NAQT You Gotta Know Mathematics

Pierre de Fermat

(1601-1665, French) is remembered for his contributions to number theory including his little theorem, which states that if p is a prime number and a is any number at all, then ap - a will be divisible by p. He studied Fermat primes, which are prime numbers that can be written as 22n + 1 for some integer n, but is probably most famous for his "last theorem," which he wrote in the margin of Arithmetica by the ancient Greek mathematician Diophantus with a note that "I have discovered a marvelous proof of this theorem that this margin is too small to contain." The theorem states that there is no combination of positive integers x, y, z, and n, with n > 2, such that xn + yn = zn, and mathematicians struggled for over 300 years to find a proof until Andrew Wiles completed one in 1995. (It is generally believed that Fermat did not actually have a valid proof.) Fermat and Blaise Pascal corresponded about probability theory.

The work of Isaac Newton

(1643-1727, English) in pure math includes generalizing the binomial theorem to non-integer exponents, doing the first rigorous manipulation with power series, and creating Newton's method for finding roots of differentiable functions. He is best known, however, for a lengthy feud between British and Continental mathematicians over whether he or Gottfried Leibniz invented calculus (whose differential aspect Newton called the method of fluxions). It is now generally accepted that they both did, independently.

Gottfried Leibniz

(1646-1716, German) is known for his independent invention of calculus and the ensuing priority dispute with Isaac Newton. Most modern calculus notation, including the integral sign and the use of d to indicate a differential, originated with Leibniz. He also did work with the binary number system and did fundamental work in establishing boolean algebra and symbolic logic.

Leonhard Euler

(1707-1783, Swiss) is known for his prolific output and the fact that he continued to produce seminal results even after going blind. He invented graph theory by solving the Seven Bridges of Königsberg problem, which asked whether there was a way to travel a particular arrangement of bridges so that you would cross each bridge exactly once. (He proved that it was impsosible to do so.) Euler introduced the modern notation for e, an irrational number about equal to 2.718, which is now called Euler's number in his honor (but don't confuse it for Euler's constant, which is different); he also introduced modern notation for i, a square root of -1, and for trigonometric functions. He proved Euler's formula, which relates complex numbers and trigonometric functions: ei x = cos x + i sin x, of which a special case is the fact that ei π = -1, which Richard Feynman called "the most beautiful equation in mathematics" because it links four of math's most important constants.

Carl Friedrich Gauss

(1777-1855, German) is considered the "Prince of Mathematicians" for his extraordinary contributions to every major branch of mathematics. His Disquisitiones Arithmeticae systematized number theory and stated the fundamental theorem of arithmetic (every integer greater than 1 has a prime factorization that is unique notwithstanding the order of the factors). In his doctoral dissertation, he proved the fundamental theorem of algebra (every non-constant polynomial has at least one root in the complex numbers), though that proof is not considered rigorous enough for modern standards. He later proved the law of quadratic reciprocity, and the prime number theorem (that the number of primes less than n is is approximately n divided by the natural logarithm of n). Gauss may be most famous for the (possibly apocryphal) story of intuiting the formula for the summation of an arithmetic sequence when his primary-school teacher gave him the task — designed to waste his time — of adding the first 100 positive integers.

William Rowan Hamilton

(1805-1865, Irish) is known for a four-dimensional extension of complex numbers, with six square roots of -1 (±i, ±j, and ±k), called the quaternions.

Kurt Gödel

(1906-1978, Austrian) was a logician best known for his two incompleteness theorems, which state that if a formal logical system is powerful enough to express ordinary arithmetic, it must contain statements that are true yet unprovable. Gödel developed paranoia late in life and eventually refused to eat because he feared his food had been poisoned; he died of starvation.

Andrew Wiles

(1953-present, British) is best known for proving the Taniyama-Shimura conjecture that all rational semi-stable elliptic curves are modular forms. When combined with work already done by other mathematicians, this immediately implied Fermat's last theorem (see above).

Archimedes

(287-212 BC, Syracusan Greek) is best known for his "eureka" moment, in which he realized he could use density considerations to determine the purity of a gold crown; nonetheless, he was the preeminent mathematician of ancient Greece. He found the ratios between the surface areas and volumes of a sphere and a circumscribed cylinder, accurately estimated pi, and developed a calculus-like technique to find the area of a circle, his method of exhaustion.

Euclid

(c. 300 BC, Alexandrian Greek) is principally known for the Elements, a textbook on geometry and number theory, that has been used for over 2,000 years and which grounds essentially all of what is taught in modern high school geometry classes. The Elements includes five postulates that describe what is now called Euclidean space (the usual geometric space we work in); the fifth postulate — also called the parallel postulate — can be broken to create spherical and hyperbolic geometries, which are collectively called non-Euclidean geometries. The Elements also includes a proof that there are infinitely many prime numbers.

