NAQT - You Gotta Know: Science
Pierre de Fermat
(1601 - 1665, French) 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 a^(p) - a will be divisible by p. He studied Fermat primes, which are prime numbers that can be written as 2^(2^n) + 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 x^(n) + y^(n) = z^(n), 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.
Isaac Newton
(1643 - 1727, English) His work in pure maths 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) 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) 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 impossible 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: e^(i x) = cos x + i sin x, of which a special case is the fact that e^(i 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) 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 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) Known for a four-dimensional extension of complex numbers, with six square roots of -1 (±i, ±j, and ±k), called the quaternions.
Max Planck
(1858 - 1947) Planck allowed quantum theory to move forward in the early 20th century by correctly modeling how an object radiates heat, solving the ultraviolet catastrophe, which was a predicted unbounded increase in the amount of radiation emitted at high frequencies. Planck's Law of Radiation superseded the Rayleigh-Jeans Law. He suggested that electromagnetic energy could only be emitted in specific packages, called quanta (singular quantum, from the Latin for "how much"), positing that the energy of this photon was equal to its frequency times a fixed value h, now known as Planck's constant.
Marie and Pierre Curie
(1867 - 1945, 1859 - 1906) Rigorously isolated and experimented on radioactive materials, forming the basis for early nuclear and particle physics
Robert Millikan
(1868 - 1953) Determined the charge of the electron by meticulously observing oil droplets in an electric field and noting the time it took them to fall a certain distance
Ernest Rutherford
(1871 - 1937) Rutherford's gold foil experiment provided the first evidence that each atom is made up of a large, positively-charged nucleus, surrounded by a cloud of negatively-charged electrons. Rutherford won the 1908 Nobel Prize in Chemistry for this work. Rutherford was also an early leader in nuclear fission techniques, having discovered the decay of carbon-14 and providing the impetus for modern carbon dating. As part of this research, he discovered the proton and neutron, the latter in cooperation with James Chadwick. He is also the only native New Zealander with an element named after him (Rutherfordium, atomic number 104).
Albert Einstein
(1879 - 1955) In one year — 1905, called his annus mirabilis, or "miracle year" — Albert Einstein authored four papers that revolutionized modern physics. The first explained the photoelectric effect in terms of quantized electromagnetic radiation. The second formed the foundation for modern statistical physics by explaining the seemingly-random motion of particles in a fluid, a behavior called Brownian motion. The third reconciled Maxwellian electrodynamics with classical mechanics by positing a finite, constant speed of light, a theory now known as special relativity. The fourth paper contained his statement that the energy of a body is equal to its mass times the speed of light squared (that is, E = mc^2). Ten years later, in 1915, Einstein published his theory of general relativity, which generalized special relativity to account for gravitational fields.
Niels Bohr
(1885 - 1962) Bohr reconciled Rutherford's results from the gold foil experiment with Max Planck's quantum theory to create a model of the atom (the Bohr model) in which electrons reside in specific energy levels at specific stable radii. This model was the basis for Johann Balmer's work with spectroscopy and Johannes Rydberg's energy formula, which explicitly stated the frequency of light that an electron would emit if it went from a higher energy to a lower energy. Bohr and his son fled to the U.S. in World War II under the pseudonym "Baker," and contributed to the Manhattan Project.
Erwin Schrodinger
(1887 - 1961) Schrödinger contributed to the early formulations of quantum theory as a foil to Werner Heisenberg, Niels Bohr, and Paul Dirac, criticizing their Copenhagen interpretation of quantum mechanics with thought experiments like his famous Schrödinger's Cat argument. He formulated both the time-independent and time-dependent Schrödinger equations, which are partial differential equations that describe how quantum systems behave. Schrödinger's work was the basis for Heisenberg's matrix formalism, Feynman's path-integral formalism, and quantum mechanical perturbation theory, which considers the effects of a small disturbance to a quantum system.
