Equation

• 1. Introduction

  ›

  An equation is a mathematical formula that joins two expressions with an equals sign and declares them equal. That small symbol, two parallel lines, was invented in 1557 by Robert Recorde. His reasoning was almost poetic. He decided nothing could be more equal than parallel straight lines of the same length. Before him, equality had no such shorthand. After him, every equation that anyone would ever write would carry his mark. But the equals sign is only the beginning. What does it mean to solve one of these statements? Why does the same word mean something narrower in French than it does in English? And how did a tool for balancing two sides of a line come to describe circles, planets, sound, and heat alike? The answers reach from a scale weighing grain to the limits of what numbers can even be.

• 2. Balance And Its Rules

  ›

  Grain in two pans of a scale is the oldest way to picture what an equation does. Place equal weights on each side, and the balance holds. Remove a quantity from one pan, and you must remove an equal amount from the other to keep it level. An equation behaves the same way. It stays balanced as long as the same operation is performed on each side. This single principle drives most elementary methods for solving equations.

  From that balance come a small set of permitted moves. You may add or subtract the same quantity on both sides. You may multiply or divide both sides by a non-zero quantity. You may apply an identity to transform just one side, such as expanding a product or factoring a sum. Two equations are called equivalent when they share the same set of solutions, and these operations carry one equivalent form to another.

  Adding or subtracting reveals something useful about every equation. By moving the right-hand side across, any equation can be rewritten so that its right-hand side is zero. This is why mathematicians so often assume the right side equals zero. It costs nothing in generality, since subtracting the right-hand side from both sides achieves it.

  Caution enters when a function is applied to both sides. The new equation keeps the original solutions, but it can sprout extra ones, called extraneous solutions. Squaring both sides is the classic trap, since it can introduce a value the original never had. Worse, a function like one over x, which is undefined at zero, can quietly lose a solution. These transformations also underpin less elementary methods such as Gaussian elimination.

• 3. Unknowns And Parameters

  ›

  Letters in an equation do not all play the same role. The unknowns are the values you are solving for, and the solutions are the values that make the equality true. Other terms are assumed to be known, and these go by the names constants, coefficients, or parameters. By long convention, unknowns take letters from the end of the alphabet, x, y, z, w, while coefficients take letters from the beginning, a, b, c, d. The general quadratic equation is written ax squared plus bx plus c equals zero, with that division of labor on full display.

  A parameter can change what an equation describes. Take an equation in x and y that also carries a parameter R. When R is set to the value of 2, the equation becomes, in Cartesian coordinates, the circle of radius 2 around the origin. Leave R unspecified, and the same equation describes the general circle of any radius. One symbol holds an entire family of shapes.

  Equations also come in two fundamental kinds. An identity is true for all values of its variables, while a conditional equation is true only for particular values. The word equation itself carries a national accent. In French, an équation must contain one or more variables, while in English any well-formed pairing of two expressions across an equals sign counts.

  Sometimes equations are considered not alone but together. A system of equations is a set of simultaneous equations, usually in several unknowns, whose common solutions are sought. One example given is a system whose unique answer is x equals minus 1 and y equals 1. The word system signals that the equations must be read collectively, not one at a time.

• 4. The Power Of Identities

  ›

  True for all values is what sets an identity apart, and that universality makes it a workhorse. In the middle of solving an equation, an identity is often used to simplify the expression and make the answer easier to reach. Algebra and calculus are full of them.

  The difference of two squares is a familiar algebraic identity, holding for all x and y. Trigonometry, though, is the richest field for identities. Many involve the sine and cosine functions, and they prove indispensable when manipulating or solving trigonometric equations.

  A worked example shows the payoff. To find the value of theta satisfying a particular trigonometric equation, with theta limited to between 0 and 45 degrees, one applies a product identity for sine and cosine. The identity rewrites the equation into a form that yields a single solution for theta. Because the sine function is periodic, an unrestricted equation would have infinitely many solutions. Restricting theta to between 0 and 45 degrees narrows that infinity down to exactly one number.

• 5. From Polynomials To Impossibility

  ›

  Two main families of equations dominate algebra. The first is polynomial equations, where both sides are polynomials, also called algebraic equations. Inside that family sits the special case of linear equations. With one variable, polynomial equations take the form P of x equals zero, while linear equations take the form ax plus b equals zero. These are classified by degree, climbing from linear at degree one through quadratic, cubic, quartic, quintic, sextic, septic, all the way to octic at degree eight.

