Studying

(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).

Andrew Wiles

was intended to be a test of the Command/Service Module in low-Earth orbit (LEO). However, a fire on the launchpad during a test killed the three astronauts aboard (Virgil "Gus" Grissom, Roger Chaffee, and Ed White). The fire was exacerbated by the pure-oxygen, positive-pressure environment inside the capsule, and the fact that the capsule door opened inward. Both of these design elements were scrapped in subsequent missions, and the second was replaced with an outward-opening hatch nominally to facilitate spacewalks. Lessons learned from Apollo 1 were also taken into account during the design of the Space Shuttle.

Apollo 1

was supposed to land in the Fra Mauro region of the Moon. An improperly-refurbished No. 2 oxygen tank and subsequent improper repairs caused the tank to rupture during a routine "cryo-stir" before entering lunar orbit. The explosion also damaged the No. 1 oxygen tank and caused further leakage. To bring back the astronauts, the orbiter was put on a free-return trajectory around the Moon. NASA engineers also solved power-management, water-conservation, and trajectory planning problems with the help of Ken Mattingly, the primary Command Module pilot who had been grounded due to exposure to German measles. All three astronauts returned to Earth safely, and the landing site was re-assigned to the subsequent Apollo 14 mission.

Apollo 13

(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.

Archimedes

A native of Latrobe, Pennsylvania, Palmer made golf popular with the masses, as his fans were known as "Arnie's Army." He won seven majors, including four Masters, and was the first golfer to earn one million dollars on the PGA Tour. Later Palmer became one of the stars of the Senior Tour, winning the Senior PGA Open in 1980 and 1981. In 2002 he played in his last competitive Masters.

Arnold Palmer (1929-2016)

are composite (i.e., non-fundamental) particles made from three quarks. The most common examples are the proton (two up quarks and one down quark) and the neutron (two down quarks and one up). All baryons are fermions. Quarks possess a characteristic called "color" (which has nothing to do with visual color) which can be either red, green, or blue (arbitrary names; again, no relation to the colors we see). A baryon must have one quark of each color so that the "total color" (analogous to mixing red, green, and blue light) is colorless (i.e., "white"). The word "baryon" comes from the Greek for "heavy." The total number of baryons is conserved (again, counting anti-baryons as -1).

Baryons

The PGA Tour's leading money winner from 1940-1942 and in 1946 and 1948, two events interrupted his playing career: service in World War II and a near-fatal 1949 head-on car accident. After each, though, Hogan rose to the top of his game; he won nine majors overall (six after the accident), including four U.S. Opens. In 1953 he accomplished a feat matched only by Tiger Woods: winning three modern major championships in one season (the Masters, U.S. Open, and British Open).

Ben Hogan (1912-1997)

are particles with integral spin. All particles are either bosons or fermions. The spin of a composite particle is determined by the total spin (i.e., the component of its intrinsic angular momentum along one axis) of its particles. For instance, an alpha particle (two protons and two neutrons) has four half-integral spin values. No matter how they are added up, the result will be an integral spin value (try it!), so an alpha particle is a (composite) boson. The Pauli Exclusion Principle does not apply to bosons (in fact, bosons prefer to be in the same quantum state). The name "boson" comes from the name of the Indian physicist Satyendra Nath Bose.

Bosons

Teams should be able to find the derivative, integral, slope at a point, local extrema, points of inflection, and critical points of polynomial, trigonometric, and other common functions.

Calculus:

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.

Carl Friedrich Gauss

is the most massive object in the asteroid belt. Ceres was discovered by Giuseppe Piazzi, an Italian priest and astronomer, in 1801, making it the first asteroid belt object to be observed and documented; it was named after the Roman goddess of agriculture. The discovery of Ceres involved the collaboration of multiple scientists, including Carl Friedrich Gauss, whose calculations of Ceres's orbital path allowed it to be found again after it was lost due to the Sun. In 2006, with the demotion of Pluto from a planet to a dwarf planet, Ceres was also reclassified as a dwarf planet, making it the only dwarf planet in the inner solar system. In 2015, NASA's Dawn probe reached Ceres, beginning an extended observation. Ceres's surface features include Ahuna Mons, a cryovolcano that erupts aqueous solutions, which quickly freeze upon exposure to Ceres's cold surface.

Ceres (formally, 1 Ceres)

The largest satellite of the dwarf planet Pluto, Charon wasn't discovered until 1978. (As of 2013, Pluto has five known moons, the last two discovered in 2011 and 2012.) Unlike Pluto, which is covered with nitrogen and methane ices, Charon appears to be covered in water ice and may also have active cryo-geysers. Because the center of mass of the Pluto-Charon system lies outside of either one, Charon doesn't truly orbit Pluto; when Pluto was reclassified as a dwarf planet in 2006, an argument was made (but not accepted) to classify Pluto-Charon as a binary system. The IAU still considers Charon, which is roughly half the size but has only 11% the mass of Pluto, to be a satellite. The New Horizons mission visited Charon and Pluto in 2015.Charon was named by its discoverer, James Christy of the Flagstaff Naval Observatory; the IAU approved the name in 1985. Internationally Charon is pronounced like the Greek mythological figure, "CARE-on", but Christy's choice of name was inspired by his wife Charlene, so NASA and New Horizons personnel use "SHARE-on".