Surjective functions, or surjections

, are functions that achieve every possible output. For instance, if you are thinking of functions whose domain and codomain are both the set of all real numbers, then f(x) = tan(x) is surjective, because every real number is an output for some input. But f(x) = x2 is not surjective, because (for instance) -3 is not an output for any real-number input. The term image is sometimes used for the set of all output values that a function actually achieves; a surjective function, then, is one whose image equals its codomain. A function that is both injective and surjective is called bijective, or a bijection. If a function is bijective, then it has an inverse. Furthermore, a function can only have an inverse if it is bijective.

Injective functions, or injections

, are functions that do not repeat any outputs. For instance, f(x) = 2x is injective, because for every possible output value, there is only one input that will result in that output. On the other hand, f(x) = sin(x) is not injective, because (for instance) the output 0 can be obtained from several different inputs (0, π, 2π, and so on). If you have the graph of a function, you can determine whether the function is injective by applying the horizontal line test: if no horizontal line would ever intersect the graph twice, the function is injective.

Logarithmic functions

, or logarithms, are functions of the form f(x) = logbx, where b is again a positive number other than 1 (and again called the base). They are the inverses of the exponential functions with the same bases. Logarithmic functions are used to model sensory perception and some phenomena in probability and statistics. The phrase "the logarithm" can refer to a logarithmic function using the base 2 (especially in computer science; this is also called the binary logarithm), e (especially in higher math), or 10 (especially in lower levels of math and physical sciences). The phrase natural logarithm refers to the logarithm base e, and the phrase common logarithm usually refers to the logarithm base 10.

Differentiable functions,

also studied in calculus, are functions for which the derivative can be found (because a particular limit, called a difference quotient, exists). Like continuity, differentiability is really a property of a specific point, and a differentiable function is one that is differentiable at every point. Every differentiable function is continuous, but some continuous functions are not differentiable; mathematicians thus say that differentiability is a stronger property than continuity. In terms of graphs, a differentiable function has a "smooth" graph with no corners or cusps (and also, because continuity is required, no holes, jumps, or asymptotes). All polynomials are differentiable, as are the sine and cosine functions, and exponential and logarithmic functions. However, the absolute value function f(x) = |x| is not differentiable because it has a "corner" at x = 0 (but recall that it is still continuous).

Polynomials

are functions made of terms added together, in which each term is a number times a product of variables raised to nonnegative-integer powers. For instance, 3x2y and -πx7y2z3 are each terms, so 3x2y - πx7y2z3 is a polynomial. (Individual terms are also considered polynomials.) Much of math is concerned with polynomials involving only one variable, such as -x3 + 2x2. The number at the beginning of each term is called a coefficient. Since simple numbers ("constants") can also be written as the same number times any variable to the zeroth power (that is, 6 is the same as 6x0), numbers are also considered terms and polynomials. So x + 6 is a polynomial, as is just -4. Polynomials can be classified according to their number of terms: a polynomial with one term, like 2x or -12x2, is called a monomial; a polynomial with two terms is called a binomial; and a polynomial with three terms is called a trinomial. Each polynomial has a degree. For polynomials of one variable, the degree is the largest exponent on the variable, so for the polynomial 4x3 - x2, the degree is 3. For polynomials of multiple variables, to find the degree you calculate the sum of the variables' exponents on each term, then choose the largest such sum, so the degree of 3x6y5 - x2y3 is calculated by adding 6+5 = 11 for the first term and 2+3 = 5 for the second and noting that 11 is larger, so the degree is 11. A polynomial that is just a constant has degree 0. A polynomial with degree 1, like 3x, is called linear; a polynomial with degree 2, like 3x2 - 8x + 4, is called quadratic; a polynomial with degree 3 is called cubic; continuing with increasing degrees, the terms are quartic, quintic, sextic, and so on, though terms corresponding to degrees larger than 5 are seldom used. For technical reasons that are beyond the scope of this article, the zero polynomial (i.e., f(x) = 0) is said to have a degree of -∞ or an undefined degree. The fundamental theorem of algebra is the statement that every single-variable polynomial, other than constants, has a root in the complex numbers, which means that if f(x) is a polynomial, then the equation f(x) = 0 has at least one solution where x is some complex number. There are formulas to find those roots for linear, quadratic, cubic, and quartic polynomials, though the latter two formulas are extremely complicated. The Abel-Ruffini theorem, also called Abel's impossibility theorem, is the statement that there is no way to find a formula for the solutions of all quintic or higher-degree polynomials, if the formula must be based on the traditional operations (addition/subtraction, multiplication/division, and exponentiation/taking roots). That impossibility is the topic that began an area of study called Galois gal-wah theory, which is part of abstract algebra.

