## variable

**-**

## Variable

# Variable

A variable is a mathematical and computer symbol that is used to symbolically represent a member of a given set and is typically denoted by a letter such as x, y, or z. The idea of a variable, invented during the late sixteenth century, is characteristic of modern mathematics and was not widely used in ancient times. Since a variable reflects a quantity that can take on different values, its use has become a critical part of nearly all disciplines that use mathematical models to represent the real world. Variables are often distinguished from constants that are known quantities and do not change in values.

The idea of using letters to represent variables was first suggested by French mathematician Francçois Viète (1540–1603) in his work, In artem analyticam isagoge (1591). Although his notations barely resemble modern day symbolism, they were an important step in the development of the concept of using a letter to represent a changing value in a mathematical equation. His ideas were further developed in the decades that followed. French mathematician and philosopher Renè Descartes (1596–1650) is generally credited for making standard, the use of the letters x, y, and z for variables.

## Characteristics of a variable

A variable is often denoted by a letter in an algebraic expression and represents a value that can be changed or varied. For example, in the expression x + 2, the letter x is a real variable and can take on the value of any real number. If x is 4, then the expression has a value of 6 because 4 + 2 = 6. Similarly, if x is 10, the expression has a value of 12. The number 2 in this expression is known as a constant because it never changes. Generally, a constant can be any number or letter in an equation whose value does not change.

In an equation, the value of a variable is often not given and is, therefore, called an unknown. In the equation y + 7 = 12, the letter y is an unknown variable and it represents some number. The value of the unknown that makes the equation true is called the solution or root of the equation. In this example, the solution of the equation is y = 5 because 5 + 7 = 12. Often, there is more than one solution to an equation so the unknown variable is equal to all of these values. The solution to the equation x^{2} = 4 is both 2 and -2 because each of these values make the equation true.

## Variables in a function

Some algebraic equations, known as functions, represent relationships between two variables. In these functions, the value of one variable is said to depend on the value of the other. For instance, the sales tax on a pair of gym shoes depends on the price of the shoes. The distance a car travels in a given time depends on its speed. In these examples, the sales tax and the distance traveled are called dependent variables because their value depends on the value of the other variable in the function. This variable, known as the independent variable, is represented by the price of the gym shoes and the speed of the car.

Using variables to represent unknowns was an important part of the development of algebra. Variables have distinct advantages over the rhetorical (written out) algebra of the ancient Greeks. They allow mathematical ideas to be communicated clearly

### KEY TERMS

**Algebraic expression** —A symbolic representation of a mathematical statement made up of numbers, letters, and operations.

**Constant** —A part of an algebraic expression that does not change, such as a number.

**Dependent variable** —A variable in a function whose value depends on the value of another variable in the function.

**Function** —A mathematical relationship between two or more variables.

**Independent variable** —A variable in a function whose value determines the value of the dependent variable.

**Rhetorical algebra** —A method of communicating mathematical ideas by using words to express relationships between values.

**Solution of the equation** —The value of a variable that makes an equation true.

**Unknown** —A term used to describe a variable whose value is not evident.

and briefly. The equation 2x^{2} + y = 6 is much clearer than the equivalent phrase “two times some number times itself, plus some other number is equal to six.” Variables also make mathematics more generally applicable. For instance, the area of a certain square with sides of 2 cm is 4 cm^{2}. The area of another square with 3 cm sides is 9 cm^{2}. By representing the side of any square with the variables, the area of any square can be represented by s^{2}.

Although any letter or character can represent any variable, over time, mathematicians and scientists have used certain letters to represent certain values. The letters x, y, and z are the most commonly used variables to represent unknown values in polynomial equations. The letter r is often used to represent the radius of a circle and the character q is used to signify an unknown angle. Other commonly used variables include t to represent time, s to represent speed, and p to represent pressure.

*See also* Solution of equation.

## Resources

### BOOKS

Bittinger, Marvin L, and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Burton, David M. Elementary Number Theory. Boston, MA: McGraw-Hill Higher Education, 2007.

Burton, David M. *The History of Mathematics: An Introduction.* New York: McGraw-Hill, 2007.