Charon

Teams should be able to compute the number of permutations and combinations of n objects taken m at a time. They should also have memorized the first six (or so) values of the factorial function to make this easier.

Combinatorics:

(also known as Churyumov-Gerasimenko) is a short-period comet with an orbital period of about 6.5 years. In 2014, the European Space Agency's Rosetta probe reached the comet, ten years after it was launched in 2004. Two months after Rosetta began to orbit the comet, its Philae lander touched down on the comet's surface on November 12, marking the first time any human spacecraft had successfully landed on a comet's nucleus. However, an accident in the landing meant that Philae landed in a spot where it could not receive solar power, and its batteries ran out after just two days of work. Among the notable scientific discoveries from the mission are that 67P contains three times the amount of deuterium found in Earth water, that the comet is surrounded by diatomic oxygen, and that the comet's composition includes organic chemical compounds. The comet will return to its perihelion, the closest point in its orbit to the Sun, in April 2028.

Comet 67P

is an extremely long-period comet that was visible to unaided observers on Earth for 18 months, giving it the longest continuous observational period of any comet. Hale-Bopp was an extremely bright comet, reaching a peak magnitude of −1.8 (smaller numbers indicate greater luminosity). The comet was named for its American co-discoverers, Alan Hale (a professional astronomer), and Thomas Bopp (an amateur stargazer). The comet's 1990s pass through the solar system greatly affected its orbit: a close encounter with the planet Jupiter and its gravitational pull shifted the comet's path such that its orbital period dropped from over 4,000 years to about 2,500 years. During its visible time, astronomers discovered that Hale-Bopp had a sodium tail. Members of the Heaven's Gate cult killed themselves in a 1997 mass suicide in an attempt to board a spaceship they believed was hidden by the comet.

Comet Hale-Bopp

collided with Jupiter in 1994, providing modern astronomers with their first opportunity to observe a collision between two major solar system objects. The comet was named for its co-discoverers, husband and wife Gene and Carolyn Shoemaker and David Levy. Shoemaker-Levy 9 was discovered in 1993, when it had already been captured by Jupiter's gravity and had passed the planet's Roche limit, the point at which a larger body's tidal forces become so powerful that a smaller body will break up. The comet's fragments collided with Jupiter in July. Subsequent observations of the clouds around the impact site revealed deeper layers of Jupiter's atmosphere never before seen; astronomers confirmed the existence of sulfur, but were disappointed by lack of evidence for a theorized layer of water clouds. Remnants of the impact were visible as disruptions in Jupiter's atmosphere for nearly a year.

Comet Shoemaker-Levy 9

Teams should be familiar with the symbol i representing an imaginary square root of -1, basic operations on complex numbers, graphing complex numbers, and converting complex numbers to polar form.

Complex numbers:

is informally defined as the ability of a function's graph to be drawn without lifting the pen. Properly, a function is continuous at a given point if the function has a limit at that point and is equal to its limit there. It is also common to speak of "continuous functions," which are functions that are continuous at every point. Polynomials are continuous, as are exponential and logarithmic functions, sine, and cosine. However, the other four trigonometric functions (cosecant, secant, tangent, and cotangent) are not continuous at some points (the ones at which their graphs have asymptotes), and rational functions generally have some points of discontinuity.

Continuity

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.

Continuous functions

is an operation that can be interpreted as giving the area under a curve, or more precisely, the signed area between the curve and the x-axis (signed area meaning that area below the axis counts as negative). An integral is denoted by a "long S" symbol, ∫. Definite integration is performed between two endpoints (e.g., the integral between 0 and 5), and those endpoints are written above and below, or next to, the integral symbol.

Definite integration

One-seventh the mass of Phobos and further away from the Martian surface, Deimos was found by Asaph Hall at the U.S. Naval Observatory six days before he discovered Phobos. Its largest and only named craters are Swift and Voltaire; Deimos's surface doesn't appear as rough as Phobos's because regolith has filled in some of the craters. A still-controversial and unproven hypothesis holds that Deimos (and possibly Phobos as well) were asteroids perturbed out of their orbit by Jupiter and then captured by the gravity of Mars.

Deimos

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).

Differentiable functions

are equations that relate a function to its derivatives, or even multiple derivatives (e.g., a function to its first and second derivatives). They are widely used throughout the sciences to model behavior like radioactive decay, population growth, predator-prey relations, waves, fluid flow, and more.

Differential equations

Teams should be able to quickly apply the divisibility rules for small integers (2 through 11) to large integers.

