# बीजगणित

बीजगणित गणितया छगू ख्यः ख। थ्व ख्यले स्ट्रक्चर, सम्बन्धमात्राया सीकेज्या जुइ। रेखागणित, गणितीय एनालाइसिस, कम्बिनेटोरिक्स, व अङ्क सिद्धान्त नापं बीजगणित गणितया छगू मू ख्यः ख। आधारभूत बीजगणितयात साधारणकथं माध्यमिक शिक्षाया पाठ्यक्रमय् स्यनिगु या। थुकिलिं बीजगणितया आधारभूत विचाःतेगु म्हसीका बिगु या। थन्यागु विचाय् ल्याखँतेगु तनेज्यागुणना, भेरिएबलया विचा, पोलिनोमियलया अर्थ, फ्याक्टोराइजेसन, रुट सीकिगु आदि ला।

बीजगणित आधारभूत बीजगणित स्वया यक्व तधं। बीजगणितय् ल्याखँ नाप प्रत्यक्ष ज्या यायेगु जक्क मखु, सिम्बोल, भेरिएबल, सेट, गणितीय इलेमेन्ट आदि नाप नं ज्या यायेगु जुइ। तनेज्या व गुणनायात साधारण अपरेसनया कथं कायेगु या। थिमिगु स्पष्ट परिभाषां ग्रुप, रिङ्ग, फिल्ड संरचना तक्क थ्यंकी।

## वर्गीकरण

बीजगणित यात थ्व कथं बायेछिं:

In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the algebraic structure. The list includes a number of areas of functional analysis:

## आधारभूत बीजगणित

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.").

### पोलिनोमियल

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 whole number exponent). For example, x2 + 2x − 3 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 (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

## एब्स्ट्र्याक्ट बीजगणित

स्वयादिसँ: Algebraic structure

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

सेट: 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 include 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.

बाइनरी अपरेसन: 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, ab is 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 ae = a and ea = 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 aa−1 = e and a−1a = 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 (ab) ∗ c = a ∗ (bc). 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 ab = ba. 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 .

### छगू बाइनरी अपरेसन दुगु सेटया ग्रुप – संरचना

मू पौ: Group (mathematics)
स्वयादिसँ: Group theory

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties:

• An identity element e exists, such that for every member a of S, ea and ae are both identical to a.
• Every element has an inverse: for every member a of S, there exists a member a−1 such that aa−1 and a−1a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (ab) ∗ c is identical to a ∗ (bc).

If a group is also commutative—that is, for any two members a and b of S, ab is identical to ba—then the group is said to be Abelian.

For example, 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)

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.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is ¼, 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.

 Set: अपरेसन Closed दसु प्राकृतिक ल्याखँ N इन्टिजर Z र्‍यासनल ल्याखँ Q (also real R and complex C numbers) इन्टेजर मोडुलो 3: Z3 = {0, 1, 2} + × (w/o zero) + × (w/o zero) + − × (w/o zero) ÷ (w/o zero) + × (w/o zero) Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Identity 0 1 0 1 0 N/A 1 N/A 0 1 इन्भर्स N/A N/A −a N/A −a N/A 1/a N/A 0, 2, 1, respectively N/A, 1, 2, respectively असोसियटिभ Yes Yes Yes Yes Yes No Yes No Yes Yes कम्युटेटिभ Yes Yes Yes Yes Yes No Yes No Yes Yes संरचना monoid monoid Abelian group monoid Abelian group quasigroup Abelian group quasigroup Abelian group Abelian group (Z2)

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. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. 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.

### Rings and fields—structures of a set with two particular binary operations, (+) and (×)

मू पौतः: ring (mathematics)field (mathematics)
स्वयादिसँ: Ring theory

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.

## अल्जेब्रा नांया वस्तु

The word algebra is also used for various algebraic structures: