Variable: Information from Answers.com
A variable (pronounced /ˈvɛərɪəbl/) is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use. The concepts of constants and variables are fundamental to all modern mathematics, science, engineering, and computer programming.
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History
Much of the basic theory for which we use variables today, such as school geometry and algebra, was developed thousands of years ago, but the use of symbolic formulae and variables is only several hundreds of years old. Arabic algebra texts such as the Al-Jabr still described mathematics in full text rather than symbolic formulae, but denoted the variable quantity in Arabic word šay' شَيْء = “thing”;. This was taken into Old Spanish with the pronunciation “šei”, which was written xei, and was soon habitually abbreviated to x; and this is still the customary variable name in many fields today. It started the habit of using letters to represent variables in algebra. Beyond mathematics, “X” has come to represent a generic placeholder variable whose value is unknown or secret, as in project X or mister X.
What it means for a variable to vary
Varying, in the context of mathematical variables, does not mean change, but rather, dependence on the values of other variables in the expression in which the variable occurs, or dependence of the value of the expression on that of the variable. However, in some languages other than English, one distinguishes between "variables" in functions and "unknown quantities" in equations ("incógnita" in Spanish, "inconnue" in French).
Examples
In the expression from physics E = m c2, E and m are variables, while c is a constant, the velocity of light in a vacuum. This famous equation describes how the value E and m are related, but that c is fixed.
In the ideal gas law, PV = nRT describes the relationship between volume V, pressure P and temperature T; here, n is a constant, but a chemist will try to juggle one or more of the variables to fit the others, knowing full they cannot beat the law.
In mathematics, in the theory of polynomials, a second-order polynomial is generally denoted as a x2 + b x + c, where a, b and c are called coefficients while x is called a variable, as what is usually being studied is how the value of the function varies with x, for given values of a, b and c.
Notation
In mathematics, single-symbol names for variables are the norm, with x, y, z, and t being most common; constants are usually denoted as a, b, c. In written mathematics, variables and constants are usually set in an italic typeface.
Specific branches and applications of mathematics usually have specific naming conventions for variables. Variables with similar roles or meanings are often assigned consecutive letters. For example, the three axes in 3D coordinate space are conventionally called x, y, and z, while random variables in statistics are usually named X, Y, Z. In physics, the names of variables are largely determined by the physical quantity they describe, but various naming conventions exist.
A convention sometimes followed in statistics is to use X, Y, Z for the names of random variables, with these being replaced by x, y, z for observations or sample outcomes of those random variables. Another convention sometimes used in statistics is to denote population values of particular statistics by lower (or upper) case Greek letters, with sample-based estimates of those quantities being denoted by the corresponding lower (or upper) case letters from the ordinary alphabet.
General introduction
Variables are used in open sentences. For instance, in the formula x + 1 = 5, x is a variable which represents an "unknown" number. Variables are often represented by Greek or Roman letters and may be used with other special symbols.
In mathematics, variables are essential because they let quantitative relationships to be stated in a general way. If we were forced to use actual values, then the relationships would only apply in a more narrow set of situations. For example:
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- State a mathematical definition for finding the number twice that of ANY other finite number:
double(x) = x + x.
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- Now, all we need to do to find the double of a number is replace x with any number we want.
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- double(1) = 1 + 1 = 2
- double(3) = 3 + 3 = 6
- double(55) = 55 + 55 = 110
- etc.
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So in this example, the variable x is a "placeholder" for any number— that is to say, a variable. One important thing we assume is that the value of x does not change, even though we do not know what x is. But in some algorithms, obviously, will change x, and there are various ways to then denote if we mean its old or new value— again, generally not knowing either, but perhaps (for example) that one is less than the other.
Name conventions
Mathematics has many conventions. Below are some of the more common. Many of the symbols have other conventional uses, but they may actually represent a constant or a specific function rather than a variable.
- ai is often used to denote a term of a sequence.
- a, b, c, and d (sometimes extended to e and f) usually play similar roles or are made to represent parallel notions in a mathematical context. They often represent constants.
- The coefficients in an equation, for example the general expression of a polynomial or a Diophantine equation are often a, b, c, d, e, and f.
- f and g (sometimes h) commonly denote functions.
- i, j, and k are often used as subscripts or index variables.
- l and w are often used to represent the length and width of a figure.
- m and n usually denote integers and usually play similar roles or are made to represent parallel notions in a mathematical context.
- n typically denotes a count of objects, or, in statistics, the number of individuals.
- p and q often represent prime numbers, or relatively prime numbers.
- p, q, and r usually play similar roles or are made to represent parallel notions in a mathematical context.
- r often denotes a remainder or modulus.
- r, s, and t usually play similar roles or are made to represent parallel notions in a mathematical context.
- u and v usually play similar roles or are made to represent parallel notions in a mathematical context, such as denoting a vertex (graph theory).
- w, x, y, and z usually play similar roles or are made to represent parallel notions in a mathematical context, such as representing unknowns in an equation.
- x, y and z correspond to the three Cartesian axes. In many two-dimensional cases, y will be expressed in terms of x; if a third dimension is added, z is expressed in terms of x and y.
- z is a common variable for a complex number.
- α, β, γ, θ and φ commonly denote angle measures.
- ε usually represents an arbitrarily small positive number.
- λ is used for eigenvalues.
- σ often denotes a sum, or the standard deviation in a statistical context.
Applied statistics
In statistics, variables refer to measurable attributes, as these typically vary over time or between individuals. Variables can be discrete (taking values from a finite or countable set), continuous (having a continuous distribution function), or neither. Temperature is a continuous variable, while the number of legs of an animal is a discrete variable. This concept of a variable is widely used in the natural, medical, and social sciences.
In causal models, a distinction is made between "independent variables" and "dependent variables", the latter being expected to vary in value in response to changes in the former. In other words, an independent variable is presumed to potentially affect a dependent one. In experiments, independent variables include factors that can be altered or chosen by the researcher independent of other factors.
So, in an experiment to test if the boiling point of water changes with altitude, the altitude is under direct control and is the independent variable, and the boiling point is presumed to depend upon it, so being the dependent variable. The results of an experiment, or information to be used to draw conclusions, are known as data. It is often important to consider which variables to allow, or directly control or eliminate, in the design of experiments.
There are also quasi-independent variables, which are used by researchers to group things without affecting the variable itself. For example, to separate people into groups by their sex does not change whether they are male or female. Or a researcher may separate people, arbitrarily, on the amount of coffee they had drunk before beginning an experiment. The researcher cannot change the past, but can use it to split people into groups.
While independent variables can refer to quantities and qualities that are under experimental control, they can also include extraneous factors that influence results in a confusing or undesired manner. In statistics the technique to work this out is called correlation.
If strongly confounding variables exist that can substantially change the result, it makes it harder to interpret. For example, a study on cancer against age will also have to take into account variables such as income, location, stress, and lifestyle. Without considering these, the results could be grossly inaccurate deductions. Because of this, controlling unwanted variables is important in research.
See also
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