1. By: Stephanie Sirna

2. 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

3. 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:

4. A quadratic equation is a
univariate polynomial
equation of the second
degree. A general
quadratic equation can
be written in the form
where 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.)

5. 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.

6. 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.”

7. 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

8. Overview of Algebra

9. Euclidean Geometry
Euclidean 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

10. 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

11. Calculus
Calculus 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.

12. Isaac Newton
Newton 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.

13. 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.

14. The Calculus Controversy

15. 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

16. Graph Theory
In 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.

17. 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

18. 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

19. 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

20. 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