Divisibility rules:

The moon, also called Luna, is the fifth-largest satellite in the solar system, the largest relative to the size of the planet it orbits, and the second densest. The USSR's Luna unmanned spacecraft first reached the moon in 1959, and Apollo 8 became the first manned mission to orbit the moon, in 1968. The 1967 Outer Space Treaty guarantees the rights of all nations to explore the moon for peaceful purposes.The flat dark lunar plains are called maria (singular: mare) and are mainly concentrated on the near side of the moon. The most famous one is Mare Tranquillitatis, the Sea of Tranquility, where Apollo 11 first landed on the moon in 1969. The Apollo program landed on the moon five more times.

Earth's moon:

is a dwarf planet that was co-discovered in 2005 by Mike Brown, Chad Trujillo, and David Rabinowitz. Eris is a scattered disc object, meaning it was "scattered" out of the Kuiper belt by the gravity of Neptune, falling into a high-eccentricity orbit that ranges between 38 and 97 AU from the Sun. Upon its discovery, Eris was thought to be both larger and more massive than Pluto; later observations concluded Pluto did have a greater surface area, but was less massive. Eris's discovery and its potential classification as another planet prompted the 2006 IAU redefinition of the term "planet" that excluded Pluto (and Eris). Eris was named after the Greek goddess of strife, in acknowledgment that its discovery had caused a great deal of strife within astronomy. Eris is orbited by the moon Dysnomia.

Eris (formally, 136199 Eris)

(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.

Euclid

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.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

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.

Exponential functions

are particles with half-integral spin. Spin is a form of "intrinsic angular momentum," possessed by particles as if they were spinning around their axis (but they aren't). The values cited for spin are not (usually) the real magnitude of that angular momentum, but the component of the angular momentum along one axis. Quantum mechanics restricts that component to being n/2 times Planck's constant divided by 2π for some integer n. If n is even, this results in "integral" spin, if it is odd, it results in "half-integral" spin. Note that the exact value of the spin itself is a real number; it's the multiplier of h/2π that determines whether it is "integral" or not. The most significant thing about fermions is that they are subject to the Pauli Exclusion Principle: No two fermions can have the same quantum numbers (i.e., same state). The name "fermion" comes from the name of the Italian-American physicist Enrico Fermi

Fermions

is the largest moon in the solar system and the only one known to have its own magnetosphere. The third of the Galilean satellites, Ganymede was also first photographed close-up by Pioneer 10 in 1973. Galileo made six flybys of Ganymede between 1996 and 2000. Based on a suggestion from Simon Marius, Ganymede (along with many of the Jovian satellites) is named for one of Jupiter's lovers in Roman mythology; Ganymede is the only such moon named for a male figure. Many of Ganymede's features, including the Enki Catena, are given names from Egyptian and Babylonian mythology, although its largest dark plain is Galileo Regio. Ganymede is scheduled to be orbited by the European Space Agency's Jupiter Icy Moon Explorer (JUICE), currently slated for a 2022 launch.

Ganymede

The most successful non-American golfer in history, this South African has won nine majors. When Player took his only U.S. Open crown in 1965, he not only became the first non-American to win that tournament in 45 years, but he also became one of three (now five) golfers (along with Nicklaus, Woods, Hogan, and Gene Sarazen) to win all four modern Grand Slam events. Nicknames include "The Black Knight" for his dress and "Mr. Fitness" for his devotion to exercise.

Gary Player (1935-present)

(sometimes called vector bosons) are fundamental bosons that carry the forces of nature. That is, forces result from particles emitting and absorbing gauge bosons. The strong nuclear force is carried by gluons, the weak nuclear force is carried by the W, Z-, and Z+ particles, the electromagnetic force is carried by the photon, and gravity is carried by the (as yet unobserved) graviton. The name comes from the role of "gauge theories" in describing the forces (which are beyond the scope of this article).

Gauge bosons

are the gauge bosons that carry the strong nuclear force and bind hadrons together. Gluons have no charge and no mass, but do have color (in the sense of quarks). This color cannot be observed directly because the gluons are part of the larger hadron. The name comes from their role in "gluing" quarks together.

Gluons

(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.

Gottfried Leibniz

are any particles made out of quarks (alternatively, any particle affected by the strong nuclear force). Generally, this means the baryons and the mesons. All hadrons are colorless (in the sense of the combined color of their constituent quarks). The name "hadron" comes from the Greek for "thick."

Hadrons

is, by far, the best-known comet. The fame of Halley's Comet rests on two main factors: first, it has a relatively short orbital period of about 75.5 years; second, it is visible to the naked eye as it passes Earth. The comet is named after Edmond Halley (1656-1742), who observed it in 1682, determined it orbited the Sun, and correctly predicted its subsequent 1758 return in his treatise Synopsis of the Astronomy of Comets. Halley's Comet is one of the most studied and depicted astronomical objects in all of human history: Chinese reports of the comet's appearance in 240 BC are part of the Records of the Grand Historian, and the comet appears in the sky of the Bayeux Tapestry, as it passed Earth in 1066 before the Battle of Hastings. Halley's Comet is the source of the Orionid meteor shower.

