Difference between revisions of "Mathematics"

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If you are reading this, follow this link into your browser, create an account, and copy and paste this article over any article you can! http://news.yahoo.com/s/ap/20071204/ap_on_go_pr_wh/us_iran_analysis_3
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[[Image:Leonhard Euler.jpg|right|thumb|225px|[[Leonhard Euler]] was an 18th-century mathematician and one of the most prolific mathematicians of all time. ]]
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'''Mathematics''' is the rigorous analysis of abstract structures, including numeric and logical systems.  The earliest known beginning of this topic is about 2400 B.C., the date of the oldest extant mathematical tablets.<ref>Davis & Hersh, ''The Mathematical Experience'' xi (Mariner Books 1981)</ref>
  
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WASHINGTON - First Iraq, now Iran. The United States has operated under a cloud of faulty intelligence in both countries.
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Mathematics includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also more abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions.
  
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In a bombshell intelligence assessment, the United States has backed away from its once-ironclad assertion that Tehran is intent on building nuclear bombs.
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== Numeracy ==
  
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Where there once was certainty, there now is doubt. "We do not know whether it currently intends to develop nuclear weapons," the new estimate said Monday.
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''See also:'' [[Numeracy]]
  
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Compare that with what then-National Intelligence Director John Negroponte told Congress in January. "Our assessment is that Tehran is determined to develop nuclear weapons."
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[[Numeracy]] is the ability to understand, interpret, and use numbers effectively in everyday life. It goes beyond basic arithmetic, encompassing logical reasoning, problem-solving, and the ability to apply mathematical concepts to real-world situations. Whether managing personal finances, interpreting data, or making informed decisions, numeracy is a fundamental skill for modern living.
  
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Just last month, President Bush, at a news conference with French President Nicolas Sarkozy, said, "We talked about Iran and the desire to work jointly to convince the Iranian regime to give up their nuclear weapons ambitions, for the sake of peace."
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==Symbols, Equations, and Theories==
  
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More ominously, Bush told a news conference Oct. 17, "I've told people that if you're interested in avoiding World War III, it seems like you ought to be interested in preventing them from having the knowledge necessary to make a nuclear weapon."
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Mathematics is expressed with [[symbol]]s. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of [[quantity]]. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.
  
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Asked then if he definitely believed that Iran wanted to build a nuclear bomb, Bush said, "Yeah, I believe they want to have the capacity, the knowledge, in order to make a nuclear weapon."
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An [[equation]] is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called [[differential equation]]s, express the equality of two [[function]]s.
  
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Bush's National Security Adviser Stephen Hadley said the president made comments like those "because he was describing the threat as the intelligence community itself had been describing the threat both publicly and in their briefings to him."
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A mathematical theory is expressed as a set of sentences, called [[axiom]]s. These axioms should be self-consistent, that is, they must not contradict with each other. From these axioms, new results can be derived adhering strictly to mathematical logic. These derived results are called [[theorem]]s. It is important to note that, according to the [[Godel's Incompleteness Theorems]], it is impossible to state a self-consistent set of axioms from which the whole mathematics can be derived.
  
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Intelligence officials advised Bush several months ago that they were reevaluating their assessments about Iran. They came to the White House last Wednesday and briefed him on their new findings.
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==Pure and Applied Mathematics==
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[[Applied mathematics]] concerns the use of mathematical methods for practical purposes. [[Pure mathematics]] involves reasoning about abstract structures.
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====Applied mathematics====
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Applied mathematics has its emphasis in applications, and is used extensively in the sciences such as [[physics]], [[chemistry]], [[medicine]], and [[biology]],  as well as [[engineering]], [[mechanics]] and [[technology]].  [[Economics]] and [[information theory]] also uses applied mathematics.  Mathematicians involved in research can and do create new theories, mathematical ideas, and new areas of study simply from their use of applied mathematics to solve various problems.
  
