## Fraction (Pecahan)

A fraction (from Latin: fractus, “broken”) represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: and 17/3) consists of an integer numerator, displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. The picture to the right illustrates or 3/4 of a cake. Read more »

## Integer

An integer is a number that can be written without a fractional or decimal component. For example, 21, 4, and ?2048 are integers; 9.75, 5½, and ?2 are not integers. The set of integers is a subset of the real numbers, and consists of the whole numbers (0, 1, 2, 3, …) and the negatives of the non-zero natural numbers (?1, ?2, ?3, …).

The name derives from the Latin integer (meaning literally “untouched,” hence “whole”: the word entire comes from the same origin, but via French[1]). The set of all integers is often denoted by a boldface Z (or blackboard bold \mathbb{Z}, Unicode U+2124 ?), which stands for Zahlen (German for numbers, pronounced [?tsa?l?n]).[2]

The integers (with addition as operation) form the smallest group containing the additive monoid of the natural numbers. Like the natural numbers, the integers form a countably infinite set. In algebraic number theory, these commonly understood integers, embedded in the field of rational numbers, are referred to as rational integers to distinguish them from the more broadly defined algebraic integers. Read more »

## Function (mathematics)

A function is a relation that uniquely associates members of one set with members of anotherset. More formally, a function from  to  is an object  such that every  is uniquely associated with an object . A function is therefore a many-to-one (or sometimes one-to-one) relation. The set  of values at which a function is defined is called its domain, while the set  of values that the function can produce is called its range. The term “map” is synonymous with function.

Examples of functions over the reals  include  (many-to-one),  (one-to-one),  (two-to-one except for the single point ), etc.

Unfortunately, the term “function” is also used to refer to relations that map single points in the domain to possibly multiple points in the range. These “functions” are called multivalued functions (or multiple-valued functions), and arise prominently in the theory of complex functions, where the presence of multiple values engenders the use of so-called branch cuts.

Several notations are commonly used to represent (non-multivalued) functions. The most rigorous notation is , which specifies that  is function acting upon a single number  (i.e.,  is a univariate, or one-variable, function) and returning a value . To be even more precise, a notation like “, where ” is sometimes used to explicitly specify the domain and codomain of the function. The slightly different “maps to” notation  is sometimes also used when the function is explicitly considered as a “map.”

Generally speaking, the symbol  refers to the function itself, while  refers to the valuetaken by the function when evaluated at a point . However, especially in more introductory texts, the notation  is commonly used to refer to the function itself (as opposed to the value of the function evaluated at ). In this context, the argument  is considered to be adummy variable whose presence indicates that the function  takes a single argument (as opposed to , etc.). While this notation is deprecated by professional mathematicians, it is the more familiar one for most nonprofessionals. Therefore, unless indicated otherwise by context, the notation  is taken in this work to be a shorthand for the more rigorous .

## Algebra

The Columbia Encyclopedia, Sixth Edition | 2008 | The Columbia Encyclopedia, Sixth Edition. Copyright 2008 Columbia University Press. (Hide copyright information) Copyright

algebra branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and multiplication) and relationships (such as equality) connecting the elements. Thus, a + a =2 a and a + b = b + a no matter what numbers a and b represent.

Principles of Classical Algebra

In elementary algebra letters are used to stand for numbers. For example, in the equation ax2 + bx +c=0, the letters a, b, and c stand for various known constant numbers called coefficients and the letter x is an unknown variable number whose value depends on the values of a, b, and c and may be determined by solving the equation. Much of classical algebra is concerned with finding solutions to equations or systems of equations, i.e., finding the roots , or values of the unknowns, that upon substitution into the original equation will make it a numerical identity. For example, x =-2 is a root of x2 -2 x -8=0 because (-2) 2 -2(-2)-8=4+4-8=0; substitution will verify that x =4 is also a root of this equation.

The equations of elementary algebra usually involve polynomial functions of one or more variables (see function ). The equation in the preceding example involves a polynomial of second degree in the single variable x (see quadratic ). One method of finding the zeros of the polynomial function f ( x ), i.e., the roots of the equation f ( x )=0, is to factor the polynomial, if possible. The polynomial x2 -2 x -8 has factors ( x +2) and ( x -4), since ( x +2)( x -4)= x2 -2 x -8, so that setting either of these factors equal to zero will make the polynomial zero. In general, if ( xr ) is a factor of a polynomial f ( x ), then r is a zero of the polynomial and a root of the equation f ( x )=0. To determine if ( xr ) is a factor, divide it into f ( x ); according to the Factor Theorem, if the remainder f ( r )—found by substituting r for x in the original polynomial—is zero, then ( xr ) is a factor of f ( x ). Although a polynomial has real coefficients, its roots may not be real numbers; e.g., x2 -9 separates into ( x +3)( x -3), which yields two zeros, x =-3 and x =+3, but the zeros of x2 +9 are imaginary numbers. Read more »