Halley's Comet

all utilized spacewalks staged from a Space Shuttle. SM1 corrected Hubble's flawed optics by installing COSTAR (a corrective optics system) while removing the High-Speed Photometer to make room. It also replaced the original Wide Field and Planetary Camera (or WFPC "wiff-pick") with WFPC2, a camera that had optical correction built in. SM2 replaced the Goddard High-Resolution Spectrograph (GHRS) and Faint Object Spectrograph (FOS) with improved successors. SM3A replaced failed gyroscope systems. SM3B replaced the Faint Object Camera (FOC) with the Advanced Camera for Surveys (ACS), and also repaired NICMOS, which was installed during SM2. SM4 was originally cancelled after the Columbia Disaster by then-NASA Administrator Sean O'Keefe, but was revived by Michael Griffin, who took over the post in 2005. SM4 installed WFC3 (made from some parts of the original WFPC) and the Cosmic Origins Spectrograph (COS), and conducted other repairs. No further repairs are planned, as much of HST's functionality will be replicated with the James Webb Space Telescope, scheduled for launch in 2021.

Hubble Space Telescope Servicing Missions (SM1, SM2, SM3A, SM3B, and SM4)

is Saturn's third-largest moon after Titan and Rhea and, like Rhea, was discovered by Giovanni Cassini in 1671. It was named based upon a suggestion from John Herschel (son of the discoverer of Uranus, William Herschel) for the Titans of Greek mythology, the brothers and sisters of Cronos (Saturn). Iapetus has a distinctive two-tone coloration; part of it is red-brown, while part is bright gray. Features on Iapetus are named for people and places from the French Song of Roland, including Charlemagne Crater and the bright northern region Roncevaux Terra. In 2004 the Cassini orbiter found an equatorial ridge running over 800 miles long and 10 miles wide that gives Iapetus some of the highest peaks in the solar system; its existence has not yet been explained.

Iapetus "eye"-AA-pih-tuss

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.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

is the innermost of the four Galilean moons of Jupiter (the moons discovered by Galileo), the fourth-largest moon in the solar system, the densest moon, and the most geologically active body in the solar system, with more than 400 volcanoes. Io's features are named for characters from the Io story in Greek mythology; fire, volcano, and thunder deities from other mythologies; and characters from Dante's Inferno. Io plays a significant role in shaping Jupiter's magnetosphere. Pioneer 10 first passed by Io in December 1973.

Io

In pure math includes generalizing the binomial theorem to non-integer exponents, doing the first rigorous manipulation with power series, and creating this person'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 this person's called the method of fluxions). It is now generally accepted that they both did, independently.

Isaac Newton

Nicknamed "The Golden Bear," he won the U.S. Amateur twice (1959 and 1961), and was the 1961 NCAA champion at Ohio State. He took his first major the following year at the U.S. Open, beating Arnold Palmer on Palmer's home course. Nicklaus became the youngest Masters champion at the time in 1963, and 23 years later became the oldest champion with a final round 65 in 1986. He has a record 18 major pro championships overall, including six Masters, five PGA Championships, four U.S. Opens, and three British Opens. Nicklaus is still somewhat active on the Senior PGA Tour, and as a golf course architect.

Jack Nicklaus (1940-present)

In 2015, the year he turned 22, Spieth won both the Masters and the U.S. Open, and finished second at the PGA Championship. His Masters win was the first in which the same golfer lead after all four days since Raymond Floyd did it in 1976, and it included an 8-under-par opening day and a record-tying 18-under-par result for the tournament. Spieth has struggled more in recent years, failing to win any PGA Tour events since his British Open win in July 2017.Leptons are one of the classes of "fundamental particles" (meaning that they cannot be broken down into smaller particles). There are six "flavors" of leptons: the electron, the muon, the tauon, the electron neutrino (usually just called "the neutrino"), the muon neutrino, and the tauon neutrino. The three neutrinos are neutral and nearly massless (they were once thought to be entirely massless), while the other three have a charge of -1. All neutrinos are fermions, and the total number of leptons is always conserved (counting regular leptons as +1 particle and anti-leptons as -1 particle). The word "lepton" comes from the Greek for "light" (the opposite of "heavy," not illumination), even though the muon and tauon are relatively massive.

Jordan Spieth (1993-present)

(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.

Kurt Gödel

Nicknamed "Supermex" for his Mexican-American heritage, Trevino came from a poor Dallas family and served in the Marines, but came from nowhere to win the 1968 U.S. Open. He won six majors: the U.S. Open, the British Open, and the PGA Championship twice each, his second PGA in 1984 at age 44. That last win was most impressive because it came after the 1975 Western Open, where Trevino was struck by lightning on the golf course.

Lee Trevino (1939-present)

(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.

Leonhard Euler

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.

Logarithmic functions

Teams should be familiar with basic operations of logarithmic math: simplifying the logarithm of a product, difference, or power, and converting from one base to another.

Logarithms:

Every team should be able to add, subtract, multiply, take the determinant of, transpose, and invert matrices, particularly 2×2 ones.