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The intelligence flip-flop recalled the embarrassing reversal that Bush was forced to make on whether Iraq possessed weapons of mass destruction. The conviction that Saddam Hussein had such weapons was one of the factors behind Bush's decision to invade Iraq. It since has been determined that Iraq did not have weapons of mass destruction.
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====Pure mathematics====
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Pure mathematics is the study of mathematics for its own sake, motivated for reasons other than application. It exhibits a trend towards increasing generality and abstraction.
  
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Democrats on Monday did not hesitate to suggest an Iran-Iraq comparison.
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==Branches of Mathematics==
  
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Senate Majority Leader Harry Reid said Democrats had requested the new Iran assessment "so that the administration could not rush this Congress and the country to another war based on flawed intelligence."
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====[[Arithmetic]]====
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Arithmetic is the study of combination of [[number]]s. Its basic operations are addition, subtraction, multiplication and division.  
  
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"I hope this administration reads this report carefully and appropriately adjusts its rhetoric and policy vis-a-vis Iran," Reid said. "The administration should begin this process by finally undertaking a diplomatic surge necessary to effectively address the challenges posed by Iran."
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====[[Algebra]]====
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Broadly, speaking algebra concerns 'addition' and 'multiplication', but in the widest possible sense. The objects that are being added or multiplied can be numbers, as in [[number theory]], but they can also can be more general structures such as [[matrix|matrices]], [[function]]s, [[polynomial]]s, [[vector]]s or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be addition of an [[additive inverse]] and division is considered to be multiplication by a multiplicative inverse. That is, subtracting 3 from 2 is rigorously defined to adding the number -2 to 3. Minus two is called the additive inverse of +2. Similarly, dividing 3 by 2 is formally defined in terms of multiplying 3 by (1/2), where 1/2 is the multiplicative inverse of 2.  Abstract algebra is the study of algebraic structures such as [[Group (mathematics)|group]]s, [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]].
  
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In the case of Iran, though, the White House has not dropped its suspicions that Tehran could pursue a nuclear bomb.
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====[[Analysis]]====
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Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of [[statistics]], and [[calculus]], including the calculus of several variables, vector calculus and tensor calculus. Also included is numerical analysis, the study of error propagation in [[algorithm]]s carried out to finite precision.  Additional topics in analysis include [[real analysis]] and [[complex analysis]].
  
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Iran continues to develop, test and deploy ballistic missiles, and its civilian uranium enrichment program is continuing. "It can readily use the same technology to produce weapons-grade uranium," Hadley said.
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==== [[Calculus]] ====
  
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In rewriting the conclusions about Iran, the new estimate said Tehran was pursuing a nuclear weapons program but halted that effort in the fall of 2003 under the weight of international pressure. Importantly, the estimate said Iran has not restarted the nuclear bomb program.
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[[Calculus]] is the mathematical subject that studies rates of change of functions. There are two main branches of calculus: [[Derivative (calculus)|differential]] calculus, and [[integral]] calculus.  There are subfields of these: single [[variable]] calculus, [[exterior calculus]], and multi [[variable]] calculus.
  
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"Tehran's decision to halt its nuclear weapons program suggests it is less determined to develop nuclear weapons than we have been judging since 2005," the new estimate said.
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====Chaos Theory====
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Chaos theory is the study of systems that are highly sensitive to initial conditions. That is, small differences in the initial conditions can produce large variations in the long time behaviour. This is known popularly as the Butterfly effect.
  
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While key facts have changed, the administration's strategy has not.
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====[[Discrete Mathematics]]====
  
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The White House says it will continue to try to build pressure on Iran to prevent it from ever acquiring nuclear bombs.
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The study of discrete structures, such as graphs, Latin squares, and block designs. Discrete mathematics can be studied from a pure, theoretical standpoint, or by studying its applications, such as those to theoretical computer science and combinatorial optimization.
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[[Image:Von.jpg|thumb|right|200px|The mathematician [[John von Neumann]] helped develop [[game theory]]. ]]
  
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"The bottom line is that for that strategy to succeed, the international community has to turn up the pressure on Iran with diplomatic isolation, United Nations sanctions and with other financial pressure," Hadley said. "And Iran has to decide that it wants to negotiate a solution."
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====[[Combinatorics]]====
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Combinatorics is the study of situations in which elements of a set or sets are combined or permuted in various ways. An example of a combinatorics problem would be "in a group of six men and four women, how many possible ways are there to choose three men and two women?" Derangements are another concept in combinatorics. A derangement is a re-ordering of a set so that no element ends up where it was originally; a derangement problem typically asks how many arrangements are possible for a given set, meeting given conditions.
  
