) when dealing with higher-dimensional data sets. Multiple regression is a straightforward generalization of linear regression to the case of multiple *independent* variables, while "multivariate" regression methods such as the general linear model allow for multiple *dependent* variables. Other statistical procedures designed to reveal relationships in multivariate data (several of which are closely tied to useful graphical depictions of the data) include principal component analysis, factor analysis, multidimensional scaling, discriminant function analysis, cluster analysis, and many others.

- ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
- ...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
- ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with
*n*+1 factors? - ...that a ball can be cut up and reassembled into two balls the same size as the original (Banach-Tarski paradox)?
- ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
- ...that Euler found 59 more amicable numbers while for 2000 years, only 3 pairs had been found before him?

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