Louis de Broglie
(1892 - 1987) Broglie's work quantifying the wave-particle duality of quantum mechanics earned him the 1929 Nobel Prize in Physics. His doctoral thesis, which proposed that all particles have a characteristic wavelength dependent on their momentum, was so groundbreaking that the reviewers passed it directly to Albert Einstein, who endorsed it. In opposition to the probabilistic interpretation of quantum mechanics, de Broglie later worked to define a purely causal interpretation, but his work remained unfinished until David Bohm refined it in the 1950s.
Enrico Fermi
(1901 - 1954) Fermi is best known to the public as a main contributor to the Manhattan Project. His work with statistical physics laid the groundwork for modern electronics and solid-state technologies. He applied the Pauli exclusion principle to subatomic particles to create Fermi-Dirac statistics, which accurately predicted the low-temperature behavior of electrons. Particles that obey Fermi-Dirac statistics are called fermions in his honor. Fermi also suggested the existence of the neutrino in order to balance nuclear beta-decay chains.
Werner Heisenberg
(1901 - 1976) Heisenberg is most known for his matrix interpretation of quantum theory, which constructs observable quantities as operators that act on a system. His famous uncertainty principle (better translated, however, as "indeterminacy principle") states that the more accurately an object's position can be observed, the less accurately its momentum can. This is because shorter wavelengths of light (used as a sort of measuring-stick) have higher energies, and disrupt a particle's momentum more strongly. Heisenberg earned the 1932 Nobel Prize in Physics for discovering the allotropic forms of hydrogen.
Paul Dirac
(1902 - 1984) One of the first to attempt a generalization of quantum theory to relativistic speeds, the result of which was the Dirac equation
Wolfgang Pauli
(1904 - 1967) His namesake exclusion principle prohibits most types of particles from occupying the same state, and forms the basis for chemical bonds
J Robert Oppenheimer
(1904 - 1967) Oversaw much of the Manhattan Project, but was later stripped of his security clearance during the McCarthy-era Red Scare, as a result of his acquaintance with communists and his enmity with Edward Teller
George Gamow
(1904 - 1968) Gamow was one of the first to explain the implications of the Big Bang theory of cosmology. He correctly predicted the abundance of hydrogen and helium in the early universe, nicknamed Alpher-Bethe-Gamow theory (an intentional pun on the first three letters of the Greek alphabet, alpha, beta, and gamma, for which the otherwise unrelated physicist Hans Bethe was included), and also theorized that the the heat from the Big Bang would still be visible as the cosmic microwave background radiation. Although Gamow received no Nobel for this prediction, the CMB's discoverers, Arno Penzias and Robert Wilson, as well as two later observers, John Mather and George Smoot, did receive Nobels.
Kurt Godel
(1906 - 1978, Austrian) 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.
Richard Feynman
(1918 - 1988) Feynman developed a mathematical formalism called the path integral formulation of quantum theory that utilized the "sum over histories," taking into account all possible paths a particle could take. This constituted the creation of quantum electrodynamics and earned him the 1965 Nobel Prize in Physics. He also used the sum over histories in developing Feynman diagrams, which illustrate the interaction of subatomic particles. Aside from being a prolific physicist, Feynman was also an accomplished bongo player and sketch artist.
Murray Gell-Mann
(1929 - 2019) Predicted the existence of quarks, which compose protons, neutrons, and other, heavier particles
Archimedes
(287 - 212 BC, Syracusan Greek) 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.
Andrew Wiles
(Born 1953, British) 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).
Euclid
(c. 300 BC, Alexandrian Greek) 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.
Capacitor
A capacitor is a charge-storing disconnection in a circuit, usually made of parallel plates separated by a medium that blocks the passage of charge. The inner medium is called a dielectric. The governing equation of a capacitor is Q = C ΔV: the charge stored on the capacitor (Q, typically measured in coulombs) equals the capacitor's capacitance (C, typically measured in farads, or more likely in smaller units like microfarads) times the voltage across the capacitor (ΔV). The total capacitance of a circuit depends on whether capacitors are arranged in parallel or series, but in the opposite fashion as for resistors: If two capacitors having capacitance C1 and C2 are placed in parallel, the total capacitance across them is C = C1 + C2; if they are placed in series, then 1/C = 1/C1 +1/ C2. This is because wiring capacitors in parallel effectively expands the area of the parallel plates, so configuring capacitors in parallel increases total capacitance.