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Some analysts believe the new conclusions will be a roadblock for Vice President Dick Cheney and other hawkish members of the administration to be more confrontational toward Iran.
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====[[Game Theory]]====
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Game theory is the mathematical study of strategic situations, in which the success of an individual making choices depend on the choices of others.
  
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"It's a good thing that we caught this before we marched headlong into another military conflict," said Jon Wolfsthal, senior fellow at the Center for Strategic and International Studies in Washington. "This isn't the timebomb the administration made it out to be for the last several years."
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====[[Geometry]]====
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Geometry is the study of shapes and special relationships. It was defined by Felix Klein as the study of [[invariant]]s under [[Group (mathematics)|group]]s of [[transformation]]s. For example, the Euclidean transformations are [[translation]], [[rotation]] and [[reflection]]. The quantities that are not altered by these transformations are things like angles and distances, so these are the subjects of interest in [[Euclidean geometry]]. Other types of transformations, such as the affine transformations, define other types of geometry. Topology is concerned with the connectedness of objects, rather than the distance between them. It is sometimes called 'rubber sheet geometry', as it concerns properties (that is, [[arc]]s between [[node]]s in [[network]]s) of objects that would be preserved even if a diagram of them were to be stretched or shrunk.  
  
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Wolfsthal said the conclusion that international pressure prompted Iran to halt its program "is the piece of information that we missed in Iraq" where Bush believed that Iraq's pursuit of WMD was continuing despite sanctions. He said the administration did not appear inclined to change its strategy toward Iran. He said that "suggests they can't take yes for an answer."
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====[[Logic]]====
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Logic is the study of [[reasoning]]. It examines general forms that [[argument]]s can take, and determines which forms are valid, and which forms are fallacies. In mathematics, it is the study of inferences within one formal language.
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====[[Set theory]]====
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Set theory is the mathematical study of collections of objects. It is one of the most fundamental areas of mathematics, since all of mathematics can be expressed in terms of [[set]]s. Sets are defined by a collection of [[axiom]]s called the [[Zermelo-Fraenkel]] axioms. One of the axioms, the [[Axiom of Choice]], has been the subject of much discussion.
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====[[Probability and Statistics]]====
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Probability can be viewed as the study of processes whose outcome cannot be predicted with certainty, and all that can be done is calculating the like hood of the different possible outcomes. Statistics is the use of numerical data from a small sample of a population to make inferences about the whole population.
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====[[Topology]]====
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Topology is the study of special properties that are preserved under continuous deformation of objects. Put more simply, it studies the properties that don't change unless you poke a hole in the object. One of the most popular examples is that of a coffee cup that can be continuously deformed into a doughnut. Then, the cup and the doughnut are said to be “topologically equivalent”. Topology began with [[Leonard Euler]]'s consideration of the Königsberg Bridges Problem, which also introduced Graph Theory. Beck's map of the London Underground in 1933 used a topological distortion of the locations of the subway stations in order to produce a more useful and artistic map.  [[Differential geometry]] is a specialized field of its own.
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====[[Trigonometry]]====
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In its more basic sense, trigonometry is the study of the relationships between the sides and the angles of [[triangle]]s. However, trigonometric functions are widely used outside their original realm of describing triangles. For example, sinusoidal functions (sine and cosine) are used to describe oscillatory motion and waves.  In fact, there are extremely deep and fruitful connections among those functions, the exponential function, complex numbers, [[Fourier transform|Fourier]] and Laplace transforms, and signal processing.
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==[[Handedness|Left-handers and mathematics]]==
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According to a study, left handers are exceptional at working out very complicated mathematical questions.<ref>{{cite web|url=https://answersingenesis.org/human-body/left-handed-complement/|title=Left-Handed Complement|date=August 13, 2018|accessdate=March 17, 2020|publisher=[[Answers in Genesis]]|author1=Troy Lacey|author2=Heather Lacey}}</ref>
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== See also ==
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*[[Quantitative reasoning]]
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*[[Data literacy]]
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*[[Mathematical modeling]]
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*[[Critical numeracy]]
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*[[Additive property of equality]]
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*[[Amédée Mannheim]]
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*[[Andrey Kolmogorov]]
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*[[Bourbaki]]
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*[[Math and the Bible]]
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== References ==
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<references/>
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[[Category:Mathematics]]

