What Is Variable In Math

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A variable 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, andcomputer programming.

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 formulas and variables is only several hundred years old.

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:

State a mathematical definition for finding the number twice that of ANY other finite number:
double(x) = x + x.
Varying, in the context of mathematical variables, does not mean change in the course of time, but rather dependence on the context in which the variable is used. This can be the immediate ...
... context of the expression in which the variable occurs, as in the case of summation variables or variables that designate the argument of a function being defined.

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.

What it means for a variable to vary

Varying, in the context of mathematical variables, does not mean change in the course of time, but rather dependence on the context in which the variable is used. This can be the immediate context of the expression in which the variable occurs, as in the case of summation variables or variables that designate the argument of a function being defined. The context can also be larger, for instance when a variable is used to designate a value occurring in a hypothesis of the discussion at hand. In some cases nothing varies at all, and alternative names can be used instead of "variable": a parameter is a value that is fixed in the statement of the problem being studied (although its value may not be explicitly known), an unknownis a variable that is introduced to stand for a constant value that is not initially known, but which may become known by solving some equation(s) for it, and an indeterminate is a symbol that need not stand for anything else but is an abstract value in itself. In all these cases the term "variable" is often still used because the rules for the manipulation of these symbols are the same; however, in some languages other than English, one distinguishes between "variables" in functions and "unknown quantities" in equations ("incógnita" in Portuguese/Spanish, "inconnue" in French).

[edit]Examples
If one defines a function f from the real numbers to the real numbers by

then x is a variable standing for the argument of the function being defined, which can be any real number. In the identity

the variable i is a summation variable which designates in turn each of the integers 1, 2, ..., n (it is also called index because its variation is over a discrete set of values) while n is a parameter (it does not vary within the formula).

In the theory of polynomials, a polynomial of degree 2 is generally denoted as ax2 + bx + c, where a, b and c are called coefficients (they are assumed to be fixed, i.e., parameters of the problem considered) while x is called a variable. When studying this polynomial for its polynomial function this x stands for the function argument. When studying the polynomial as an object in itself, x is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.

Formulas from physics such as E = mc2 or PV = nRT (the ideal gas law) do not involve the mathematical notion of a variable, because the quantities E, m, P, V, n, and T are instead used to designate certain properties (energy, mass, pressure, volume, quantity, temperature) of the physical system.

Variables in 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 gender 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.

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