Reid, Constance. *From Zero to Infinity:* What Makes Numbers Interesting. Wellesley, MA: A.K. Peters, 2006.

Weisstein, Eric W. *The CRC Concise Encyclopedia of Mathematics.* Boca Raton, FL: Chapman & Hall/CRC, 2003.

Perry Romanowski

## Variable

# Variable

A variable is a mathematical symbol which is used to represent a member of a given set and is typically denoted by a letter such as x, y, or z. The idea of a variable, invented during the late sixteenth century, is characteristic of modern **mathematics** and was not widely used in ancient times. Since a variable reflects a quantity which can take on different values, its use has become a critical part of nearly all disciplines which use mathematical models to represent the real world.

The idea of using letters to represent variables was first suggested by the sixteenth century mathematician François Viète (1540-1603) in his work, *In artem analyticam isagoge* (1591). Although his notations barely resemble our modern day symbolism, they were an important step in the development of the concept of using a letter to represent a changing value in a mathematical equation. His ideas were further developed in the decades that followed. Rene Descartes (1596-1650) is generally credited for making standard, the use of the letters x, y, and z for variables.

## Characteristics of a variable

A variable is often denoted by a letter in an algebraic expression and represents a value which can be changed or varied. For example, in the expression x + 2, the letter x is a real variable and can take on the value of any real number. If x is 4 then the expression has a value of 6 because 4 + 2 = 6. Similarly, if x is 10 the expression has a value of 12. The number 2 in this expression is known as a constant because it never changes. Generally, a constant can be any number or letter in an equation whose value does not change.

In an equation, the value of a variable is often not given and is therefore called an unknown. In the equation y + 7 = 12, the letter y is an unknown variable and it represents some number. The value of the unknown which makes the equation true is called the solution or root of the equation. In this example, the solution of the equation is y = 5 because 5 + 7 = 12. Often, there is more than one solution to an equation so the unknown variable is equal to all of these values. The solution to the equation x2 = 4 is both 2 and -2 because each of these values make the equation true.

## Variables in a function

Some algebraic equations, known as functions, represent relationships between two variables. In these functions, the value of one variable is said to depend on the value of the other. For instance, the sales tax on a pair of gym shoes depends on the price of the shoes. The **distance** a car travels in a given time depends on its speed. In these examples, the sales tax and the distance travelled are called dependent variables because their value depends on the value of the other variable in the **function** . This variable, known as the independent variable, is represented by the price of the gym shoes and the speed of the car.

Using variables to represent unknowns was an important part of the development of **algebra** . Variables have distinct advantages over the rhetorical (written out) algebra of the ancient Greeks. They allow mathematical ideas to be communicated clearly and briefly. The equation 2x2 + y = 6 is much clearer than the equivalent phrase "two times some number times itself, plus some other number is equal to six." Variables also make mathematics more generally applicable. For instance, the area of a certain square with sides of 2 cm is 4 cm2. The area of another square with 3 cm sides is 9 cm2. By representing the side of any square with the variable s, the area of any square can be represented by s2.

Although any letter or character can represent any variable, over time, mathematicians and scientists have used certain letters to represent certain values. The letters x, y, and z are the most commonly used variables to represent unknown values in polynomial equations. The letter r is often used to represent the radius of a **circle** and the character q is used to signify an unknown **angle** . Other commonly used variables include t to represent time, s to represent speed, and p to represent **pressure** .

See also Solution of equation.

## Resources

### books

Bittinger, Marvin L., and Davic Ellenbogen. *Intermediate Algebra: Concepts and Applications.* 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Paulos, John Allen. *Beyond Numeracy.* New York: Alfred A. Knopf, Inc., 1991.

Weisstein, Eric W. *The CRC Concise Encyclopedia of Mathematics.* New York: CRC Press, 1998.

Perry Romanowski

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Algebraic expression**—A symbolic representation of a mathematical statement made up of numbers, letters, and operations.

**Constant**—A part of an algebraic expression which does not change, such as a number.

**Dependent variable**—A variable in a function whose value depends on the value of another variable in the function.

**Function**—A mathematical relationship between two or more variables.

**Independent variable**—A variable in a function whose value determines the value of the dependent variable.

**Rhetorical algebra**—A method of communicating mathematical ideas by using words to express relationships between values.