  Solvability has a hard ceiling. Some polynomial equations with rational coefficients can be solved algebraically, meaning a finite number of operations on the coefficients produces an exact answer. This works for every equation of degree one, two, three, or four. But equations of degree five or higher cannot always be solved this way, a limit proven by the Abel-Ruffini theorem. Beyond that wall, mathematicians turn to computing accurate approximations of the real or complex solutions.

  Linear systems form their own deep subject. A system of linear equations is a collection of linear equations in one or more variables, and a solution assigns numbers that satisfy every equation at once. The theory of these systems is a fundamental part of linear algebra. Computational algorithms for solving them belong to numerical linear algebra, and they play a prominent role in physics, engineering, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear one, a technique called linearization that helps in building mathematical models and computer simulations.

• 6. Geometry Made Algebraic

  ›

  René Descartes changed mathematics in the 17th century by inventing Cartesian coordinates. His system provided the first systematic link between Euclidean geometry and algebra. A geometric problem could now be turned into an analysis problem, which is exactly why the field is named analytic geometry. The point of view he outlined enriched and modified the geometry the ancient Greek mathematicians had conceived.

  Coordinates let figures become equations. A plane in three-dimensional space is the solution set of a single linear equation, and the coefficients of that equation give a vector perpendicular to the plane. A line appears as the intersection of two planes. A conic section is the intersection of a cone with a plane, which means every conic is defined by an equation of a cone together with an equation of a plane. This formalism even pins down the positions and properties of a conic's focuses.

  The goal shifts when equations have infinitely many solutions. Implicit equations and parametric equations cannot have their solutions listed or counted, so instead one uses them to study the properties of figures. That shift is the starting idea of algebraic geometry. There the fundamental objects are algebraic varieties, the geometric face of solutions to systems of polynomial equations. The most studied include plane algebraic curves, a family spanning lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals.

  A different tool describes a curve by motion rather than by an implicit relation. A parametric equation expresses the coordinates of points on a curve as functions of a variable called a parameter. The unit circle, for instance, has a parametric representation in which a single parameter t traces it out. The idea generalizes to surfaces and manifolds, with the number of parameters matching the dimension of the object.

• 7. Integers And The Edge Of Number

  ›

  Diophantus of Alexandria, a Hellenistic mathematician of the 3rd century, gave his name to one of the oldest hard problems in mathematics. He studied equations whose unknowns must take integer values, and he was one of the first mathematicians to introduce symbolism into algebra. The field he began is now called Diophantine analysis.

  These problems share a peculiar shape. A Diophantine equation has fewer equations than unknown variables, and the task is to find integers that satisfy them all. In technical terms, such an equation defines an algebraic curve or surface and asks about the integer lattice points lying on it. They are difficult in general. Often one searches only for whether a solution exists at all, and if so, how many there are.

  Integers also help define what kind of number a number is. An algebraic number is a solution of a non-zero polynomial equation in one variable with rational coefficients, equivalently with integer coefficients once denominators are cleared. Numbers that are not algebraic, such as pi, are called transcendental. The striking fact is that almost all real and complex numbers are transcendental, leaving the algebraic numbers as a vanishingly small minority.

• 8. Equations That Describe Change

  ›

  A function and its derivatives, related in a single statement, define a differential equation. The functions usually stand for physical quantities, and their derivatives represent rates of change, so the equation captures how one depends on the other. Such equations model processes driven by change, across physics, chemistry, biology, and economics.

  Exact answers are rare. Only the simplest differential equations are solvable by explicit formulas. When no self-contained formula exists, the solution can be numerically approximated using computers. The theory of dynamical systems instead emphasizes qualitative analysis, studying the behavior of solutions without writing them down exactly.

  The family splits by how many variables are involved. An ordinary differential equation, or ODE, contains a function of one independent variable and its derivatives. Linear ODEs, whose solutions can be added together and scaled by coefficients, are well understood and yield exact closed-form answers. Nonlinear ODEs rarely submit to elementary functions, and their exact solutions tend to appear in series or integral form. A partial differential equation, or PDE, instead involves unknown multivariable functions and their partial derivatives. PDEs describe sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, and quantum mechanics, formalizing these distinct phenomena in a shared language. Their own generalization is the stochastic partial differential equation, where a term becomes a stochastic process, one branch in a wider classification that also includes integral, integro-differential, functional, and difference equations.