Mathematics

Mathematicians solve problems by using logic. Mathematicians often use deduction. Deduction is a special way of thinking to discover and prove new truths using old truths. To a mathematician, the reasons you know something is true are just as important as the fact that it is true. Using deduction is what makes mathematical thinking different from normal thinking. //Mathematics // includes the study of: Mathematics uses logic to study these things and to create general rules, which are an important part of mathematics. These rules leave out information that isn't important so that a single rule can cover many situations. By finding general rules, mathematics solves many problems at the same time. A //proof// gives a reason why a rule in mathematics is correct. This is done by using certain other rules that everyone agrees are correct, which are called //axioms//. A rule that has a proof is sometimes called a //theorem//. Experts in mathematics perform research to create new theorems. Sometimes experts find an idea that they think is a theorem but can not find a proof for it. That idea is called a //conjecture// until they find a proof. Sometimes, mathematics finds and studies rules or ideas that have not yet been found in the real world. Often in mathematics, ideas and rules are chosen because they are considered simple or beautiful. Sometimes these ideas and rules are found in the real world after they are studied in mathematics. This has happened many times in the past. This means that studying the rules and ideas of mathematics can help us know the world better. Mathematics includes the study of number, or //quantity//. Natural numbers || Integers || Rational numbers || Real numbers || complex numbers || π,//e// ||   ||   ||   || Arithmetic || Mathematical constant || Ordinal number || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Cardinal number ||   || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Some areas of mathematics study the structure that an object possesses. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Number theory || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Abstract algebra || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Linear algebra || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Order theory || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Graph theory || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Operand theory || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Some areas of mathematics study the arrangement of things. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Topology || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Geometry || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Trigonometry || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Differential geometry || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Fractal geometry || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Ways to express and handle change in mathematical functions, at changes between numbers. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Calculus || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Vector calculus || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Analysis || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Differential equations || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Dynamical systems || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Chaos theory || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Applied mathematics uses mathematics to solve problems of other areas such as engineering and physics. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Numerical analysis – Optimization – Probability theory – Statistics – Mathematical finance – Game theory – Mathematical physics – Fluid dynamics <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">These theorems have interested mathematicians and people who are not mathematicians. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Pythagorean Theorem – Fermat's last theorem – Goldbach's conjecture – Twin Prime Conjecture – Gödel's incompleteness theorems – Poincaré conjecture – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's Identity – Church-Turing thesis <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">These are theorems and conjectures that have greatly changed mathematics. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Riemann hypothesis – Continuum hypothesis – P=NP – Pythagorean theorem – Central limit theorem – Fundamental theorem of calculus – Fundamental theorem of algebra – Fundamental theorem of arithmetic – Fundamental theorem of projective geometry – classification theorems of surfaces – Gauss-Bonnet theorem – Fermat's last theorem <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Progress in understanding the nature of mathematics also influences the way mathematicians study their subject. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Philosophy of mathematics – Mathematical intuitionism – Mathematical constructivism – Foundations of mathematics – Set theory – Symbolic logic – Model theory – Category theory – Logic – Reverse Mathematics – Table of mathematical symbols <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Mathematics in history, and the history of mathematics. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Maths and gender <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The word "mathematics" comes from the Greek word "μάθημα" (máthema). The Greek word "μάθημα" means "science, knowledge, or learning". <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Often, the word "mathematics" is made shorter into //maths// (in British English) or //math// (in American English). The short words //math// or //maths// are often used for arithmetic, geometry or simple algebra by young students and their schools. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">There is no Nobel prize in mathematics. Mathematicians can receive the Abel prize and the Fields Medal for important works. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The Clay Mathematics Institute have said they will give one million dollars to anyone who solves one of the Millennium Prize Problems <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Tools that are used to do mathematics or find answers to mathematics problems. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Old: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">New:
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Mathematics **<span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">, or math (sometimes also shortened as "maths"), is the study of numbers, shapes and patterns. Mathematicians are people whose job is to learn and discover such things in mathematics. The biggest part of mathematics is useful for solving problems that occur in the real world, so many people besides mathematicians study and use these parts of mathematics. Today in many kinds of jobs -- in business, science, engineering, and construction, for example -- knowledge about certain aspects of mathematics is needed.
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">About **
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Number: how many things there are.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Structure: how things are organized.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Place: where things are and their arrangement.
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Change: how things become different over time.
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Number **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Structure **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Place **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Change **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Applied mathematics **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Famous theorems **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Foundations and methods **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">History and the world of mathematicians **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Name **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Awards in mathematics **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Mathematical tools **
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Abacus
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Napier's bones, slide rule
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Ruler and Compass
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Mental calculation
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Calculators and computers
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Programming languages
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Computer algebra systems (listing)
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Internet shorthand notation
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">statistical analysis software
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">SPSS
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">SAS programming language
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">R programming language