Variable (mathematics)

• 1. Introduction

  ›

  A variable, in mathematics, is a symbol that points at a mathematical object nobody has yet named. Usually it is a single letter. People say casually that the letter represents the object, and that any valid candidate for that object becomes its value. Often those values are numbers. Sometimes they form a whole set, like the set of real numbers. But here is the strange part. The object the symbol points at may not exist at all. It might even be uncertain whether any valid candidate exists. Take two integers, call them p and q, and demand that the square of p equals twice the square of q. Proving that no nonzero integers can satisfy this is not obvious. Yet the impossibility has been known since ancient times, and it shaped mathematics ever after. How did a single letter come to carry that much weight? Who decided that x would mean the unknown? And why does the same symbol sometimes mean a fixed thing and sometimes a thing that varies?

• 2. Aha, The Unknown

  ›

  The Moscow Mathematical Papyrus, dated around 1500 BC, holds some of the earliest known uses of an unknown quantity. The Ancient Egyptians described these problems rhetorically, in words rather than symbols. They called them the Aha problems, where aha meant something like stack, the unknown quantity to be found. The setup was simple. You were told the sum of the quantity and some part or parts of it, and asked to recover the quantity itself. The Rhind Mathematical Papyrus carries four problems of this same type. One of them, problem 19, asks you to take a quantity a certain number of times, add 4, and reach 10. Around the same era in Mesopotamia, the mathematics of the Old Babylonian period, roughly 2000 BC to 1500 BC, ran further ahead. Babylonian scholars were already studying quadratic and cubic equations. The idea of a hidden number waiting to be uncovered was, even then, thousands of years old.

• 3. Letters Before Letters

  ›

  Euclid's Elements, written around 300 BC, treated mathematics as geometry rather than symbol-pushing. In proposition 1 of Book II, Euclid states that if one of two straight lines is cut into any number of segments, the rectangle on the two lines equals the rectangles on the uncut line and each segment. To a modern eye that is the distributive law, but Euclid describes it entirely through shapes. He and other Greek geometers did use single letters, only to label points and figures, an approach now called Greek geometric algebra. Diophantus of Alexandria pushed closer to symbols in his Arithmetica, around 200 AD. His syncopated algebra let him manipulate expressions with unknowns and powers, though he had no modern signs for equality or inequality and no exponents. He had a name for the unknown number, and separate names for its square, its cube, its fourth power, and its fifth. Far to the east in the 7th century BC, Brahmagupta took a different route in the Brahmasphutasiddhanta. He used different colours to stand for the unknowns in his equations. One section of that book carries the title Equations of Several Colours. After such bursts of invention, notation often froze for long stretches, with few revolutions until the early modern period.

• 4. Vowels, Consonants, And X

  ›

  Francois Viete, at the end of the 16th century, proposed letting letters stand in for both known and unknown numbers, and computing with them as though they were numbers themselves. The result would emerge through simple replacement. Viete drew a line down the alphabet. Consonants carried the known values, and vowels carried the unknowns. In 1637, Rene Descartes overturned that scheme. He introduced the convention of writing unknowns with the letters at the end of the alphabet and knowns with the letters at the beginning. Unlike Viete's arrangement, Descartes' choice survives in everyday use. The peculiar career of the letter x in mathematics drew enough interest to be discussed in an 1887 Scientific American article. This is also why letters early in the alphabet still tend to mark parameters and coefficients, while letters near the end mark unknowns and the variables of functions.

• 5. From Flowing Quantities To Static Symbols

  ›

  Isaac Newton and Gottfried Wilhelm Leibniz, working independently from the 1660s, built the infinitesimal calculus. The idea was to study how a tiny change in a time-varying quantity, which Newton called a fluent, forces a matching change in another quantity that depends on the first. Nearly a century later, Leonhard Euler settled the terminology of the field and introduced notation for a function, its variable, and its value. Until the close of the 19th century, the word variable referred almost entirely to the arguments and values of functions. Trouble surfaced in the second half of the 19th century, when the foundations of calculus proved too loose to handle paradoxes such as a continuous function that is nowhere differentiable. Karl Weierstrass answered by replacing the intuitive notion of a limit with a strict formal definition. The old phrasing said that when the variable varies and tends toward something, the function tends toward something, with no precise meaning for tends. Weierstrass recast that loose sentence as a formula in which none of the variables is treated as varying at all. From this static formulation came the modern variable: simply a symbol for a mathematical object that is either unknown or may be replaced by any element of a given set.

• 6. Many Roles, One Syntax

  ›

  A general cubic equation can be read as holding five variables, four of them given numbers and one of them an unknown to be found. To keep these straight, the variable to be solved for is called an unknown, and the rest are called parameters or coefficients. An unknown is precisely a variable in an equation that must be solved for. A parameter is a quantity, usually a number, that forms part of the input of a problem and stays constant throughout its solution; in mechanics, the mass and size of a solid body are parameters when studying its motion. In computer science, by contrast, parameter means an argument of a function. An indeterminate is a symbol appearing in a polynomial or a formal power series. Strictly speaking it is not a variable but a constant in the polynomial ring or the ring of formal power series, though the tight link between polynomials and the functions they define leads many authors to treat it as a special kind of variable. A random variable belongs to probability theory and its applications. All these names are matters of semantics. The way of computing with them, the syntax, stays the same across every one.

• 7. Who Depends On Whom

  ›

  Calculus, and its use in physics and the sciences, often sets one variable's possible values to depend on another's. The dependent variable then represents the value of a function of the other. It is frequently useful to write the dependent variable and its mapping function with the same symbol. The state of a physical system depends on measurable quantities like pressure, temperature, and spatial position, and all of these shift as the system evolves over time. In the formulas, such quantities become variables that depend on time, treated implicitly as functions of it. A dependent variable, then, is one that is implicitly a function of another variable or several. An independent variable is one that is not dependent. Whether a variable counts as dependent or independent is rarely intrinsic; it shifts with the point of view. In a notation with three variables, all three may be independent, giving a function of three variables. But if two of them depend on the third, the same notation becomes a function of a single independent variable.

• 8. The Space Of Parabolas

  ›

  The ideal gas law offers a clean demonstration of how viewpoint decides what is variable. The equation is usually read as having four variables and one constant, the constant being the Boltzmann constant. One variable, the number of particles, is a positive integer and therefore discrete, while pressure, volume, and temperature are continuous. Rearrange the equation and pressure becomes a function of the others, the dependent variable, with the rest as its independent arguments. To study how pressure depends on just one of them, an experimenter fixes all but one, and the others now behave as constants, a partial application of the earlier function. Push this thinking further and it leads to moduli spaces. Consider the equation for a parabola, with several quantities all taken to be real. Hold some of them fixed and they specify a single parabola, while the remaining two are the variables tracing its graph. Now turn the fixed quantities into variables themselves. Each set of three values picks out a different parabola, so those values become coordinates on the space of all parabolas. That space of parabolas is a moduli space, and it begins with nothing more than choosing which symbols you let move.