English For MathematicsEducation

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Polynomials

• A polynomial is an expression that is the sum of a finite number of non-zero terms,each term consisting of the product of a constant and a finite number of variablesraised to whole number powers. For example, 𝑥2+ 2𝑥 − 3 is a polynomial in thesingle variable 𝑥.

• A polynomial expression is an expression that may be rewritten as a polynomial,by using commutativity, associativity and distributivity of addition andmultiplication. For example, (𝑥 − 1)(𝑥 + 3) is a polynomial expression, that,properly speaking, is not a polynomial.

• A polynomial function is a function that is defined by a polynomial, or,equivalently, by a polynomial expression. The two preceding examples define thesame polynomial function.

• Two important and related problems in algebra are the factorization ofpolynomials, that is, expressing a given polynomial as a product of otherpolynomials that can not be factored any further, and the computation ofpolynomial greatest common divisors. The example polynomial above can befactored as (𝑥 − 1)(𝑥 + 3). A related class of problems is finding algebraicexpressions 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. Here are listed fundamentalconcepts in abstract algebra :

• Sets: Rather than just considering the different types of numbers, abstract algebradeals with the more general concept of sets: a collection of all objects (calledelements) selected by property specific for the set. All collections of the familiartypes of numbers are sets. Other examples of sets include the set of all two-by-twomatrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all twodimensional vectors in the plane, and the various finite groups such as the cyclicgroups, which are the groups of integers modulo n. Set theory is a branch of logicand not technically a branch of algebra.

• Binary operations: The notion of addition (+) is abstracted to give a binaryoperation, ∗ say. The notion of binary operation is meaningless without the set onwhich the operation is defined. For two elements a and b in a set S, a ∗ b isanother element in the set; this condition is called closure. Addition (+),subtraction (-), multiplication (×), and division (÷) can be binary operations whendefined on different sets, as are addition and multiplication of matrices, vectors,and polynomials.

Abstract algebra

• Identity elements: The numbers zero and one are abstracted to give the notion of anidentity element for an operation. Zero is the identity element for addition and one isthe identity element for multiplication. For a general binary operator ∗ the identityelement 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. Not all sets and operatorcombinations have an identity element; for example, the set of positive naturalnumbers (1, 2, 3, ...) has no identity element for addition.

• Inverse elements: The negative numbers give rise to the concept of inverse elements.For addition, the inverse of a is written −a, and for multiplication the inverse is writtena−1. A general two-sided inverse element a−1 satisfies the property that a ∗ a−1 = 1 anda−1 ∗ a = 1 .

• Associativity: Addition of integers has a property called associativity. That is, thegrouping 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 bymost binary operations, but not subtraction or division or octonion multiplication.

• Commutativity: Addition and multiplication of real numbers are both commutative.That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2.In general, this becomes a ∗ b = b ∗ a. This property does not hold for all binaryoperations. For example, matrix multiplication and quaternion multiplication are bothnon-commutative.

GroupsCombining the previous concepts gives one of the most important structures inmathematics: a group. In mathematics, a group is an algebraic structure consisting of a setof elements together with an operation that combines any two elements to form a thirdelement. The operation satisfies four conditions called the group axioms, namely closure,associativity, identity and invertibility. One of the most familiar examples of a group is theset of integers together with the addition operation; the addition of any two integers formsanother integer. 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, e ∗ a and a ∗ e are

both identical to a.• 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.• The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical

to a ∗ (b ∗ c).If a group is also commutative—that is, for any two members a and b of S, a ∗ b is identicalto b ∗ a—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. Theassociativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)

GroupsThe nonzero rational numbers form a group under multiplication. Here, the identityelement 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 isbecause, in general, the multiplicative inverse of an integer is not an integer. Forexample, 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 theclassification of finite simple groups, mostly published between about 1955 and 1983,which separates the finite simple groups into roughly 30 basic types.Semigroups, quasigroups, and monoids are structures similar to groups, but moregeneral. They comprise a set and a closed binary operation, but do not necessarilysatisfy the other conditions. A semigroup has an associative binary operation, butmight not have an identity element. A monoid is a semigroup which does have anidentity but might not have an inverse for every element. A quasigroup satisfies arequirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however the binary operation might not beassociative. All groups are monoids, and all monoids are semigroups.

Example Algebraic properties satisfied by the natural numbersThe addition (+) and multiplication (×) operations on natural numbers asdefined above have several algebraic properties:• Closure under addition and multiplication: for all natural numbers a and

b, both a + b and a × b are natural numbers.• Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c

and a × (b × c) = (a × b) × c.• Commutativity: for all natural numbers a and b, a + b = b + a and a × b =

b × a.• Existence of identity elements: for every natural number a, a + 0 = a

and a × 1 = a.• Distributivity of multiplication over addition for all natural numbers a,

b, and c, a × (b + c) = (a × b) + (a × c).• No nonzero zero divisors: if a and b are natural numbers such that a × b

= 0, then a = 0 or b = 0.

Example

Algebraic properties satisfied by the integers

Rings and fieldsGroups just have one binary operation. To fully explain the behaviour of the differenttypes of numbers, structures with two operators need to be studied. The mostimportant of these are rings, and fields.A ring has two binary operations (+) and (×), with × distributive over +. Under the firstoperator (+) 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 required. Theadditive (+) identity element is written as 0 and the additive inverse of a is written as−a.Distributivity generalises the distributive law for numbers. 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 +.The integers are an example of a ring. The integers have additional properties whichmake it an integral domain.A field is a ring with the additional property that all the elements excluding 0 form anabelian group under ×. The multiplicative (×) identity is written as 1 and themultiplicative inverse of a is written as a−1.The rational numbers, the real numbers and the complex numbers are all examples offields.

Combining Like TermsLike terms have the same variables and the sameexponents of the variables. For example, 3𝑥 and5𝑥 are like terms because they have the samevariables, whereas 5𝑥 and 8𝑦 are unlike termsbecause they have different variables. Left Tablewill help you to identify like terms.Removing Parentheses and Combining LikeTerms. Sometimes in algebra it is necessary touse the distributive property to remove

parentheses andthen combineterms. Thisprocedure is alsocalled ‘‘simplifying’’an algebraicexpression.

Ratio and Proportion

A ratio is a comparison of two numbers. For example, consider the statement,‘‘About 14 out of every 100 people in the United States do not have healthinsurance.’’A ratio can be expressed by a fraction or by using a colon. In the preceding

example, the ratio14

100is the same as 14 to 100 or 14 : 100. Fractions are usually

reduced to lowest terms, so the ratio becomes7

50or 7 : 50.

It is important to understand that whatever number comes first in the ratiostatement is placed in the numerator of the fraction and whatever number comessecond in the ratio statement is placed in the denominator of the fraction. In

general, the ratio of a to b is written as𝑎

𝑏.

A proportion is a statement of equality of two ratios. For example,3

4=

6

8is a

proportion. Proportions can also be expressed using a colon, as 3 : 4 ¼ 6 : 8. Aproportion consists of 4 terms. Usually it is necessary to find the value of one

term given the other three terms. For example,5

8=

𝑥

24.

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Individual Task :Download Allan G. Bluman, pre-Algebra Demystified, McRaw-Hill, New York, 2004.