Matrices:

are composite particles generally made from a quark and an anti-quark. There are dozens of examples including the pion, kaon, J/Psi, Rho, and D. All mesons are bosons. The quark and anti-quark must have the same color (such as red and anti-red) so that the resulting meson is colorless (or "white"). It is also possible to make mesons out of two (or more) quarks and the same number of anti-quarks, but this kind of particle (a "tetraquark") is rare, both in nature and in quiz bowl.

Mesons

are an older name that was used for the "internal parts" of hadrons before the discovery and widespread acceptance of the quark model. Models based on partons are still used but, for the most part, it was determined that partons were quarks and the term is rarely used at the high school level except in historical contexts.

Partons

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.

Periodic functions

Known as "Lefty" due to his swing (though he is naturally right-handed), Phil Mickelson won The Masters in 2004, 2006, and 2010, and has also won a PGA Championship and British Open. Though he has never reached number 1 in the World Golf Ranking, Mickelson was a constant force during the 2000s, finishing second in the U.S. Open six times. In 2018, Mickelson won "The Match," a pay-per-view event against Tiger Woods using the match play format; the win earned Mickelson $9 million.

Phil Mickelson (1970-present)

Both Phobos ("fear") and Mars' smaller moon Deimos ("dread") were discovered by Asaph Hall III in 1877. At just 3,700 miles above the Martian surface, Phobos orbits more closely to its planet than any other moon in the Solar System. Because it orbits Mars faster than Mars rotates, each day it appears (from the Martian surface) to set twice in the east. Geological features on Phobos, including the Stickney Crater, are primarily named for either astronomers (Stickney was the maiden name of Asaph Hall's wife) or characters from Jonathan Swift's Gulliver's Travels. In 1971 the U.S.'s Mariner IX became the first spacecraft to provide close-up photos of Phobos.

Phobos

(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.

Pierre de Fermat

Teams should be able to calculate the areas of triangles, trapezoids, parallelograms, rhombi, regular polygons, and circles using several methods.

Plane figures:

was, from its 1930 discovery until 2006, considered to be the ninth planet from the Sun. However, in 2006, it was reclassified as a dwarf planet due to its inability to clear the neighborhood of its orbit (meaning there are other bodies of comparable size that share its orbit). Pluto was discovered by 24-year-old astronomer Clyde Tombaugh, who found it by looking at whether or not objects had moved in subsequent photos of the same area of the sky. Pluto was visited by NASA's New Horizons probe (which carried Tombaugh's ashes) in 2015; among its most striking features was a heart-shaped area in its southern hemisphere, which was christened the Tombaugh Regio. Pluto is part of the Kuiper belt, a disk of outer Solar System objects ranging from around 30-50 AU from the Sun. Pluto has five natural satellites, the largest of which is Charon.

Pluto (formally, 134340 Pluto)

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 this term. (Individual terms are also considered this term) Much of math is concerned with this term 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 this term. So x + 6 is this term, as is just -4.This term can be classified according to their number of terms: a this term with one term, like 2x or -12x2, is called a monomial; this term with two terms is called a binomial; and a this term with three terms is called a trinomial.Each this term 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.

Polynomials

Teams should be able to quickly add, subtract, multiply, divide, factor, and find the roots of low-degree polynomials, and understand how the degree behaves under the first four operations.

Polynomials:

Sets of small integers that satisfy the equation of the Pythagorean theorem, a2 + b2 = c2, and could therefore be the side lengths of a right triangle. The simplest ones are {3, 4, 5}, {5, 12, 13}, {7, 24, 25}, and {8, 15, 17}. Note that any multiple of a Pythagorean triple is also a Pythagorean triple, so (for instance) {6, 8, 10}, {15, 20, 25}, and {300, 400, 500} are also ones by virtue of {3, 4, 5} being one.

Pythagorean triples:

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).

Quadratics

are another class of fundamental particle. They also come in six flavors: up, down, charm, strange, top (occasionally called "truth"), and bottom (occasionally "beauty"). The up, charm, and top quarks have a charge of +2/3, while the down, strange, and bottom have a charge of -1/3. All quarks are fermions, and they combine in pairs to form mesons and in triples to form baryons. The enormous mass of the top quark (178 GeV) made it difficult to create in particle accelerators, but its discovery in 1995 confirmed an essential element of the Standard Model of particle physics. The name "quark" comes from the line "Three quarks for Muster Mark" in Finnegans Wake that appealed to Murray Gell-Mann. The study of quarks (and of the strong nuclear force) is quantum chromodynamics.

Quarks

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.

Rational functions

are a way to formalize the definition of an integral. A Riemann sum approximates the area under a curve by partitioning the area into narrow rectangles that extend between the x-axis and the function's graph, finding the areas of the rectangles, and adding those up. For well-behaved functions, the limit can be taken as the rectangles get infinitely thin, giving the definite integral.