**Solution of the equation**—The value of a variable that makes an equation true.

**Unknown**—A term used to describe a variable whose value is not evident.

## variable

var·i·a·ble / ˈve(ə)rēəbəl/ • adj. 1. not consistent or having a fixed pattern; liable to change: *the quality of hospital food is highly variable* *awards can be for variable amounts.* ∎ (of a wind) tending to change direction. ∎ Math. (of a quantity) able to assume different numerical values. ∎ Bot. & Zool. (of a species) liable to deviate from the typical color or form, or to occur in different colors or forms.
2. able to be changed or adapted: *the drill has variable speed.* ∎ (of a gear) designed to give varying ratios or speeds.
• n. an element, feature, or factor that is liable to vary or change: *there are too many variables involved to make any meaningful predictions.* ∎ Math. a quantity that during a calculation is assumed to vary or be capable of varying in value. ∎ Comput. a data item that may take on more than one value during or between programs. ∎ Astron. short for variable star. ∎ (variables) the region of light, variable winds to the north of the northeast trade winds or (in the southern hemisphere) between the southeast trade winds and the westerlies.
DERIVATIVES: var·i·a·bil·i·ty / ˌve(ə)rēəˈbilitē/ n.var·i·a·ble·ness n.var·i·a·bly / -blē/ adv.

## variable

**variable** In the physical sciences, variables are the characteristics of entities which are physically manipulated, such as the heat or volume of a substance. In the social sciences, the term refers to attributes which are fixed for each person or other social entity, but which are observed to be at different levels, amounts, or strengths across samples and other aggregate groups. Variables measure a social construct (such as social class, age, or housing type) in a way which renders it amenable to numerical analysis. Thus, the key feature of a variable is that it is capable of reflecting variation within a population, and is not a constant.

There are various difficulties involved in the process of creating variables from constructs (operationalization). The central considerations here are validity (that the variable is a true measurement of the construct it is aimed at) and reliability (that its measurement is reliable).

Variables may be measured at different levels of measurement, but the basic distinction is between continuous variables such as income, and categoric or discrete variables such as class. Relatively few social variables are continuous—forming interval scales such as income or age. Most are discrete, forming ordinal and nominal scales, such as highest educational qualification obtained, or sex, respectively. The different levels of measurement have implications for the types of analysis that can be undertaken.

## variable

**variable** **1.** A unit of storage that can be modified during program execution, usually by assignment or read operations. A variable is generally denoted by an identifier or by a name.

**2.** The name that denotes a modifiable unit of storage.

**3.** See parameter.

**4.** In logic, a name that can stand for any of a possibly infinite set of values.

## variable

**variable ( vair-i-ă-bŭl) n.** a characteristic relating to an individual or group that is subject to variation.

*continuous v.*a variable found at any point on a numerical scale (e.g. weight).

*dependent v.*the variable in a study or experiment that is controlled by the independent variable and measured to determine the outcome of the study.

*discrete v.*a variable found only at a fixed point on a numerical scale (e.g. body weight).

*independent v.*the variable in a study or experiment that is manipulated in order to define the conditions of the study.

*qualitative v.*a descriptive characteristic, such as sex, race, or occupation.

*quantitative v.*a variable relating to a numerical scale, which may be continuous or discrete.

## variable

**variable** liable to vary. XIV. — (O)F. — L. *variābilis*, f. *variāre* VARY; see -ABLE.

Hence **variability** XVIII. So **variance** variation, difference XIV; discrepancy; dissension XV (*at v.* XVI). — OF. — L. *variantia*. **variant** †inconstant, not uniform; diverse, differing (*from*) XIV; sb. XIX. — (O)F. **variation** †difference, divergence XIV; fact or instance of varying XVI. — (O)F. or L.

## variable

**variable** In mathematics, symbol used to represent an unspecified quantity. Variables are used to express a range of possible values. For example, in the expression x^{2} + x + 1, the quantity *x* may be assigned the value of any real number; here *x* is said to be an independent variable. If *y* is defined by y = x^{2} + x + 1, then *y* is a dependent variable because its value depends on the value of *x*.