Mathematics
is the study of
numbers
,
shapes
, and
patterns
. The word comes from the
Greek
μ?θημα
(mathema), meaning "
science
,
knowledge
, or
learning
", and is sometimes shortened to
math
or
maths
.
It is the study of:
- Numbers
: including how things can be
counted
.
- Structure
: including how things are
organized
, but also how they can be or could have been. This subfield is usually called
algebra
.
- Place: where things are, and spatial arrangement, including arrangements of spaces themselves. This subfield is usually called
geometry
.
- Change: how things become different. This subfield is usually called
analysis
.
Applied math
is useful for solving problems in the real world. People working in
business
,
science
,
engineering
, and
construction
use mathematics.
[1]
[2]
Mathematics solves problems by using
logic
. One of the main tools of logic used by mathematicians is
deduction
. Deduction is a special way of thinking to discover and prove new truths using old truths. To a mathematician, the reason something is true (called a
proof
) is just as important as the fact that it is true, and this reason is often found using deduction. Using deduction is what makes mathematical thinking different from other kinds of scientific thinking, which might rely on
experiments
or on
interviews
.
[3]
Logic and reasoning are used by mathematicians to create general
rules
, which are an important part of mathematics. These rules leave out
information
that is not important so that a single rule can cover many situations. By finding general rules, mathematics solves many
problems
at the same time as these rules can be used on other problems.
[4]
These rules can be called
theorems
(if they have been proven) or
conjectures
(if it is not known if they are true yet).
[5]
Most mathematicians use non-logical and creative reasoning in order to find a logical proof.
[6]
Sometimes, mathematics finds and studies rules or ideas that we don't understand yet. Often in mathematics, ideas and rules are
chosen
because they are considered simple or neat. On the other hand, sometimes these ideas and rules are found in the real world after they are studied in mathematics; this has happened many times in the past. In general, studying the rules and ideas of mathematics can help us
understand
the world better. Some examples of math problems are addition, subtraction, multiplication, division, calculus, fractions and decimals.
Algebra
problems are solved by evaluating certain
variables
. A
calculator
answers every math problem in the four basic
arithmetic
operations.
- Mathematics includes the study of numbers and quantities. It is a branch of science that deals with the logic of shape, quantity, and arrangement. Most of the areas listed below are studied in many different fields of mathematics, including
set theory
and
mathematical logic
. The study of
number theory
usually focuses more on the structure and behavior of the integers rather than on the actual foundations of numbers themselves, and so is not listed in this given subsection.
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Natural numbers
|
Integers
|
Rational numbers
|
Real numbers
|
Complex numbers
|
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Ordinal numbers
|
Cardinal numbers
|
Arithmetic operations
|
Arithmetic relations
|
Functions
, see also
special functions
|
- Structural mathematics studies objects' and constructions' shape and integrity. These are areas of
algebra
and
calculus
.
- Some areas of mathematics study the shapes of things. Most of these areas are part of the study of
geometry
.
- Some areas of mathematics study the way things change. Most of these areas are part of the study of
analysis
.
- Applied math
uses symbolic logic to solve problems in areas like
engineering
and
physics
.
- Numerical analysis
?
Optimization
?
Probability theory
?
Statistics
?
Mathematical finance
?
Game theory
?
Mathematical physics
?
Fluid dynamics
-
Computational algorithms
These
theorems
and
conjectures
have interested mathematicians and amateurs alike:
- Pythagorean theorem
?
FLT
?
Goldbach's conjecture
?
Twin Prime Conjecture
?
Godel's incompleteness theorems
?
Poincare conjecture
?
Cantor's diagonal argument
?
Four color theorem
?
Zorn's lemma
?
Euler's Identity
?
Church-Turing thesis
These theorems and
hypotheses
have greatly changed mathematics:
- Central limit theorem
classification theorems of surfaces
?
Continuum hypothesis
?
Fourier Theorem
?
Fundamental theorem of calculus
?
Fundamental theorem of algebra
?
Fundamental theorem of arithmetic
?
Fundamental theorem of projective geometry
?
Gauss-Bonnet theorem
-
Kantorovich theorem
?
P Versus NP
?
Pythagorean theorem
?
Riemann hypothesis
These are a few conjectures that have been called "revolutionary":
- Beal Conjecture
(a generalization of FLT) ?
Birch and Swinnerton-Dyer Conjecture
?
Collatz Conjecture
?
Goldbach's Conjecture
?
Hodge Conjecture
?
Poincare Conjecture
- Set theory
?
Symbolic logic
?
Model theory
?
Category theory
?
Logic
?
Table of mathematical symbols
History and the world of mathematicians
[
change
|
change source
]
- History of mathematics
?
Timeline of mathematics
?
Mathematicians
?
Fields Medal
?
Abel Prize
?
Millennium Prize Problems (Clay Math Prize)
?
International Mathematical Union
?
Mathematics competitions
?
Lateral thinking
?
Mathematics and gender
There is no
Nobel Prize
in mathematics. Mathematicians can receive the
Abel Prize
and the
Fields Medal
for important works.
[7]
[8]
The
Clay Mathematics Institute
has said it will give one million dollars to anyone who solves one of the
Millennium Prize Problems
.
There are many tools used to do math or find answers to math problems.
- Older tools
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