Diode
A diode permits the flow of current in only one direction. Ideally, a diode permits any amount of current in one direction but has a very high resistance to current in the opposite direction. In practice, there is a minimum voltage that must be provided before current will flow, called the bias voltage, and there is a voltage past which the resistance of the diode reduces and current flows backward, called the breakdown voltage. In a Zener diode, the breakdown voltage is well-defined with as sharp a transition as possible, so engineers can design a circuit around the breakdown voltage characteristic.
Ground
A ground or earth is a point on a circuit that is connected to the ground. This is to prevent accidental discharge of electricity from an unexpected connection—a person who touches an ungrounded circuit might cause electricity to pass through their body. This lets ground act as a known reference point of voltage for analyzing circuits: any grounded point in the circuit is typically understood to have zero voltage.
Rectifier
A rectifier is a device that converts AC or another variable signal into a positive-voltage signal. A half-wave rectifier simply zeroes out negative voltage, while a full-wave rectifier essentially acts as an absolute value function, converting negative voltage to positive voltage. A rectifier can be combined with a capacitor to approximately convert AC into DC.
Resistor
A resistor is an element that impedes the flow of current, creating a voltage drop. Resistance is measured in ohms, named for Georg Ohm and symbolized by a capital omega (Ω). Resistors are governed by Ohm's law, V = IR. In other words, the voltage drop across a resistor (V) equals the current passing through the resistor (I) times the resistor's resistance (R). According to Joule's first law, the power dissipated by a resistor is P = I2R. Combining that fact with Ohm's law yields P = IV. The total resistance of a circuit depends on whether resistors are arranged in parallel (different branches connecting to the same pair of nodes) or series (same branch). If two resistors having resistance R1 and R2 are placed in series, the total resistance across them is R = R1 + R2; if they are placed in parallel, then 1/R = 1/R1 +1/ R2. A rheostat, or variable resistor, can be made by taking a material of uniform resistivity and changing how much of the material is in the circuit. A potentiometer is a rheostat with two terminals and a connection in the middle, so the middle terminal can set a precise, variable dividing point between the ends; therefore, a potentiometer can be used as a voltage divider. Rheostats and potentiometers are often operated by knobs or sliders. A Wheatstone bridge is a diamond-shaped arrangement of two resistors of known resistance, one rheostat, and a resistor of unknown resistance. With a DC voltage source between the two known resistors, and also between the rheostat and the unknown resistor, this setup can be used to find the resistance of the unknown resistor: an ammeter (device to measure current) is connected across the two other nodes and the rheostat is adjusted until no current flows. The formula for solving a Wheatstone bridge uses Kirchhoff's laws: the total current flowing into any point in a circuit equals the total current flowing out of the point, and the total voltage change around any loop of a circuit is 0 V.
Inductor
A solenoid is a coil of wire. When electric current through the wire changes, a magnetic field is generated in the core (interior) of the solenoid. When a solenoid is used in a circuit, it is called an inductor and serves to oppose changes in the current in the circuit. The strength of an inductor is its inductance, symbolized L and measured in henries (after Joseph Henry, who discovered the principles of self-inductance and mutual inductance, though Michael Faraday published his findings on electromagnetic induction first). The voltage across an inductor only depends on the change in current, so inductors are particularly used with alternating current. For standard (sinusoidal) alternating current, an inductor only changes the phase of the current through it. Inductors function the same way as resistors when arranged in series or parallel.
