Variable (math)

A variable is a name for a space in a logical or mathematical expression. The term is derived from the Latin adjective variabilis (changeable). The terms placeholders or changeable are used equally. Words or symbols used to serve as “variables”; today, letters are usually used as symbols in mathematical notation . If a concrete object is used instead of the variable, it must be ensured that the same object is used wherever the variable occurs.

In physics and engineering, a formula symbol stands for a physical quantity or number that is not necessarily numerically determined or at least initially still variable . The formula symbols for sizes are generally individual letters, supplemented by indices or other modifying symbols if necessary.

Up until the 1960s, variables that appear in an equation were also called unknowns or indeterminates in mathematics textbooks . When several variables come together, a distinction is made between dependent and independent variables, but only if there is a functional relationship between the variables. All independent variables belong to a set of definitions or a domain, the dependent variables to a set of values or a range of values.

History of origin

The concept of a variable comes from the mathematical branch of algebra (see also elementary algebra ). Already about 2000 years BC Babylonians and Egyptians used words as word variables. Around 250 AD, Diophantus of Alexandria saw the transition from word algebra to symbol algebra. He already uses symbols for the unknown and its powers as well as for arithmetic operations. Diophant's spelling was developed by the Indians through a more efficient number spelling and through the use of negative numbers, e.g. B. by Aryabhata in the 5th century AD or Brahmagupta in the 7th century AD, further developed. For calculations with several unknowns, they used a letter in different colors. The knowledge of the Greeks and Indians reached the late medieval western world through the Arabs. However, Arabic algebra was again a word algebra. In the Liber Abaci by Leonardo von Pisa , published in 1202, letters are used as symbols for any number and negative solutions are also allowed. Jordanus Nemorarius (13th century) solved equations with general coefficients. In Germany at the beginning of the 16th century z. B. Christoph Rudolff and Michael Stifel the formal foundations of modern algebra. In general, François Viète, with his book In artem analyticam isagoge , published in 1591, is considered a pioneer and founder of our modern symbol algebra. With René Descartes we find our modern symbol spelling. He uses another symbol only for the equal sign. He introduced the terms variable, function and right-angled coordinate system. The concept of a variable and the notion of a variable is fundamental to the infinitesimal calculus , which was developed in the 17th century by both Isaac Newton and Gottfried Wilhelm Leibniz .

Types of variables

A distinction can be made according to the type of use of a variable:

Independent variable

One usually speaks of an independent variable if its value can be freely chosen within its definition range. In mathematical generality, the sign is often used. On the concrete object of a diameter of an imaginary circle (or for its dimension number for a unit of length ) every positive real value comes into consideration. ${\displaystyle x}$ ${\displaystyle d}$

In a right-angled coordinate system , the independent variable is usually plotted as the abscissa on the horizontal coordinate axis.

Dependent variable

Often the value of one variable depends on the values ​​of other variables. She often receives the mark in the general case . Specifically, the circumference of a circle with the diameter is about the definition of the circle number by the relationship ${\displaystyle y}$ ${\displaystyle U}$ ${\displaystyle d}$ ${\displaystyle \pi }$

${\displaystyle U=\pi \cdot d}$

given. As soon as the diameter (independent variable ) is known, the circumference is clearly defined (dependent variable ). This approach is arbitrary: you can just as easily specify the circumference as an independent variable, but then you have to specify the circle diameter accordingly ${\displaystyle d}$ ${\displaystyle U}$ ${\displaystyle U}$

${\displaystyle d={\frac {U}{\pi }}}$

view it as a dependent variable.

The dependency can be illustrated in a line diagram. In the right-angled coordinate system, the dependent variable is usually plotted as the ordinate on the vertical axis.

parameter

A parameter or a shape variable is an independent variable in itself, which, however, at least in a given situation is more likely to be understood as a fixed quantity.

Example 1: The braking distance of a vehicle depends primarily on its speed : ${\displaystyle s}$ ${\displaystyle v}$

${\displaystyle s=f\cdot v^{2}}$

Here is a parameter, the value of which depends on further parameters such as the grip of the road surface and the tread depth of the tires on closer inspection. ${\displaystyle f}$

Example 2: The quadratic equation

${\displaystyle x^{2}+px+q=0}$

contains the three variables and . The variables and are form variables that are used as placeholders for concrete real numbers. The equation thus becomes the determining equation for , see below. ${\displaystyle p,\,q}$ ${\displaystyle x}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle x}$

Example 3: the equation

${\displaystyle y=mx+b}$

contains 4 variables: as an independent variable, and as a parameter as well as a variable dependent on these 3 variables. In a coordinate system, exactly one straight line is obtained for each parameter pair , and a family of parallel straight lines with the slope and the -axis section , which in this case is the family parameter. ${\displaystyle x}$ ${\displaystyle m}$ ${\displaystyle b}$ ${\displaystyle y}$ ${\displaystyle x,y}$ ${\displaystyle (m,b)}$ ${\displaystyle m}$ ${\displaystyle m|}$ ${\displaystyle y}$ ${\displaystyle b}$

If the influence of a parameter is to be illustrated in a line diagram, this is possible using a family of curves , with each curve belonging to a different parameter value.