Latest revision as of 19:03, April 8, 2026

Leonhard Euler was an 18th-century mathematician and one of the most prolific mathematicians of all time.

Mathematics is the rigorous analysis of abstract structures, including numeric and logical systems. The earliest known beginning of this topic is about 2400 B.C., the date of the oldest extant mathematical tablets.[1]

Mathematics includes many practical results concerning quantity and measure, such as calculations involving numbers, financial accounting, geometric construction of building, astronomical calculations, calendar dating, telling time, engineering, physics, chemistry, etc. but also more abstract issues such as establishing the conditions under which certain kinds of equations and formulas have solutions.

Numeracy

See also: Numeracy

Numeracy is the ability to understand, interpret, and use numbers effectively in everyday life. It goes beyond basic arithmetic, encompassing logical reasoning, problem-solving, and the ability to apply mathematical concepts to real-world situations. Whether managing personal finances, interpreting data, or making informed decisions, numeracy is a fundamental skill for modern living.

Symbols, Equations, and Theories

Mathematics is expressed with symbols. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of quantity. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.

An equation is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called differential equations, express the equality of two functions.

A mathematical theory is expressed as a set of sentences, called axioms. These axioms should be self-consistent, that is, they must not contradict with each other. From these axioms, new results can be derived adhering strictly to mathematical logic. These derived results are called theorems. It is important to note that, according to the Godel's Incompleteness Theorems, it is impossible to state a self-consistent set of axioms from which the whole mathematics can be derived.

Pure and Applied Mathematics

Applied mathematics concerns the use of mathematical methods for practical purposes. Pure mathematics involves reasoning about abstract structures.

Applied mathematics

Applied mathematics has its emphasis in applications, and is used extensively in the sciences such as physics, chemistry, medicine, and biology, as well as engineering, mechanics and technology. Economics and information theory also uses applied mathematics. Mathematicians involved in research can and do create new theories, mathematical ideas, and new areas of study simply from their use of applied mathematics to solve various problems.

Pure mathematics

Pure mathematics is the study of mathematics for its own sake, motivated for reasons other than application. It exhibits a trend towards increasing generality and abstraction.

Branches of Mathematics

Arithmetic

Arithmetic is the study of combination of numbers. Its basic operations are addition, subtraction, multiplication and division.

Algebra

Broadly, speaking algebra concerns 'addition' and 'multiplication', but in the widest possible sense. The objects that are being added or multiplied can be numbers, as in number theory, but they can also can be more general structures such as matrices, functions, polynomials, vectors or many others. Concentrating on addition and multiplication does not exclude subtraction or division, since subtraction is formally considered to be addition of an additive inverse and division is considered to be multiplication by a multiplicative inverse. That is, subtracting 3 from 2 is rigorously defined to adding the number -2 to 3. Minus two is called the additive inverse of +2. Similarly, dividing 3 by 2 is formally defined in terms of multiplying 3 by (1/2), where 1/2 is the multiplicative inverse of 2. Abstract algebra is the study of algebraic structures such as groups, rings, and fields.