Source
A source is any device that provides voltage or current. Direct Current (DC) sources like batteries provide constant current, while Alternating Current (AC) sources like household electric outlets provide current that changes at a regular frequency with a constant average voltage. In alternating-current circuits, capacitors and inductors can help tune a circuit's efficiency by contributing to a phenomenon called impedance, which is the complex analogue of resistance. A circuit's reactance is the imaginary part of its impedance and depends on the frequency of the circuit. When reactance is balanced, we find the frequency at which the circuit loses the least energy, or its resonant frequency.
Switch
A switch is simply a device that toggles between an open connection and a closed connection. Switches can also be used to toggle between two (or more) different closed connections.
Transformer
A transformer is a pair of solenoids connected to a central core, which permits the circulation of magnetic flux. Because the flux in the core is constant, an alternating current in one coil produces a changing magnetic field, inducing an alternating current in the second coil. If the coils have different numbers of turns of wire (as is typical), the current in the second coil will have a different voltage than the current in the first coil. Therefore, transformers are used to "step up" or "step down" voltage, for example in power transmission lines and in adapters for household electronics.
Infrared Spectroscopy
Acquires information about the chemical groups present in a compound based on which wavelengths of infrared light the bonds in those groups absorb. When IR-active bonds absorb infrared light, they undergo a change in dipole moment and are excited to a higher-energy vibrational mode, which have names like "stretching," "wagging," and "scissoring." The output of an IR experiment (called an "IR spectrum") is a graph of absorbance on the y-axis against wavenumbers, measured in cm−1 (inverse centimeters), on the x-axis. The most distinguishable IR peak is that of a carbonyl group, which displays a very strong absorbance at 1700 cm−1, while peaks below 1500 cm−1 produce a complex pattern unique to the compound being analyzed, called the fingerprint region. Samples are typically prepared by grinding them into a potassium bromide pellet or by creating a mull with the oil Nujol.
Cnidaria (10,000 Species)
Also called Coelenterata [see-LEN-tur-AH-tuh], the cnidarians develop from a diploblastic (two-layered) embryo, and have two separate tissue layers and radial body symmetry. Many cnidarians have two life stages, the mobile, usually bell-like medusa and the sessile polyp. All cnidarians have nematocysts, or stinging cells, for capturing prey, and some can inflict painful stings on swimmers. Examples include the hydras, sea anemones, corals, jellyfishes, and Portuguese man-o-war (which is actually an aggregation of colonial cnidarians).
LED
An LED, or light-emitting diode, is a diode that lights up when current passes through in the forward direction
Partons
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.
Op-Amp
An op-amp (operational amplifier) is a five-port device that changes a voltage difference across two ports, multiplied by a factor (gain), into a voltage on an output port, using power supplied to two power ports. There are various ways to wire an op-amp, often feeding an output back into an input, to accomplish interesting tasks like subtracting and multiplying voltages. An ideal op-amp is assumed to have infinite gain, so when solving a circuit, the two inputs must reach a point with no voltage difference across them. Real op-amps have finite gain, impedance across their inputs, resistance on their output, and a finite bandwidth or spread of operating frequencies when used with alternating current.
Quarks
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.
Hadrons
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."
Phobos
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.
Carbon (C, 6)
By definition, it is in all organic compounds. It is the fourth most abundant element in the Universe. It has three major isotopes: isotope 12, which is stable; isotope 13, which is used in NMR spectroscopy; and isotope 14, which is radioactive and is the basis of carbon dating. Carbon's ability to form four chemical bonds means that it has many different allotropes. The best-characterized natural allotropes are diamond, which consists of a tetrahedral network of carbon atoms, and graphite, which consists of planes of carbon atoms arranged in hexagons. Fullerenes such as buckyballs and carbon nanotubes, on the other hand, are generally produced synthetically; buckyballs are roughly spherical. More recently, graphene, which is a single layer of atoms shaped like graphite, has proven to have remarkable properties; for example, it is nearly transparent while being about 200 times stronger than an equivalent mass of steel.