Riemann sums

An Atlanta native, and the greatest amateur golfer of all time, Jones never turned pro, but won thirteen major championships in eight years, including four U.S. Amateurs. In 1930 he won what was then considered the Grand Slam, taking both the British and U.S. Amateur and Open Championships. After that season, Jones retired from golf to practice law, but helped design a golf course in Augusta, Georgia that became the permanent site of the Masters in 1934.

Robert Tyre "Bobby" Jones (1902-1971)

McIlroy won his first major championship at the U.S. Open in 2011, and despite being just 22 years old, he finished at 16 strokes under par, a record for the tournament. He won three more majors in the following three years, and became the top-ranked golfer in the world in 2012; after doing so, he and Luke Donald took turns at the top spot for over a year. He briefly regained the top ranking in 2020, a year in which he and Dustin Johnson raised over £1.5 million for coronavirus relief in a televised skins game. Though he is from Northern Ireland, McIlroy has expressed interest in competing for Ireland in international competition instead of the U.K.

Rory McIlroy (1989-present)

are two Space Shuttle (or Space Transportation System) launches known mostly for their accidents and subsequent losses of life. The Challenger disaster was attributed by the Rogers Commission (the investigative body set up after the accident) to poor performance of the solid rocket booster (SRB) O-rings. The O-rings lost integrity and became brittle at low temperatures, such as those present on the morning of the launch. The failure of the O-rings caused "blow-by," where hot gasses escaped the booster joint, ultimately resulting in the destruction of Challenger. The Rogers Commission also cited both NASA and SRB contractor Morton Thiokol for a failure to redesign the SRB joint, known to be dangerous—manifestations of "go fever."According to the Columbia Accident Investigation Board, the disaster on STS-107 was due to a piece of foam from the external fuel tank hitting and breaching the left wing of the Orbiter during launch. The breach damaged the heat shielding, allowing hot gas to enter the Orbiter during re-entry. Ultimately, that damage caused the vehicle to disintegrate over Texas.Both missions resulted in the complete loss of crew (seven crew were on each mission) and orbiters. Both modes of failure had been previously observed in earlier missions, but went un-fixed because those missions were successful.

STS-51-L and STS-107 (the Challenger and Columbia accidents respectively)

No golfer has won more PGA Tournaments than Snead's 81, and he amassed 135 victories worldwide. Nicknamed "Slammin' Sammy," he won seven major professional championships between 1942 and 1954, but he is known more for the one he never won: the U.S. Open. In 1939 Snead led the Open for 71 holes but lost on the last hole when he took an eight. In the 1960s and '70s he won a record six Senior PGA Championships.

Sam Snead (1912-2002)

The areas of similar figures are related by the square of any corresponding length, and the volumes are related by the cube of any corresponding length. For instance, if a square has a diagonal that is 30% longer than another square, it has an area that is 1.30 × 1.30 = 1.69 times as great (69% greater). Similar reasoning applies to perimeters, side lengths, diameters, and so forth.

Similar figures:

Teams should be able to calculate the volume and surface area of simple geometric figures including the sphere, cone, cylinder, pyramid, hemisphere, prism, and parallelepiped.

Solids:

was the first commercial mission to resupply the International Space Station (ISS). A structural failure in one of the nine Merlin engines that make up the Falcon 9 rocket first stage necessitated a longer burn with the remaining eight engines. That correction resulted in a proper orbital insertion for the primary payload (a Dragon resupply vehicle), but an unstable, decaying orbit for its secondary payload (an ORBCOMM satellite). This was taken as a proof of concept by SpaceX for the redundant, multiple-engine design. The craft successfully berthed with the ISS and successfully carried out its primary mission, re-supplying the space station and returning cargo to Earth.

SpaceX CRS-1 (also called SpX-1 or CRS-1)

is a way of approximating a differentiable function using an infinite sum of monomials (which can be truncated to get a polynomial). Coefficients of Taylor series are found using nth derivatives and the factorial function. Taylor series can be based around any point of a function, and when the chosen point is x = 0, it is called a Maclaurin series. Taylor's theorem gives a bound on the error resulting from truncating the Taylor series to a finite number of terms; the error can be expressed as a remainder term in the Lagrange form, Cauchy form, or integral form.

Taylor series

marked a definite end to the Space Race, and was a symbol of the de-escalation of tensions between the US and the USSR. Deke Slayton, an original Mercury 7 astronaut grounded for medical reasons until 1973, was accompanied by Tom Stafford and Vance Brand on the last launch of Apollo before the advent of the Space Shuttle. The mission demonstrated that two dissimilar spacecraft could rendezvous and dock while in space (an Apollo Command Module and Soyuz 19 docked), and also enabled the crew of Soyuz to photograph the Sun's corona through an artificial eclipse created by the Apollo spacecraft. Each spacecraft also carried out independent experiments.

The Apollo-Soyuz Test Project

is used to find the derivative of the composition of two functions, such as sin(x²). It states that the derivative of f(g(x)) is f′(g(x)) · g′(x). In Leibniz notation, it is written as df/dx = df/dg dg/dx and thus resembles canceling two fractions. Integrating both sides of this result gives a corresponding rule for integration by substitution.

