Variable (mathematics)

A variable is a name for an object that can take various values ​​from a set of elements; in other words, the term variable refers to a name for a space in a logical or mathematical expression. The term derives from the Latin adjective variabilis ( variable ). Equivalently, the terms placeholders or variables are used. As a " variable" was understood earlier words or symbols, today you mean by that formula characters. If a concrete object used instead of variables, so it must be ensured that wherever the variable occurs, and the same object is used.

A Symbol is in physics and engineering for an unnecessary numerically specified or for an at least initially changing physical size or number. The symbols for quantities are generally single letters, if necessary supplemented by indices or other modifying characters.

Variables that occur in an equation, also called Unknown or Indefinite. When they meet several variables, a distinction is dependent and independent variable. All independent variables belong to a set of definitions or a definition of the field, the dependent of it to a set of values ​​or a range of values ​​.

• 3.1 Linear equations determining
• 3.2 Functional Dependencies
• 3.3 Terms with variables as proof principle

Genesis

The concept of a variable comes from the mathematical field of algebra (see also Elementary algebra ). Already about 2000 BC Babylonians and Egyptians used words as a word variable. To 250 AD, the transition from the word algebra symbol algebra can be seen in Diophantus of Alexandria. He already used characters for the unknown and their potencies as well as for arithmetic operations. Diophantus' notation was developed by the Indians by a more efficient number notation and using negative numbers, for example, by Aryabhata in the 5th century AD or Brahmagupta in the 7th century AD. For invoices with several unknowns, they used a letter in different colors. About the Arabs, the knowledge of the Greeks and Indians came to the late medieval West. However, the Arabic algebra was a word algebra again. In the published in 1202 Liber Abaci of Leonardo of Pisa letters are used as a sign for any numbers and also allowed negative solutions. Jordanus Nemorarius (13th century ) solved equations with general coefficients. In Germany created at the beginning of the 16th century, for example, Christoph and Michael Stifel Rudolff the formal foundations of modern algebra. Generally François Viète with his published in 1591 book In artem analyticam Isagoge as a pioneer and founder of our modern symbol algebra. When René Descartes we find our modern symbol notation. Only for the equal sign he still uses a different symbol. He introduced the concepts of variable, function, and rectangular coordinate system. The notion of a variable and the idea of one variable is fundamental to the Calculus, which was developed in the 17th century by both Isaac Newton and Gottfried Wilhelm Leibniz.

Types of variables

By type of use of a variable can be distinguished:

Independent variable

One usually speaks of an independent variable if its value can be chosen freely within its domain of definition. In mathematical generality, the sign is often used. On concrete object with a diameter of an imaginary circle ( or on his measure to a length unit) every positive real value comes into consideration.

In a rectangular coordinate system, the independent variable is typically plotted as abscissa on the horizontal coordinate axis.

Dependent variable

Often the value of a variable depends on the values ​​of other variables. She gets in the general case often the sign. Specifically, the periphery of a circle with the diameter of the circle defining the number by the relationship

Given. Once the diameter (independent variable ) is known, the scope is clearly defined (dependent variable ). This approach is arbitrary: one can just as well determine the extent as an independent variable, but must then circle diameter according

Watch as the dependent variable.

The dependence can be illustrated in a line chart. In the rectangular coordinate system, the dependent variable is typically plotted as ordinate on the vertical axis.

Parameter

A parameter or a form variable is an independent variable in itself, but which is seen more at least in a given situation as a pinned size.

Example 1: The stopping distance of a vehicle is primarily dependent on the speed:

This is a so-called proportionality - a parameter whose value depends on closer examination of other parameters such as the grip of the road surface and the tread depth of the tires. But applies to any fixed value of that an increase of the independent variable, for example, to 10% (that is, on ) is an extension of braking distance at or near 21%, according to.

In a line chart with a set of curves, a parameter usually identifies the individual curve copies of each other.

Example 2: The quadratic equation

Contains the three variables. In its preferred application but are defined specifically as real numbers. The equation thus becomes the determining equation for, see below. For a real solution of the condition must be met.

Constants

Concrete immutable numbers, downer sizes or by measuring deviations unsafe or incorrect measurement values ​​with a formula characters are often provided, which can be used instead of the numerical indication now. The symbol stands for the usually unknown true value. Examples are the circuit number = 3.1415 ... or the elementary charge = 1.602 ... · 10-19 As.

Other variables

In specialty areas other meanings occur, for example, Statistical variable or free variable and bound variable.

Elementary applications in examples

Linear determining equations

Often an equation is not universally valid, but there are certain values ​​from the domain for which the equation yields a true statement. Then there is a task to determine these values ​​.

Example 1: Bernhard is now twice as old as Anna; together they are 24 years old. In this context the age is an integer specified so that the unknown age may also be only one item of integers. If the age of Anna describes as Bernhard years old. Together they are years old. This equation with unknowns, a first independent variable, enables the value of determining, as a third must be 24. So Anna and Bernhard eight 16 years old.

Example 2: The equation is valid for the two solutions.

Functional Dependencies

Mathematically specifiable contexts, such as physical and technical principles are usually described by equations that contain some quantities as variables. The number of variables is not limited to two.

For example, the DC electrical resistance of a metallic wire is given by its cross-sectional area, its length, and a constant of the material as

Immediately to the equation it can be seen that a resistance is increased by using a longer or a thinner or a different alloy wire with higher. Is not included in the equation, that is, the resistance also increases as its temperature increases, as remain depending on the temperature, but is handled here as a parameter in a rule.

Terms with variables as proof principle

Looking around for the natural numbers ( including zero ) their sequence of squares (0, 1, 4, 9, 16, ... ), it is noticeable that the respective distances between two adjacent squares exactly the sequence of odd numbers (1, 3, 5, 7, ... ) result. For a finite number of sequence elements can be simply recalculate the; in this way but it provides no proof. With the help of these variables but very easy succeed. The starting point is the binomial formula

Proof: The square of natural number, the next. The difference between two adjacent squares is thus

For the sequence of natural numbers that describes the sequence of odd numbers.

Demarcation

A random variable or random variable random variable or stochastic variable is not a variable but a function whose function values ​​depend on the random results of the associated random experiment.