Exponential functions

are those of the form f(x) = bx, where b (called the base) is a positive number other than 1. Exponential functions are used to model unrestricted growth (such as compound interest, and animal populations with unlimited food and no predators) and decay (such as radioactive decay). The phrase "the exponential function" refers to the function f(x) = ex, where e is a specific irrational number called Euler's number, about equal to 2.718. Exponential functions have the interesting property that their derivatives are proportional to themselves.

Periodic functions

are those whose graph repeats a pattern (specifically, the graph has translational symmetry). Technically speaking, a function of one variable f is periodic if f(x+p) = f(x) for every x in the domain of the function and some positive number p, which is called the period, because the graph repeats itself every p units. While the trigonometric functions are the periodic functions most commonly encountered by high school math students, some other functions like triangle waves are also periodic; in general, functions representing waves tend to be periodic. A Fourier fur-ee-ay series is a way to rewrite (almost) any periodic function in terms of only sine and cosine functions.

Quadratics

are, as mentioned above, polynomials of degree 2. The graph of a quadratic equation will be in the shape of a parabola that opens straight up (if the coefficient on the x2 term is positive) or straight down (if that coefficient is negative). It is possible to find the roots of a quadratic by graphing it, factoring it, completing the square on it, or using the quadratic formula (itself derived by completing the square) on it. If the quadratic is in the form ax2 + bx + c, then the expression b2 - 4ac, which appears in the quadratic formula, is called the discriminant. If the discriminant is positive, the quadratic will have two real roots; if the discriminant is zero, the quadratic will have one real root (said to have a multiplicity of 2); and if the discriminant is negative, the quadratic will have two non-real complex roots (and if the coefficients of the quadratic are real numbers, the complex roots will be conjugates of each other).

The inverse trigonometric functions

are, as one might expect, the inverse functions of the trigonometric functions. (Note that "inverse" in this case refers to a function that "undoes" another, not to "multiplicative inverse," also called "reciprocal.") They are sometimes just called "inverse sine," etc, and are also given with the prefix "arc": arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent. They are sometimes notated in the form sin-1 for arcsine, but that notation can be confusing, so NAQT tends to prefer "arcsine," etc. Because the trigonometric functions are not bijective (see below), to have inverses it is necessary to restrict the domains of the inverse trigonometric functions; for instance, arcsin(x) is only defined for x between -1 and 1, inclusive.

Rational functions

consist of one polynomial divided by another polynomial. The denominator polynomial cannot be the zero polynomial, because dividing by zero is undefined. Examples therefore include 1/x, x2/(x - 3), and (x2 + 1)/(x2 - 1). Every polynomial can be considered to be a rational function because 1 is a polynomial and dividing by 1 doesn't change an expression (so to consider the polynomial x3 as a rational function, think of it as x3/1). It is often instructive to study the asymptotes of rational functions, which are places in which their graphs approach a line (or occasionally other shape), usually getting infinitely close to but not crossing it. That analysis may require performing long division on the numerator and denominator polynomials to find their greatest common factor.

The trigonometric functions

represent relations between angles and sides of triangles. They are often illustrated using points and segments related to a circle of radius 1 centered at the origin, called the unit circle. By far the most commonly discussed trigonometric functions are the sine, cosine, tangent, cosecant, secant, and cotangent functions. There are many interesting relationships between these: the graphs of sine and cosine are translations of each other; the tangent function equals the sine function divided by the cosine function; the cosecant, secant, and cotangent functions are the reciprocals of the sine, cosine, and tangent functions, respectively; and there are many other relationships called trigonometric identities.

Odd functions

satisfy the rule f(-x) = -f(x) for every x in the domain of the function. The graph of an odd function remains the same when it is rotated 180° around the origin. Odd functions are so named because if a polynomial's exponents (on the variable) are all odd, then the polynomial is an odd function; for instance, x3, 4x7, and -x5 + 2x3 are all odd. There are other odd functions, such as the sine and cube root functions. Many functions are neither even nor odd, such as x2 + x. Only one function is both even and odd: the zero function, f(x) = 0.

Even functions

satisfy the rule f(-x) = f(x) for every x in the domain of the function. The graph of an even function has reflection symmetry over the y-axis. Even functions are so named because if a polynomial's exponents (on the variable) are all even, then the polynomial is an even function; for instance, x2, 3x6, and -x8 + 7x4 are all even. There are other even functions, though, such as the cosine and absolute value functions.

Continuous functions,

studied in calculus, are functions where the limit approaching each point equals the function's value at that point. In particular, there are no holes, jumps, or asymptotes "in the middle of the graph." Continuity is really a property of a specific point; a continuous function is a function that is continuous at every point. Continuity is often explained as a function's graph being drawable in one motion without lifting the writing utensil from the paper. All polynomials are continuous, as are the sine and cosine functions, exponential and logarithmic functions, and the absolute value function. Some examples of non-continuous functions are many rational functions, as they often have holes or asymptotes; the tangent, cosecant, secant, and cotangent functions, which have asymptotes; and the floor and ceiling functions, which have jumps.