Oxygen (O, 8)
By mass, the most common element in Earth's crust. It was discovered independently by Carl Scheele and Joseph Priestley; Priestley originally called it "dephlogisticated air." Oxygen normally exists in elemental form as a diatomic gas (O2), but it can also exist in a triatomic form, ozone (O3), which is known for its role in blocking UV rays in Earth's stratosphere. Diatomic oxygen is — despite having an even number of electrons — paramagnetic, meaning it has unpaired electrons. This points out a problem with traditional valence bond theories, which predict that oxygen should be diamagnetic; molecular orbital theory correctly explains this behavior. Because oxygen is easily capable of accepting electrons, reactions in which a species gives up electrons are known as oxidation reactions.
Titrations
Calculate the concentration of a solution by adding in small volumes of a reactant of known concentration until a chemical change, like a pH indicator changing color, occurs. Acid-base titrations are usually performed in a thin glass tube called a buret and use pH indicators like phenolphthalein and bromothymol blue. The Henderson-Hasselbalch equation can be used to calculate the pH at any point during a titration. The equivalence point is the point at which equal amounts of acid and base have been mixed and there is a sharp inflection point in the pH curve. Redox titrations use an oxidation-reduction reaction instead of an acid-base reaction. In complexometric titrations, the analyte forms a coordination complex with the titrant. Karl Fischer titrations use electrolysis to determine the amount of water in a substance.
Calorimetry
Calculates the heat or enthalpy change of a chemical or physical process by using specialized vessels to measure a change in temperature. A very simple calorimeter can be made by placing a thermometer in an insulating polystyrene coffee cup and sealing it with a lid, causing the reaction inside to occur at constant pressure. Bomb calorimeters are very sturdy containers used to conduct combustion reactions at constant volume. Isothermal titration calorimetry (ITC) is a variant which can be used to determine the binding affinities and stoichiometry of proteins and enzymatic reactions. Differential scanning calorimetry (DSC) is another variant that measures how a compound's heat capacity changes with temperature; it is commonly used to determine properties of polymers such as their melting point or glass transition temperature.
Echinodermata (6,500 Species)
Characteristics of this phylum include an endoskeleton composed of many ossicles of calcium and magnesium carbonate, a water vascular system, a ring canal around the esophagus, and locomotion by tube feet connected to the water vascular system. Unique to echinoderms is the five-fold radial symmetry obvious in sea stars (starfish), sea urchins, and sea lilies. Others, like sea cucumbers, have varying degrees of bilateral symmetry. In the echinoderm body plan, a true head is absent; the anatomical terms oral (mouth-bearing) and aboral (away from the mouth) are used to describe orientation of the body surfaces. Feeding adaptations include particle feeding through the water vascular system, inverting the stomach to engulf prey (sea stars), and a scraping device called Aristotle's lantern (sea urchins).
Baryons
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).
Mesons
Composite particles generally made from a quark and an antiquark. There are dozens of examples including the pion, kaon, J/Psi, Rho, and D. All mesons are bosons. The quark and antiquark 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 antiquarks, but this kind of particle (a "tetraquark") is rare
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, x^2/(x - 3), and (x^2 + 1)/(x^2 - 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 x^3 as a rational function, think of it as x^3/1). 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.
Flame Tests
Detect the presence of elements by dipping a wooden splint or nichrome wire in a sample of the element or its salt, then placing the sample over a Bunsen burner. The unique emission spectrum of the element present then causes the flame to briefly change color. The D lines of sodium produce one of the strongest flame test colors: a bright yellow. Because of this, sodium can contaminate samples, so flames are usually viewed through a cobalt blue glass to filter out the yellow. Other notable flame test colors include red, produced by lithium, calcium and strontium; lilac, produced by potassium; green, produced by barium; and blue, produced by copper, selenium, arsenic, and indium.
Even/Odd Functions
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. 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 one hundred and eighty degrees around the origin. Odd functions are so named because if a polynomial's exponents are all odd, then the polynomial is an odd function.
Differentiable Functions
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).
Exponential Functions
Functions in the form of f(x) = b^x, where b (called the base) is a positive number other than one. 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) = e^x, 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.
