Graph of y = cos x

y

It is important to notice that cosine will never be larger than 1 or smaller than -1. This will be useful on occasion in a calculus class.

Groups

What is a Group?
A group is a combination of a set S and a binary operation '*' ( defined in any way you choose, but) with the following properties:
1)An identity element e exists,
such that for every member a of S, e * a and a * e are both identical to a.
2)Every element has an inverse: for every member a of S, there exists a member a-1 such that a * a-1 and a-1 * a are both identical to the identity element.
3)The operation is associative:
if a, b and c are members of S, then
(a * b) * c is identical to a * (b * c).
Note:
If a group is also commutative - that is, for any two members a and b of S, a * b is identical to b * a – then the group is said to be Abelian.
Examples
1)the set of integers under the operation of addition is a group.
In this group, the identity element is 0 and the inverse of any element a is its negation, -a.
The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)
2)The nonzero rational numbers form a group under multiplication.
Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a.
The inverse of a is 1/a, since a × 1/a = 1.

3)The integers under the multiplication operation do not form a group. Because, in general, the multiplicative inverse of an integer is not an integer.
a)4 is an integer, but its multiplicative inverse is 1/4, which is not an integer.

The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.

Elementary algebra

Elementary algebra,
in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra), also called second year and third year algebra;

Elementary algebra

Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic.
In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur.

• In algebra, numbers are often denoted by symbols (such as a, x, or y).

This is useful because: It allows the general formulation of arithmetical laws
(such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance, "Find a number x such that 3x + 1 = 10"). It allows the formulation of functional relationships (such as "If you sell x tickets, then your profit will be 3x - 10 dollars, or f(x) = 3x - 10, where f is the function, and x is the number to which the function is applied.").

Polynomial
A polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant positive whole number exponent).

• For example, 2x+5 is a polynomial in the single variable x.

• An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials.
  
• The example polynomial above can be factored as A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.

Sets:
Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property, specific for the set.
All collections of the familiar types of numbers are sets.
Other examples of sets
the set of all two-by-two matrices,
the set of all second-degree polynomials (ax2 + bx + c),
the set of all two dimensional vectors in the plane
and the various finite groups such as the cyclic groups which are the group of integers modulo n.

Set theory is a branch of logic and not technically a branch of algebra.
Binary operations:

The notion of addition (+) is abstracted to give a binary operation, * say. The notion of binary operation is meaningless without the set on which the operation is defined.

• For two elements a and b in a set S a*b gives another element in the set; this condition is called closure.

Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as is addition and multiplication of matrices, vectors, and polynomials.

Identity elements:

The numbers zero and one are abstracted to give the notion of an identity element for an operation.

• Zero is the identity element for addition and one is the identity element for multiplication.
  

• For a general binary operator * the identity element e must satisfy a * e = a and e * a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a.
  
• However, if we take the positive natural numbers and addition, there is no identity element.

Inverse elements:

The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is -a, and for multiplication the inverse is 1/a.

• A general inverse element a-1 must satisfy the property that a * a-1 = e and a-1 * a = e. Associativity:

Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum.

• For example: (2+3)+4=2+(3+4).

In general, this becomes (a * b) * c = a * (b * c).

This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

Commutativity:

Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum.

• For example: 2+3=3+2. In general, this becomes a * b = b * a. Only some binary operations have this property.
  
• It holds for the integers with addition and multiplication, but it does not hold for matrix multiplication or quaternion multiplication .

Types of Groups

Semigroups, quasigroups, and monoids are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions.

1.A semigroup has an associative binary operation, but might not have an identity element.
2.A monoid is a semigroup which does have an identity but might not have an inverse for every element.
3.A quasigroup satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative. All groups are monoids, and all monoids are semigroups.
4.Rings and fields—structures of a set with two particular binary operations, (+) and (×) ring and field Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied.
The most important of these are rings, and fields. Distributivity generalised the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence).

• For the integers (a + b) × c = a×c+ b×c and c × (a + b) = c×a + c×b, and × is said to be distributive over +. A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an Abelian group.
  
• Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not allowed. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.
  
• The integers are an example of a ring. The integers have additional properties which make it an integral domain. A field is a ring with the additional property that all the elements excluding 0 form an Abelian group under ×.
  
• The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a-1.
  
• The rational numbers, the real numbers and the complex numbers are all examples of fields.

Idea of Algebra

• Algebra is used to make statements of mathematical problems in a brief

form.

• Algebra is the Shorthand of O Math. In arithmetic we are using numerals like 5,-8,3/8 etc.
• In Algebra we are using letters in place of numbers.These letters are 'number SYMBOLS' which are unknown or unspecified.They are called VARIABLES.Use of variables gives us a generalised result.
• Example look the mathematical statement '1+2=3'.
  
• In words 'the sum of 1 and 2 is 3'. In general the sum of two numbers is a third number. 5+6=11,8+7=15 . . . and so on.
• We write this using variables as 'a+b=c'. This is the generalized form of adding 2numbers. Here a,b,c are variables(any number).
  
• Your AGE is a variables.Age is changing every year.
  
• But Year of birth is a CONSTANT.

Algebra

• Algebra is a branch of mathematics concerning the study of structure, relation and quantity.
• The name is derived from the treatise written in Arabic by the Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.
• Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics.
  
• Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.
• Algebra is much broader than elementary algebra and can be generalized.
  
• In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.