Analysis

Analysis is concerned with limits and other infinite processes. This subject includes the theory of limits of sequences and series and all forms of statistics, and calculus, including the calculus of several variables, vector calculus and tensor calculus. Also included is numerical analysis, the study of error propagation in algorithms carried out to finite precision. Additional topics in analysis include real analysis and complex analysis.

Calculus

Calculus is the mathematical subject that studies rates of change of functions. There are two main branches of calculus: differential calculus, and integral calculus. There are subfields of these: single variable calculus, exterior calculus, and multi variable calculus.

Chaos Theory

Chaos theory is the study of systems that are highly sensitive to initial conditions. That is, small differences in the initial conditions can produce large variations in the long time behaviour. This is known popularly as the Butterfly effect.

Discrete Mathematics

The study of discrete structures, such as graphs, Latin squares, and block designs. Discrete mathematics can be studied from a pure, theoretical standpoint, or by studying its applications, such as those to theoretical computer science and combinatorial optimization.

 
The mathematician John von Neumann helped develop game theory.

Combinatorics

Combinatorics is the study of situations in which elements of a set or sets are combined or permuted in various ways. An example of a combinatorics problem would be "in a group of six men and four women, how many possible ways are there to choose three men and two women?" Derangements are another concept in combinatorics. A derangement is a re-ordering of a set so that no element ends up where it was originally; a derangement problem typically asks how many arrangements are possible for a given set, meeting given conditions.

Game Theory

Game theory is the mathematical study of strategic situations, in which the success of an individual making choices depend on the choices of others.

Geometry

Geometry is the study of shapes and special relationships. It was defined by Felix Klein as the study of invariants under groups of transformations. For example, the Euclidean transformations are translation, rotation and reflection. The quantities that are not altered by these transformations are things like angles and distances, so these are the subjects of interest in Euclidean geometry. Other types of transformations, such as the affine transformations, define other types of geometry. Topology is concerned with the connectedness of objects, rather than the distance between them. It is sometimes called 'rubber sheet geometry', as it concerns properties (that is, arcs between nodes in networks) of objects that would be preserved even if a diagram of them were to be stretched or shrunk.

Logic

Logic is the study of reasoning. It examines general forms that arguments can take, and determines which forms are valid, and which forms are fallacies. In mathematics, it is the study of inferences within one formal language.

Set theory

Set theory is the mathematical study of collections of objects. It is one of the most fundamental areas of mathematics, since all of mathematics can be expressed in terms of sets. Sets are defined by a collection of axioms called the Zermelo-Fraenkel axioms. One of the axioms, the Axiom of Choice, has been the subject of much discussion.

Probability and Statistics

Probability can be viewed as the study of processes whose outcome cannot be predicted with certainty, and all that can be done is calculating the like hood of the different possible outcomes. Statistics is the use of numerical data from a small sample of a population to make inferences about the whole population.

Topology

Topology is the study of special properties that are preserved under continuous deformation of objects. Put more simply, it studies the properties that don't change unless you poke a hole in the object. One of the most popular examples is that of a coffee cup that can be continuously deformed into a doughnut. Then, the cup and the doughnut are said to be “topologically equivalent”. Topology began with Leonard Euler's consideration of the Königsberg Bridges Problem, which also introduced Graph Theory. Beck's map of the London Underground in 1933 used a topological distortion of the locations of the subway stations in order to produce a more useful and artistic map. Differential geometry is a specialized field of its own.

Trigonometry

In its more basic sense, trigonometry is the study of the relationships between the sides and the angles of triangles. However, trigonometric functions are widely used outside their original realm of describing triangles. For example, sinusoidal functions (sine and cosine) are used to describe oscillatory motion and waves. In fact, there are extremely deep and fruitful connections among those functions, the exponential function, complex numbers, Fourier and Laplace transforms, and signal processing.

Left-handers and mathematics

According to a study, left handers are exceptional at working out very complicated mathematical questions.[2]

See also

References

  1. ↑ Davis & Hersh, The Mathematical Experience xi (Mariner Books 1981)
  2. ↑ Left-Handed Complement. Answers in Genesis (August 13, 2018). Retrieved on March 17, 2020.
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