Essay—Mathematics in Ten Minutes (Issue 31)

Summary—Mathematics in Ten Minutes summarizes the branches of mathematics including geometry, arithmetic, algebra, analytic geometry, trigonometry, calculus, fractals, number theory, group theory, and probability and statistics.

The Greek Plato’s (427-347 BC) theories of knowledge and Forms holds that true or a priori knowledge must be certain and infallible, and it must be of real objects or Forms.  Thales and Pythagoras laid the foundation for Plato by founding geometry as the first mathematical discipline.  Mathematics is the systematic treatment of Forms and relationships between Forms.  It is the science of drawing conclusions and is the primordial foundation of all science.

Geometry.  Geometry is the division of mathematics that deals with space and time.  In its simplest outward appearance geometry is concerned with problems such as determining the areas and diameters of multi-dimensional figures and the volumes of solids as well as the areas of surfaces.  Other foundations of geometry include analytic geometry, topology, fractal geometry (ie. fractions of dimensions) and non-Euclidean geometry (ie. based on less than the five Euclidean axioms or rules).  The goal of geometry is to calculate properties such as area, boundary and diameter.  The Greek mathematician Pythagoras (582-500 BC) laid down the foundation for analytic geometry by showing that the various erratic and disconnected laws of geometry could be proved to follow the logical endings of a limited number of axioms.

Arithmetic.  Arithmetic is the knack of counting.  Numbers used in counting are called positive integers.  They are created by adding one to each number unto infinity—so that each number in the sequence is one more than its predecessor.  Civilizations through history have developed different types of mathematical systems.  One of the most common usages is that used in modern cultures in which components are counted in groups of ten.  This so-called decimal system is used in modern mathematics.  Arithmetic is concerned with the ways that numbers are joined via the operators of addition, subtraction, multiplication, division and roots.  Numbers also includes negative numbers, fractional numbers and irrational numbers (eg. p = 3.1416).  The base-ten integers are represented by numbers that are articulated by the powers of ten.

Algebra.  Algebra is the branch of mathematics which employs letters to correspond to essential arithmetic relationships.  As with arithmetic, the basic operations are addition, subtraction, multiplication, division and roots.  However, as with arithmetic, algebra cannot generalize mathematical relations such as the Pythagorean Form—which states that the sum of the squares of the sides of any right triangle is also a square.  Algebra is concerned with solving equations by using symbols instead of numbers—and using arithmetic operations in order to establish ways of handling symbols.  Modern algebra has evolved from classical algebra by increasing its focus on the structures of mathematics.  Modern mathematics considers modern algebra to be a set of objects with rules connecting them.  In its basic form algebra may be depicted as the language of mathematics.

Analytic Geometry.  It was Descartes (1596-1650) who first realized that arithmetic, geometry and algebra are the same thing—ie. analytic geometry.  In analytic geometry lines, curves and geometric figures are represented by algebraic expressions using a set of coordinates.  Analytic geometry has contributed to the advancement of mathematics because it has unified the concepts of analysis and geometry (both spatial and temporal).  The study of non-Euclidean geometries of space and time having more than three dimensions would not be possible without analytic geometry.  As well, the modus operandi of analytic geometry made possible the representation of numbers and algebraic expressions in geometric terms which have cast new light on calculus and other problems in mathematics.

Trigonometry.  Trigonometry is the division of mathematics concerned with the relationships between the sides and angles of triangles. The two branches of trigonometry are plane trigonometry (concerned with figures lying in a single plane) and spherical trigonometry (concerned with triangles that are spherical sections).  The initial purpose of trigonometry was in the playing fields of astrophysics, surveying and navigation.  The main problem was to determine the distance such as that between the moon and the earth.  The notion of the trigonometric angle is fundamental to the study of trigonometry.  Other purposes of trigonometry include engineering, physics, and chemistry—particularly in the field of episodic phenomena such as the study of bridge-building and the stream of electrical current.