Polynomials
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, 3x^(2)y and -πx^(7)y^(2)z^(3) are each terms, so 3x^(2)y - πx^(7)y^(2)z^(3) is a polynomial. Much of math is concerned with polynomials involving only one variable, such as -x^(3) + 2x^(2). The number at the beginning of each term is called a coefficient. Polynomials can be classified according to their number of terms: a polynomial with one term is called a monomial; a polynomial with two terms is called a binomial; 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 of the variable. For polynomials of multiple variables, you calculate the sum of the variables' exponents on each term, then choose the largest sum. A polynomial of degree one is linear; a polynomial of degree two is quadratic; a polynomial with degree three is cubic; continuing with increasing degrees, the terms are quartic, quintic, sextic, and so on. 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 now 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 Theory, which is part of abstract algebra.
Logarithmic Functions
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.
Surjective Functions
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) = x^2 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
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.
Trigonometric Functions
Functions that 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.
Continuous Functions
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.
Gauge Bosons
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).
Mass Spectrometry
Identifies an unknown unknown compound by ionizing it, fragmenting it into pieces, then passing it through electromagnetic fields to separate the pieces based on their mass-to-charge ratio (m/z). A mass spectrum plots the abundance of each fragment against the m/z—the spectrum features a base peak (the peak of highest intensity) and a series of peaks whose spacing tells you what elements are present—for instance, a peak spacing of 14 typically indicates a CH2 or methylene unit. The different isotopes of elements can also produce characteristic peak patterns—for instance, two equal-intensity peaks spaced 2 units apart and 79 units away from the next peak indicates bromine. When analyzing proteins, fragmentation is often undesirable, so "soft" ionization methods such as matrix-assisted laser desorption/ionization (MALDI) or electrospray ionization (ESI) are used. In forensics, mass spectrometry is often "coupled" to gas chromatography in GC-MS; gas chromatography first separates the unknown mixture, then mass spec identifies the individual components of the mixture.
Gold (Au, 79)
Known to the ancients as a relatively inert metal. Its atomic symbol comes from its Latin name, aurum. It is resistant to attack by most acids, but it (along with platinum) will dissolve in aqua regia, a mixture of concentrated nitric acid and hydrochloric acid. Among all metals, it has the highest electronegativity and electron affinity; it occasionally is found in a -1 oxidation state as Au^-. Widely used in jewelry, it also has a number of scientific uses. Ernest Rutherford's gold foil experiment demonstrated the existence of a positively charged nucleus. Scanning electron microscopy (SEM) often requires that specimens be "sputtered," or thinly coated, with gold atoms to allow imaging. Suspensions of gold compounds have been used to treat rheumatoid arthritis.
Monocots vs Dicots
Most (but not all) angiosperms fall into one of two classes based on the number of cotyledons, or embryonic seed-leaves, in the plant embryo. Monocots, or Monocotyledonae, have one cotyledon, while dicots, or Dicotyledonae, have two. While there are no other hard-and-fast distinguishing characteristics between the two groups, plants in each category tend to share other characteristics: Monocots produce pollen grains that have a single furrow (monosulcate); flower parts in multiples of three; numerous, fibrous roots; parallel leaf veins; and stems with scattered vascular bundles. They also lack secondary growth, remaining herbaceous throughout their lives. Dicots, on the other hand, tend to have pollen with three furrows (tricolpate); flower parts in multiples of four or greater; taproot systems; stems with rings of vascular tissue; and branching leaf veins. Many of them exhibit secondary growth that produces wood.
