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Mathematics

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Math and nature.
For help with math formatting on the CreationWiki see: Math Help.

Mathematics is the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. Mathematics is one of the great gifts of God and greatest discoveries of the human race. Mankind has used it to bring about our most important scientific and technological advances. Mathematics works hand in hand with science. Carl Friedrich Gauss once called mathematics "the Queen of the Sciences".

Mathematics is one of the oldest and most fundamental sciences. Mathematicians use mathematical theory, computational techniques, algorithms, and the latest computer technology to solve economic, scientific, engineering, physics, and business problems. The work of mathematicians falls into two broad classes—theoretical (pure) mathematics and applied mathematics. These classes, however, are not sharply defined and often overlap.[1]

Contents

Mind or Matter?

The laws of mathematics, like the laws of Logic have their foundation in the mind, not the natural world. Mathematics is a language to describe the natural world but the concepts themselves only exist in the human mind. The Piraha people[2] have no math or numerics in there entire language. Human minds do not naturally think in mathematical terms, but must be trained in mathematics. Once trained, all human minds understand mathematics in the same way. Some understand it better, of course and take it to higher levels, but even those at the peak of mathematical understanding explain and teach their concepts using a common substrate or foundation of knowledge. For example, the concept of "2" does not exist in the natural world. It is a quantity assigned by the mind. Likewise "2+2" is not a property of matter, but a mental assignment of the concepts of "2".

That these align with the natural world for purposes of understanding and measurement of the physical world, is the clearest evidence of the existence of God. Only human minds that were prescribed and designed to function in a common way, could ever hope to intersect on such abstract and ultimately very complex concepts. If the human mind is a product of chance, or even "directed" chance, how can anyone be certain that their mind is fully functional (and does not need another million years of evolution before it can even start asking the right questions)?

Rather, the laws of math and Logic are identical for all people-groups, languages, timeframes and locations. Math works the same on the Moon, Rigel, Fomalhaut and Proxima. But math does not physically exist in any of these places. It is a language that describes the natural world. As such, if observations in the natural world do not fit the mathematical descriptions, scientists work to reconcile the two with the presumption that they should reconcile. Christian believers can presume that certain elements of math will be very consistent with the natural world because they have a theistic basis for this presumption. Secularists have no justification whatsoever for this presumption, and must arbitrarily borrow from the truth of creation to justify it.

Fields

Theoretical

Theoretical mathematicians advance mathematical knowledge by developing new principles and recognizing previously unknown relationships between existing principles of mathematics. Although these workers seek to increase basic knowledge without necessarily considering its practical use, such pure and abstract knowledge has been instrumental in producing or furthering many scientific and engineering achievements. Many theoretical mathematicians are employed as university faculty, dividing their time between teaching and conducting research.[1]

Applied

Applied mathematicians, on the other hand, use theories and techniques, such as mathematical modeling and computational methods, to formulate and solve practical problems in business, government, engineering, and the physical, life, and social sciences. For example, they may analyze the most efficient way to schedule airline routes between cities, the effects and safety of new drugs, the aerodynamic characteristics of an experimental automobile, or the cost-effectiveness of alternative manufacturing processes.

Applied mathematicians working in industrial research and development may develop or enhance mathematical methods when solving a difficult problem. Some mathematicians, called cryptanalysts, analyze and decipher encryption systems—codes—designed to transmit military, political, financial, or law enforcement-related information.

Applied mathematicians start with a practical problem, envision its separate elements, and then reduce the elements to mathematical variables. They often use computers to analyze relationships among the variables and solve complex problems by developing models with alternative solutions.

Individuals with titles other than mathematician do much of the work in applied mathematics. In fact, because mathematics is the foundation on which so many other academic disciplines are built, the number of workers using mathematical techniques is much greater than the number formally called mathematicians. For example, engineers, computer scientists, physicists, and economists are among those who use mathematics extensively. Some professionals, including statisticians, actuaries, and operations research analysts, are actually specialists in a particular branch of mathematics.[1]

Concepts

Quantity

Mathematics in quantity deals with numbers. It is used in simple methods such as addition, subtraction, multiplication, and division. Numbers used begin with the natural numbers ({1, 2, 3, …}, also known as the counting numbers), whole numbers ({0, 1, 2, 3, …}), integers (the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, …) and the number zero), and branches out into more complex areas.

Integers

The set of integers (called Z for their German name zahlen represented by the symbol \mathbb{Z}) is the most fundamental of all sets, because it is the most "natural." This set is closed under addition, subtraction, and multiplication. (Any set is closed under any operation if that operation, taking only elements of the set as arguments, always yields a result that is also a member of the set.[3]) There is a subset of Integers called the natural numbers. Is defined by the mathematicians as the set of the positive integers or the set of non-negative integers depending on the presence or not of the zero element in the set. The natural numbers are represented by the symbol \mathbb{N}.

Rational numbers

A rational number is any number of the form \frac{a}{b}, where a and b are integers and b \neq 0. It is represented by the symbol \mathbb{Q}.

Rational numbers are closed under division, whereas integers are not. Rational numbers are also dense, meaning that between any two rational numbers, another rational number must exist.