Constants

Concrete, unchangeable numbers , fixed values or measured values ​​that are uncertain or incorrect due to measurement deviations are often provided with a formula symbol that can now be used instead of the numerical information. The symbol stands for the usually unknown true value . Examples are the circle number  = 3.1415… or the elementary  charge = 1.602… · 10 ⁻¹⁹  As. ${\displaystyle \pi }$ ${\displaystyle e}$

Another variable

Other meanings occur in specialty areas, e.g. B. statistical variables in stochastics or free variables and bound variables in mathematical logic .

Elementary applications in examples

Linear defining equations

→ Main article : determining equation

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

Example 1: Bernhard is twice as old as Anna today; together they are 24 years old. If the age of Anna describes, then Bernhard is years old. Together they are years old. This equation with the variable enables the value of to be determined because one third of 24 must be. So Anna are 8 and Bernhard 16 years old. ${\displaystyle a}$ ${\displaystyle 2a}$ ${\displaystyle 3a=24}$  ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle a}$

Example 2: The equation is valid for the two solutions and . ${\displaystyle x^{2}+5x+6=0}$ ${\displaystyle x_{1}=-2}$ ${\displaystyle x_{2}=-3}$

Functional dependencies

Mathematically definable relationships, for example physical-technical laws, are usually described by equations that contain some quantities as variables. The number of variables is by no means limited to two.

For example, the electrical direct current resistance of a metallic wire is given by its cross-sectional area , its length and a material constant as ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle l}$ ${\displaystyle \varrho }$

${\displaystyle R=\varrho \cdot {\frac {l}{A}}}$.

The three independent variables , and include the dependent variable . ${\displaystyle \varrho }$ ${\displaystyle l}$ ${\displaystyle A}$ ${\displaystyle R}$

Terms with variables as the proof principle

For example, if you look at the sequence of their squares (0, 1, 4, 9, 16, ...) for the natural numbers (including zero) , it is noticeable that the respective distances between two neighboring squares exactly correspond to the sequence of the odd numbers (1 , 3, 5, 7, ...) results. This can easily be recalculated for a finite number of sequence terms; but no proof is obtained this way . With the help of variables, this can be done very easily. The starting point is the binomial formula

${\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}$.

Proof: The square of the natural number is the closest . The difference between two neighboring squares is ${\displaystyle n}$ ${\displaystyle n^{2}}$ ${\displaystyle (n+1)^{2}}$

${\displaystyle (n+1)^{2}-n^{2}=(n^{2}+2n+1)-n^{2}=2n+1}$.

For the sequence of natural numbers, this describes the sequence of odd numbers. ${\displaystyle n}$ ${\displaystyle 2n+1}$

Demarcation

A random variable - also a 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 .

literature

• G. Malle: Didactic Problems of Elementary Algebra. Vieweg Verlag Braunschweig, 1993, ISBN 3-528-06319-X .
• W. Popp: Subject Didactics Mathematics. Aulis Verlag Cologne, 1999, ISBN 3-7614-2125-7 .
• W. Popp: Ways of exact thinking. Verlag Ehrenwirth Munich, 1981, ISBN 3-431-02416-5 .
• Student Duden. Die Mathematik I. Dudenverlag Mannheim, 1990, ISBN 3-411-04205-2 .

See also

Wiktionary: Variable  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ^ ^{A b} Norbert Henze , Günter Last: Mathematics for industrial engineers and for scientific-technical courses, volume 1. Vieweg, 2003, p. 7.
2. ↑ EN ISO 80000-1: 2013, sizes and units - Part 1: General. No. 7.1.1.
3. ↑ DIN 1304-1: 1994 Formula symbols - General formula symbols.
4. ^ Vollrath: Algebra in Secondary School. BI Wissenschaftsverlag, Mannheim / Leipzig / Vienna / Zurich 1994, p. 68, ISBN 3-411-17491-9 .
5. ^ Arnfried Kemnitz: Mathematics at the beginning of the course. Vieweg + Teubner, 2010.
6. ↑ Jürgen Koch, Martin Stämpfle: Mathematics for engineering studies. Carl Hanser, 2013.
7. ^ W. Popp: Ways of exact thinking. P. 122 ff.
8. ^ W. Popp: Subject Didactics Mathematics. P. 164 f.
9. ^ W. Popp: Ways of exact thinking. P. 129.
10. ^ W. Popp: Subject Didactics Mathematics. Pp. 165-168.
11. ^ W. Popp: Ways of exact thinking. P. 135 f.
12. ^ W. Popp: Subject Didactics Mathematics. P. 169 f.