Variable (mathematics)
The Moscow Mathematical Papyrus from around 1500 BC contains problems that ancient Egyptian scribes called Aha problems. These puzzles required finding an unknown quantity, which the text referred to as aha or stack. Problem 19 in the document asks the solver to calculate a quantity taken seven times and added to four to make ten. The Rhind Mathematical Papyrus also holds four similar examples of these rhetorical unknowns. Around the same time in Mesopotamia, Old Babylonian mathematicians studied quadratic and cubic equations between 2000 BC and 1500 BC. Greek geometers like Euclid described mathematical relationships entirely through shapes rather than symbols. Proposition 1 of Book II in Euclid's Elements states that if two straight lines exist and one is cut into segments, the rectangle formed by them equals the sum of rectangles formed by the uncut line and each segment. This geometric description corresponds to modern algebraic identities but lacks symbolic notation for equality or exponents.
François Viète introduced a system at the end of the 16th century where consonants represented known values and vowels represented unknowns. René Descartes changed this convention in 1637 by using letters near the start of the alphabet for knowns and letters near the end for unknowns. His choice of x, y, and z remains standard today while Viète's vowel method faded from use. Isaac Newton and Gottfried Wilhelm Leibniz developed infinitesimal calculus starting in the 1660s to study how variations in quantities induced changes in functions. Leonhard Euler later fixed the terminology of calculus around 1748 by introducing the notation f(x) for a function with variable x and value f(x). Until the late 19th century, the word variable referred almost exclusively to arguments and values of functions. The concept evolved from rhetorical descriptions in ancient Egypt to precise symbolic manipulation in early modern Europe.
Karl Weierstrass addressed paradoxes in calculus during the second half of the 19th century by replacing intuitive limits with formal definitions. He replaced the vague phrase when the variable x varies and tends toward a with a formula involving epsilon and delta. This static formulation meant none of the five variables were considered as varying within the definition itself. The older notion of limit lacked accurate definitions for what it meant to tend toward a specific value. Weierstrass introduced this new formalism to resolve issues like nowhere differentiable continuous functions that challenged existing foundations. This shift led to the modern notion where a variable is simply a symbol representing an object that may be unknown or replaceable by any element of a set. The transition from fluid intuition to rigid logic marked a turning point in mathematical rigor.
A general cubic equation contains five distinct roles for symbols: four coefficients and one unknown number. The variable x is called an unknown because it must be solved for while other symbols are parameters or constants. An indeterminate appears in polynomials but acts as a constant in polynomial rings despite common usage calling them variables. Parameters remain fixed throughout the solution of a problem even though they appear as part of the input. Free variables differ from bound variables which are quantified over in logical statements. Random variables serve as a special kind used in probability theory to describe outcomes of experiments. These semantic distinctions allow mathematicians to assign specific names to symbols based on their function rather than just their appearance.
Letters at the beginning of the alphabet such as a, b, c denote constants while later letters like u, v, w, x, y, z represent variables. Greek letters including alpha, beta, gamma often follow similar patterns for parameters and coefficients. In three-dimensional coordinate space, axes conventionally carry the labels x, y, and z. Physics determines variable names largely by the physical quantity being described rather than arbitrary choice. Probability and statistics use p, q, r for random variables while keeping m, n, k for better-defined values. Printed mathematics sets these symbols in italic typeface to distinguish them from regular text. Subscripts can add numbers, words, or expressions to clarify meaning when single letters become insufficient.
In calculus, a dependent variable represents the value of a function whose possible values depend on another independent variable. The state of a physical system varies with measurable quantities like pressure, temperature, and spatial position over time. These quantities are represented by variables that implicitly act as functions of time within formulas describing the system. An independent variable is one that does not depend on other variables in the given context. The property of dependence depends often on the point of view taken rather than intrinsic characteristics of the symbol itself. For example, if x and y both depend on t, then f(x,y) becomes a function of the single independent variable t. Rearranging equations allows any variable to be treated as dependent depending on which argument is fixed during analysis.
Considering constants and variables together leads to the concept of moduli spaces where coefficients become coordinates. A parabola equation involves five real numbers: three coefficients and two variables representing points in the plane. When coefficients a, b, c are treated as variables instead of constants, each set of triples corresponds to a different parabola. This creates a space of parabolas known as a moduli space where coordinates specify the shape and orientation of curves. The ideal gas law illustrates how fixing all but one variable transforms a multi-variable relationship into a partial application. Treating parameters as variables reveals deeper structures in mathematical objects beyond their immediate numerical values.
Common questions
What is the origin of variables in ancient Egyptian mathematics?
The Moscow Mathematical Papyrus from around 1500 BC contains problems that ancient Egyptian scribes called Aha problems. These puzzles required finding an unknown quantity, which the text referred to as aha or stack.
When did François Viète introduce vowel-based variable notation?
François Viète introduced a system at the end of the 16th century where consonants represented known values and vowels represented unknowns. René Descartes changed this convention in 1637 by using letters near the start of the alphabet for knowns and letters near the end for unknowns.
How did Karl Weierstrass redefine variables in calculus during the second half of the 19th century?
Karl Weierstrass addressed paradoxes in calculus during the second half of the 19th century by replacing intuitive limits with formal definitions involving epsilon and delta. This static formulation meant none of the five variables were considered as varying within the definition itself.
Which symbols represent constants versus variables in modern algebraic equations?
Letters at the beginning of the alphabet such as a, b, c denote constants while later letters like u, v, w, x, y, z represent variables. Greek letters including alpha, beta, gamma often follow similar patterns for parameters and coefficients.
What is the difference between dependent and independent variables in physical systems?
In calculus, a dependent variable represents the value of a function whose possible values depend on another independent variable. An independent variable is one that does not depend on other variables in the given context.