Real numbers

A real number is any number that can stand for a real, physical quantity. It is represented by the symbol \mathbb{R}.

The set of real numbers includes all the rational numbers and also certain numbers that are irrational. Irrational numbers are not expressible as ratios \frac{a}{b} of integers. They are represented by the symbol \mathbb{I}. Irrational numbers include:

  • Algebraic irrational numbers: the results of solutions of polynomial equations (see below). All rational numbers are considered algebraic.
  • Transcendental numbers: all other irrational numbers. The most famous of these are π, e, and the results of trigonometric, hyperbolic, and logarithmic functions and their inverses. Another example of irrational number is:
\sqrt{2}

The set of real numbers is closed under any operation, except that the taking of even-indexed roots of negative numbers is not allowed.

Complex numbers

Complex numbers include real and imaginary numbers and their sums. It is represented by the symbol \mathbb{C}. The imaginary numbers are all real multiples of the imaginary unit, i:

\,\!\mathit{i}^2 = -1

Complex numbers have uses in sinusoidal analysis (which treats the action of certain electric circuits subjected to alternating currents) and in some fields of theoretical physics. Certain polynomial equations that have no real solution have complex solutions.

Operations

Basic operations

The four basic operations of mathematics are addition, subtraction, multiplication, and division. All other operations depend on these four. Addition and subtraction, and multiplication and division, are inverses.

Exponents and radicals

Classically, an exponent (written as a superscript) denotes how many times to multiply a number (called the base) by itself. The evaluation of a base and its exponent is called a power. A radical (from the Latin radix root) is the inverse of a power; thus

\sqrt[n]{x}

gives that number which, when taken to the nth power, yields x.

Polynomials and rational expressions

Polynomials are expressions that exclusively contain various powers (but not roots) of a variable or variables, always multiplied by some constant, classically an integer. Each such power, with its integer multiplier (cofactor), is called a term.

Rational expressions are ratios of polynomials.

A polynomial equation is any polynomial having an equals sign in it. A polynomial inequality has an operator of inequality.

The degree of any term is the sum of the exponents applied to the variables within it. The degree of a polynomial equation, inequality, or other expression is the highest degree of any of its terms. Constant terms have a degree of zero. First-degree expressions are called linear; second-degree expressions are called quadratic.

Space

Non-negative real numbers can describe a space in any number of dimensions. Classically one describes space in one dimension (length or displacement), two (area), or three (volume).

Time

Real numbers, either positive or negative, can describe time. Positive numbers describe a look forward in time; negative numbers describe a look backward. Classically, time has only one dimension.

Change

The classic expression of change is as a difference between two measurements of the same thing.

Isaac Newton introduced the concept of an instantaneous rate of change of a function, and showed how to predict such a rate of change. He called this system of prediction of rates of change calculus (from the Latin word for a stone). The two major branches of calculus are:

  • Differential calculus: predicts the rate of change of a function at a specified value of that function's domain.
  • Integral calculus: the inverse of differential calculus, it predicts the accumulation of a function value if its instantaneous rates over its domain are known.

Scientists often use rates of change of certain spatial values as functions of time. For example, one can differentiate displacement with respect to time and predict instantaneous speed, and then differentiate speed with respect to time to predict instantaneous acceleration.

Structure

Sir Isaac Newton assumed that space and time were fundamentally distinct. But Albert Einstein showed that space and time are actually continuous with one another, at least as a local effect. He described a space-time continuum with four dimensions. The first three were the classic dimensions of volume, while the fourth is the multiple of elapsed time by the speed of light.

Dr. Moshe Carmeli, in 1996, proposed a different continuum in which the fourth dimension was not time multiplied by the speed of light, but the speed of universal expansion (retreat from the original point of expansion) multiplied by a new "universal time constant." This constant has units of time and is the approximate nominal time for light from the farthest observed objects in the universe to reach the earth:

\tau \approx 13.5 \times 10^9 yr

Dr. John Hartnett, in 2007, proposed a refinement to Carmeli's model. To Carmeli's four dimensions, he added back the dimension of time. Hartnett used his new space-time-velocity continuum to account for the rotations of spiral galaxies (and their groups, clusters, and superclusters), and the perceived acceleration of the universe, without resorting to the fanciful concepts of dark matter and dark energy.

Notable Mathematicians

Using Math on CreationWiki

Main Article: Math help

MediaWiki Math is a feature that allows for the direct rendition of tagged mathematical symbols in an article. With this feature, an editor can make mathematical formulas look far more impressive with the TeX formatting that it provides.

MediaWiki Math is now available on CreationWiki. The recent transfer to a dedicated server enabled the installation of the support software that MediaWiki math requires.

References

  1. 1.0 1.1 1.2 Mathematicians U.S. Department of Labor Occupational Outlook Handbook, 2008-09.
  2. http://www.slate.com/blogs/lexicon_valley/2013/10/16/piraha_cognitive_anumeracy_in_a_language_without_numbers.html
  3. Gallian, Joseph A (1994). Contemporary Abstract Algebra (3rd ed.). Lexington, Massachusetts: D.C.Heath and Company. p. 25,35. ISBN 0-669-33907-5. 

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