The chain rule

roughly states that integration and differentiation are opposite operations. This means that most basic rules for integration can be found by reversing the basic rules for differentiation given above. The formal statement of the theorem is usually given in two parts. The first part states that integrating a function gives an antiderivative of the function (i.e., a function whose derivative is the original function). The second part states that the definite integral of f(x) from a to b can be calculated as F(b) − F(a), where F is an antiderivative of f.

The fundamental theorem of calculus

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.

The inverse trigonometric functions

the first artificial satellite, by the USSR kicked off the so-called "Space Race." The Space Race was manifestation of the Cold War where successes in launching objects and people into space was seen as a proxy for capacity to build intercontinental ballistic missiles. Sputnik 1 was originally intended to carry many scientific instruments, but those instruments were descoped, in favor of a simple radio transmitter. That transmitter broadcasted a "beep" at a specified interval, allowing scientists to map its deceleration as a result of atmospheric drag.Sputnik 1 was followed by Sputnik 2, which carried a dog named Laika (meaning "barker") into low-earth orbit, and was destroyed upon re-entry.

The launch of Sputnik 1

is used to find the derivative of the product of two functions, such as x sin x. It states that the derivative of f(x) g(x) is f′(x) g(x) + f(x) g′(x). Integrating both sides of this result gives a corresponding rule for integrating the product of two functions, called integration by parts. It can be combined with the chain rule to find the quotient rule (the way to differentiate a function written as one function divided by another).

The product rule

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.

The trigonometric functions

is a value that a function approaches as the input approaches another value, even if the function is not equal to that output value (or even defined there). For example, the limit of 1/x as x gets infinity large ("approaches infinity") is 0, even though the function never actually reaches 0. The definition of a limit is formalized in a way traditionally written with the Greek letters epsilon and delta, a technique developed by Augustin-Louis Cauchy.

limit

is a hypothetical solar system body that, between 4 and 5 billion years ago, collided with Earth; the debris ejected by this collision subsequently coalesced to create the Moon, a process known as the "Giant Impact Hypothesis." Theories regarding the size of Theia vary, with some estimates placing it around the size of Mars. Evidence for the existence of Theia—and its collision with Earth—include observations of shock heating in meteorites of approximately the same age as the hypothesized impact and the fact that the Earth is much more abundant in light chlorine, while the Moon contains more heavy chlorine—a result that fits with what would be expected if two objects collided, broke apart, and re-formed. In 2021, scientists at Arizona State University proposed that the Large Low Shear Velocity Provinces found in the lowest part of Earth's mantle may be remnants of Theia. Theia is named after the mythical Greek Titaness who was the mother of the Moon goddess Selene.Apollo 11 saw the first Moon landing and moonwalk by astronauts Neil Armstrong and Buzz Aldrin (astronaut Michael Collins piloted the Command Module in lunar orbit and never walked on the moon).After the Lunar Module landed in the Sea of Tranquility, Armstrong said, "Houston, Tranquility Base here. The Eagle has landed." Later stepping onto the lunar surface, he said, "That's one small step for [a] man, one giant leap for mankind." According to Chris Kraft, NASA officials chose Armstrong as the first to walk on the Moon because he was more humble than Aldrin, and because he was the Commander. However, the stated reason was that Armstrong's seat was closer to the door. Apollo 11, like other Apollo missions, launched atop a Saturn V booster.

Theia

Born to an African-American father and a Thai mother, he appeared on The Mike Douglas Show with a golf club at age two. Woods won three straight U.S. Junior Amateurs, and then became the only golfer to win three straight U.S. Amateurs (1994-1996). In 1997 Woods became the youngest ever to win the Masters—by a whopping 12 strokes. At the 2000 U.S. Open, when he won by 15 strokes, Woods began a remarkable run of four straight major championships: British Open (by eight strokes, making him the youngest ever to complete the career Grand Slam), PGA Championship, and the 2001 Masters. He accumulated 14 major wins by 2008, when his career was derailed by injuries and personal issues, including infidelities leading to a highly publicized separation from his former wife. His long-awaited comeback reached its peak in 2019 when he won the Masters, his 15th major title.

Tiger Woods (1975-present)

is the largest moon of Saturn and the second largest in the solar system. Until Voyager 1 visited in 1980, it was thought to be larger than Ganymede. It is the only known satellite with a dense atmosphere — so dense that it makes observation of surface features nearly impossible except from close up — and also the only known satellite for which there is evidence of stable bodies of surface liquid. Discovered in 1655 by Christiaan Huygens, it was visited by the Cassini-Huygens mission in 2004. Titan's albedo features, such as the highly reflective area Xanadu, are named for sacred or enchanted places from world literature and mythology. Because of its nitrogen-rich atmosphere and the presence of surface liquid, Titan is often thought to be the most likely place in the solar system for microbial life to exist outside of Earth.

