The Mathematics Portal


Mathematics is the study of numbers, quantity, space, pattern, 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.


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Example of a four color map
Image credit: User:Inductiveload

The four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. "Color by Number" worksheets and exercises, which combine learning art and math for people of young ages, are a good example of the four color theorem.

It is often the case that using only three colors is inadequate. This applies already to the map with one region surrounded by three other regions (even though with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to color a map.

The four color theorem was the first major theorem to be proven using a computer, and the proof is disputed by some mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof). Ultimately, in order to believe the proof, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof.

The lack of mathematical elegance was another factor, and to paraphrase comments of the time, "a good mathematical proof is like a poem — this is a telephone directory!"

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graph of an increasing curve showing cumulative share of income earned versus cumulative share of people from lowest to highest income

A Lorenz curve shows the distribution of income in a population by plotting the percentage y of total income that is earned by the bottom x percent of households (or individuals). Developed by economist Max O. Lorenz in 1905 to describe income inequality, the curve is typically plotted with a diagonal line (reflecting a hypothetical "equal" distribution of incomes) for comparison. This leads naturally to a derived quantity called the Gini coefficient, first published in 1912 by Corrado Gini, which is the ratio of the area between the diagonal line and the curve (area A in this graph) to the area under the diagonal line (the sum of A and B); higher Gini coefficients reflect more income inequality. Lorenz's curve is a special kind of cumulative distribution function used to characterize quantities that follow a Pareto distribution, a type of power law. More specifically, it can be used to illustrate the Pareto principle, a rule of thumb stating that roughly 80% of the identified "effects" in a given phenomenon under study will come from 20% of the "causes" (in the first decade of the 20th century Vilfredo Pareto showed that 80% of the land in Italy was owned by 20% of the population). As this so-called "80–20 rule" implies a specific level of inequality (i.e., a specific power law), more or less extreme cases are possible. For example, in the United States in the first half of the 2010s, 95% of the financial wealth was held by the top 20% of wealthiest households (in 2010), the top 1% of individuals held approximately 40% of the wealth (2012), and the top 1% of income earners received approximately 20% of the pre-tax income (2013). Observations such as these have brought income and wealth inequality into popular consciousness and have given rise to various slogans about "the 1%" versus "the 99%".

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