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1 Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. Historically, it has been referred to as "the calculus of infinitesimals", or "infinitesimal calculus". Most basically, calculus is the study of change, in the same way that geometry is the study of space.

Calculus has widespread applications in science, economics, and engineering and is used to solve problems for which algebra alone is insufficient. Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus, that are related by the fundamental theorem of calculus. In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions.

2 Geometry (Ancient Greek: γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.

3 plane curve is a curve in a Euclidian plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.

A smooth plane curve is a curve in a real Euclidian plane R2 is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation f(x,y) = 0, where f is a smooth function of two variables, and the partial derivatives fx and fy are not simultaneously equal to 0. In other words, a smooth plane curve is a plane curve which "locally looks like a line" with respect to a smooth change of coordinates.

4 In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation

F(x,y,z) = 0

applied to homogeneous coordinates x:y:z for the projective plane; or the inhomogeneous version for the affine space determined by setting z = 1 in such an equation. Here F is a non-zero linear combination of the third-degree monomials

x3, y3, z3, x2y, y2x, y2z, z2x, z2x, z2y, xyz.

5 n mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1]

i^2=-1.\,

Every complex number can be written in the form a + bi, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively.

6 In mathematics, a cubic function is a function of the form

f(x)=ax^3+bx^2+cx+d,\,

where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function.

7 in mathematics, a polynomial is an expression constructed from variables (also known as indeterminates) and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and also because its third term contains an exponent that is not a whole number.

8 Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulas and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra


9 In mathematics, a natural number (also called counting number) can mean either an element of the set {1, 2, 3, ...} (the positive integers) or an element of the set {0, 1, 2, 3, ...} (the non-negative integers). The latter is especially preferred in mathematical logic, set theory, and computer science.

Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), and they can be used for ordering ("this is the 3rd largest city in the country").

10 Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic.[1] The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Mathematical logic is often divided into the subfields of set theory, model theory, recursion theory, and proof theory and constructive mathematics. These areas share basic results on logic, particularly first-order logic, and definability.

11. In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards complete formalisation brings diminishing returns in certainty, and a lack of readability for humans. Therefore discussion of axiomatic systems is normally only semi-formal. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical proof within a formal system.


12 In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The first twenty-five prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (sequence A000040 in OEIS).

See the list of prime numbers for a longer list. The number 1 is by definition not a prime number. The set of prime numbers is sometimes denoted by \mathbb{P}.

13 Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study.

Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. (See the list of number theory topics.)

The term "arithmetic" is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves).

14 In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction \frac{a}{b}, where b is not zero. a is called the numerator, and b the denominator.

Each rational number can be written in infinitely many forms, such as \frac{3}{6} = \frac{2}{4} = \frac{1}{2}, but it is said to be in simplest form when a and b have no common divisors except 1 (i.e., they are coprime).

15 The decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system.

16 Decimal notation is the writing of numbers in a base-10 numeral system. Examples are Roman numerals, Brahmi numerals, and Chinese numerals, as well as the Arabic numerals used by speakers of English. Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). Brahmi numerals had symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100 and another for 1000. Chinese has symbols for 1–9, and fourteen additional symbols for higher powers of 10, which in modern usage reach 1044.