Titan

Uranus's largest moons, Titania and Oberon, are named for characters from Shakespeare's A Midsummer Night's Dream. (Other Uranian moons are named for characters from either Shakespeare or Alexander Pope.) They were discovered on the same day in 1787 by William Herschel, who also discovered Uranus itself in 1781. In 1986 Voyager 2 became the only spacecraft to date to visit the Uranian moons. Because Uranus orbits the sun almost on its side and Titania and Oberon orbit Uranus in the same plane as its equator, the moons have extreme seasons: Titania's poles spend over 42 years in nonstop sunlight followed by 42 years of darkness. Most of Titania's features are named for settings or female characters from Shakespeare — its largest crater is Gertrude Crater, after Hamlet's mother — while most of Oberon's are named after settings or male characters from Shakespeare. However, Oberon's largest feature is Mommur Chasma, which is named from a French epic poem.

Titania and Oberon

He became the major rival to Jack Nicklaus in the second half of the Golden Bear's career. Watson's greatest achievements were at the British Open, a tournament he won five times between 1975 and 1983. He took eight major championships overall.

Tom Watson (1949-present)

The largest moon of Neptune and the only large moon with a retrograde orbit (that is, an orbit opposite to the rotation of its planet), Triton is the seventh-largest moon in the solar system and is thought to have been captured from the Kuiper Belt. For over 100 years after its 1846 discovery, Triton was thought to be Neptune's only moon; Nereid wasn't discovered until 1949. (There are fourteen known satellites now.) Triton is geologically active and has geysers that are assumed to erupt nitrogen. Because of the activity, impact craters on Triton are relatively scarce; most of the larger craters were formed by volcanic activity. Triton orbits around Neptune in almost a perfect circle. Voyager 2 visited Triton in 1989 and is the only space probe to have done so (and no more are currently planned). Much of Triton's western hemisphere consists of an unexplained series of fissures and depressions sometimes called "cantaloupe terrain." Triton's features are named after various water spirits, monsters, and sacred waters from mythology.

Triton

Every team should be able to find the magnitude (length) of a vector, and add, subtract, find the angle between, find the dot product of, and find the cross product of two vectors.

Vectors:

is the second-most massive object in the asteroid belt, after Ceres. Vesta was discovered by Heinrich Olbers, who also discovered the asteroid Pallas (which is approximately as large as Vesta, but less massive); it was named by Carl Friedrich Gauss after the Roman goddess of the hearth. Vesta—like Ceres—was visited by the Dawn probe, which orbited Vesta from 2011 to 2012 before leaving orbit and heading for Ceres. Notable features of Vesta include the Rheasilvia crater, an enormous impact crater whose diameter is over 90% of the diameter of Vesta itself and at whose center is a mountain over 12 miles tall. Much of what is currently known about the composition of Vesta is based on the study of HED meteorites (so named because they are classified as howardites, eucrites, or diogenites), which originated from Vesta before colliding with Earth.

Vesta (formally, 4 Vesta)

were the only two missions of the Voskhod program, and were superseded quickly by the Soyuz program. The spacecrafts made use of Sergei Korolev's designs. Korolev himself was preoccupied with the Moon race, again in a position of favor after the fall of Khrushchev (though Korolev died before the designs were used).Voskhod 1 was the first flight to contain multiple astronauts, and Voskhod 2 was the platform for the first EVA (extra-vehicular activity or "spacewalk"). Alexey Leonov, who conducted the spacewalk on Voskhod 2, also participated in the later Apollo-Soyuz Test Project.

Voskhod 1 and Voskhod 2

transported the first human and the first woman into space (respectively, Yuri Gagarin and Valentina Tereshkova). Gagarin's April 12, 1961 flight is still celebrated as Yuri's Night. During Vostok 1, Gagarin completed a single orbit around Earth before re-entering and parachuting out of his capsule.Vostok 6 was largely uneventful, though Tereshkova did note minor physical pains and also could not reach the scientific experiments aboard. Both Vostok 1 and 6 used Vostok-K boosters.

Vostok 1 and Vostok 6

(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.

William Rowan Hamilton

is an operation that takes a function and results in another function that gives the original function's rate of change. Taking a derivative is called differentiation and is carried out with respect to the input variable that is changing. The derivative is calculated as the limit of a tangent line between two points on the function's graph as the two points get arbitrarily close together. At any given point, the derivative's value is equal to the slope of the tangent line to the function's graph at that point. The derivative of a function f(x) is notated as f′(x) (Lagrange's notation, pronounced "f prime of x") or df/dx (Leibniz notation, pronounced "d f d x). Differentiation can be repeated, e.g,. differentiating a function twice gives its second derivative (f′′(x) or d²f/dx²).

derivative

is called differentiable if its derivative can be evaluated. Like for continuity, this can be stated about a specific point or the whole function. The absolute value function is an example of a function that is differentiable in some places but not others: it is differentiable everywhere except at an input of 0, where its graph has a sharp corner. Differentiability is a stronger property than continuity: a function can only be differentiable if it is continuous, but it could be only continuous and not differentiable (the absolute value function being an example of that). If a function's derivative can be taken infinitely many times, it is called smooth.

function