By: Stephanie Sirna

Slope Intercept is the equation of a straight line in the form y = mx + b where m is the slope of the line and b is its y-intercept

The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation:

A quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formwhere x represents a variable or an unknown, and a, b, and c are constant with a ≠ 0. (If a = 0, the equation is a linear equation.)

Pythagorean Theorem

A2+B2=C2

To find the length of the line opposite the hypotenuse (the longest line) all you have to do is square the lengths of the two longest sides and add them together. Their sum will be the length of the longest line squared. Similarly, one can subtract the length of lines A2 or B2 from C2 to fine the length of one of the smaller lines.

The acronym PEMDAS is useful when learning how to utilize the order of operations, a basic yet vital skill throughout the field of mathematics. While seemingly trivial, most of the toughest math problems boil down to PEMDAS. The “P” stands for “parenthesis.” Work in the parenthesis is always done first. Next, the “E” is for “exponents” followed by “M” and “D” or “multiplication” and “division” respectively. Finally, the “A” and “S” stand for “addition” and “subtraction.”

Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. Algebra can essentially be considered as doing computations similar to that of arithmetic with non-numerical mathematical objects. Initially, these objects were variables that either represented numbers that were not yet known (unknowns) or represented an unspecified number (indeterminate or parameter), allowing one to state and prove properties that are true no matter which numbers are substituted for the indeterminate.

Algebra

Overview of Algebra

Euclidean GeometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions(theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical systemhe Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, couched in geometrical language.

Oxyrhynchus papyrus (P.Oxy. I 29) showing fragment of Euclid's Elements

Trigonometry (from Greek trigōnon "triangle" + metron "measure") is a branch of mathematic that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies. It is also the foundation of the practical art of surveying.

Trigonometry

CalculusCalculus is the mathematical study of change,in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus(concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Calculus has widespread uses in science, economics, and engineering and can solve many problems that algebra alone cannot.

Isaac NewtonNewton completed no definitive publication formalizing his Fluxional Calculus; rather, many of his mathematical discoveries were transmitted through correspondence, smaller papers or as embedded aspects in his other definitive compilations, such as the Principia and Opticks. Newton would begin his mathematical training as the chosen heir of Isaac Barrow in Cambridge. His incredible aptitude was recognized early and he quickly learned the current theories. By 1664 Newton had made his first important contribution by advancing the binomial theorem, which he had extended to include fractional and negative exponents. Newton succeeded in expanding the applicability of the binomial theorem by applying the algebra of finite quantities in an analysis of infinite series. He showed a willingness to view infinite series not only as approximate devices, but also as alternative forms of expressing a term.

Gottfried Leibniz While Newton began development of his fluxional calculus in 1665-1666 his findings did not become widely circulated until later. In the intervening years Leibniz also strove to create his calculus. In comparison to Newton who came to math at an early age, Leibniz began his rigorous math studies with a mature intellect. He was a polymath, and his intellectual interests and achievements involved metaphysics, law, economics, politics, logic, and mathematics. In order to understand Leibniz’s reasoning in calculus his background should be kept in mind. Particularly, his metaphysics which considered the world as an infinite aggregate of indivisible monads, and his plans of creating a precise formal logic whereby, “a general method in which all truths of the reason would be reduced to a kind of calculation.” In 1672 Leibniz met the mathematician Huygens who convinced Leibniz to dedicate significant time to the study of mathematics. By 1673 he had progressed to reading Pascal’s Traité des Sinus du Quarte Cercleand it was during his largely autodidactic research that Leibniz said "a light turned on"[Like Newton, Leibniz, saw the tangent as a ratio but declared it as simply the ratio between ordinates and abscissas. He continued this reasoning to argue that the integral was in fact the sum of the ordinates for infinitesimal intervals in the abscissa; in effect, the sum of an infinite number of rectangles. From these definitions the inverse relationship or differential became clear and Leibniz quickly realized the potential to form a whole new system of mathematics. Where Newton shied away from the use of infinitesimals, Leibniz made it the cornerstone of his notation and calculus.

The Calculus Controversy

Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. If an individual coin toss or the roll of dice is considered to be a random event, then if repeated many times the sequence of random events will exhibit certain patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.

Probability Theory

Graph TheoryIn mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices or nodes and lines called edges that connect them. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see graph (mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

Number theory (or arithmetic) is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).

Number theory

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanic, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

Differential Equation

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation and study of mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Applied mathematics

Game theory is a study of strategic decision making. More formally, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."[ An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory. Game theory is mainly used in economics, political science, and psychology, as well as logic and biology. The subject first addressed zero-sum game, such that one person's gains exactly equal net losses of the other participant(s). Today, however, game theory applies to a wide range of behavioral relations, and has developed into an umbrella term for the logical side of decision science, to include both human and non-humans, like computers. Classic uses include a sense of balance in numerous games, where each person has found or developed a tactic that cannot successfully better his/her results, given the strategies of other players.

Game Theory