Calculus.  Calculus is the mathematics of motion.  It is the division of mathematics concerned with the study of such notions as the slope of a curve, the rate of change, the computation of the maximum and minimum values of functions, and the calculation of area bounded by curves.  Calculus is used in engineering, biological, material and societal sciences.  It is also used in the physical sciences by studying falling bodies, the rate of decomposition of radioactive substances, and the pace of transformation of chemical reactions.  Calculus is broadly used in the study of probability and statistics.  It can also be used in many problems involving the concept of acute quantities such as the fastest, largest, slowest and roots.

Fractals.  A fractal is a geometric phenomenon that maintains a detailed structure under any level of scaling.  They are self-similar in that they have the property such that each small portion of the fractal can be viewed as a reduced-scale replica of the sum total.  Examples of fractals include broccoli snowflakes, mountains, broccoli, lightning and galaxy clusters.  In 1975 the French mathematician Benoit Mandelbrot adopted an abstract definition of fractional dimensions that were used in Euclidean geometry.  Mandelbrot posed the question—How long is the coastline of Britain?  Appealing to relativity, Mandelbrot pointed out that the answer depends on one’s perspective.  From space, the coastline is shorter than it is to someone who is walking.  That is because on foot the observer is exposed to greater detail and must travel farther.  Ultimately, according to Mandelbrot, when the shape of each pebble is taken into account, the coastline turns out to have infinite length.

Number Theory.  Number theory is the division of mathematics that deals with the belongings and relations between numbers.  The theory of numbers consists mainly of mathematical analysis.  It is concerned with the study of integers.  A perfect number is a positive integer that is equal to the sum of all its positive divisors.  For example, 6 equals 1 + 2 + 3—and 28 equals 1 + 2 + 4 + 7 + 14.  A positive number that is not perfect is imperfect and is deficient or abundant according to whether the sum of its positive, proper divisors is smaller or larger than the number itself.  Thus 8 with proper divisors (1 + 2 + 4 = 7) is deficient—while 12 with proper divisors (1 + 2 + 3 + 4 + 6 = 16) is abundant.

Group Theory.  Group theory is the fundamental configuration of algebra consisting of a set of elements and an operator.  The operator uses two elements of the set and forms another element of the set in such a way as to meet with certain conditions.  Group theory is the subject of powerful study contained within the field of mathematics.  In relativity theory (1905—ie. the fundamental law of space and time) group theory plays a pivotal role—also known as the Lorentz contractor from relativity—which is also coincident with the Pythagorean Form.  A case in point of group theory is found in the set of all numbers with the operator of addition.  The operator + takes two numbers (eg. 3 and 5) and forms their sum of 3 + 5 = 8.

Probability and Statistics.  Probability is the division of mathematics that deals with the quantitative measuring of likelihood that an incident or an experiment will have a specific outcome.  It is based on the study of permutations and combinations and is the necessary foundation for statistics.  Permutations and combinations are the arrangement of elements and are the basis of probability and statistics.  Probability and statistics are used in actuarial science and began in an attempt to answer questions arising from games of chance.  Statistics is the division of mathematics that deals with the set, association and analysis of numerical data.  It deals with problems such as decisionmaking and experiment design.  Simple forms of statistics have been in use since the beginning of civilization when symbolic representations were used to record numbers of people and animals.

Conclusion.  Ancient mankind lived in the fear and dread of natural events because he could not explain them.  Fables and enchantments dominated his thought process.  Increasingly, man began to understand the laws of nature.  A second illumination is needed so that we can live in peace with our unearthing and constructions.  A movement must take place in the emphasis from an old-style humanity to a new-style science in our educational system.  This shift cannot take place through normal educational means.  The scientific material available is only read by a small group of people.  We clearly require material with sufficient appeal and persuasive power to include the enlighteningly as the scientifically uninformed multitudes.  There is promising evidence that the new enlightenment may come to pass—that the cultural gap between our new technology and its meaning in terms of human values that in the end may be bridged.