Bryophytes vs Pterophytes
Not all plants produce seeds. Seedless plants are divided into bryophytes (mosses, liverworts, and hornworts) and pterophytes (ferns, club mosses, quillworts, and horsetails). Both of these groups, like all other plants, reproduce by producing sperm and eggs on a structure called the gametophyte. The gametes fuse to form another structure called the sporophyte, which produces spores that disperse and grow into new gametophytes. Both groups produce flagellated sperm that require water for fertilization. Note that these uses of "bryophyte" and "pterophyte" here are informal, and should not be confused with the actual phyla Bryophyta (true mosses) and Pteridophyta (true ferns). Bryophytes are small enough that water and nutrients can diffuse to all parts of the plant without any specialized vascular tissue. They lack true leaves and roots, instead fastening themselves to the ground with rhizoids. Unlike other land plants, bryophytes have a prominent gametophyte stage that is usually dioecious, meaning that an individual plant produces only one type of gamete (either sperm or egg). The short-lived sporophyte grows from the female gametophyte. The more complex pterophytes can grow taller thanks to vascular tissues (see number 4 below) that provide structural support and transport water and other materials throughout the plant. Many of them do have true leaves and roots. Pterophytes have a prominent sporophyte stage that grows from a small, short-lived gametophyte. Pterophyte gametophytes may be dioicous or monoicous, producing both sperm and egg on the same plant.
Mercury (Hg, 80)
One of just two elements that is a liquid at standard temperature and pressure (the only other one is bromine). It has been known since antiquity, and is found in ores such as cinnabar. Older names for it, reflecting its liquid nature, include hydrargyrum (the source of its symbol) and quicksilver. Because it is a very dense liquid, it is commonly used in barometers to measure atmospheric pressure; the pressure exerted by the atmosphere equals the pressure exerted by a column containing 760 millimeters of mercury. Alloys of mercury with other metals are called amalgams, some of which have been used as dental fillings. Chronic exposure to mercury can cause psychological problems; its use in hatmaking led to the expression "mad as a hatter." More recently, concerns about mercury exposure have led to the banning of mercury in thermometers.
Leptons
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.
Deimos
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.
Fermions
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.
Bosons
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.
Quadratics
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 x^2 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 ax^2 + 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).
Ultraviolet-Visible Spectrophotometry
Quantifies the presence of compounds by shining light in the ultraviolet-to-visible range on a molecule, then measuring which wavelengths are absorbed. Parts of a molecule which absorb visible light are referred to as chromophores; since different molecules have different chromophores, the wavelength of maximum absorbance, or lambda-max, can be used to identify a compound. Beer's law is then used to quantify the concentration of the present compound. Samples are held in small square plastic or quartz tubes called cuvettes. The Woodward-Fieser rules empirically estimate the lambda-max from the types of bonds and functional groups in a molecule. UV-Vis has many uses in biology—for example: calculating the OD600 to measure bacterial growth rate, determining nucleic acid quality with the 260/280 ratio, measuring protein concentration in the Lowry and Bradford protein assays, and monitoring protein folding by measuring tryptophan or tyrosine absorbance.
Angiosperms vs Gymnosperms
Seed-producing plants can be divided into gymnosperms (cycads, ginkgos, conifers, and gnetophytes) and angiosperms (phylum Anthophyta, or flowering plants). Most of these plants produce male gametophytes that grow into the female, allowing fertilization to take place in relatively dry conditions. Many of them also exhibit secondary growth of woody tissues, allowing them to grow even taller than the pterophytes. The word gymnosperm means "naked seed," referring to the fact that their gametophytes develop on the surface of leaves or on the scales of cones. In contrast, angiosperm means "receptacle seed." Their gametophytes develop enclosed within flowers. Angiosperms are further classified based on their seed structure, described below.
Chromatography
Separates a complex mixture into its individual components, commonly illustrated by the separation of pen ink into many colors. Chromatography involves two components: a mobile phase, which moves, and a stationary phase, which interacts differently with different components of the mobile phase to produce a separation. For example, in thin-layer chromatography, a mixture is spotted on one end of a plate of silica gel (the stationary phase), then a solvent (the mobile phase) carries the components across the plate and separates them based on their polarity—polar substances will strongly interact with the polar silica gel and not move very far, while nonpolar substances will move very far. In gas chromatography (GC), substances are vaporized and run through a packed column, where the time it takes each component to travel through—the retention time—is determined. High-performance liquid chromatography (HPLC) is like gas chromatography, but the sample remains in the liquid phase and is pushed through using pressures of up to 400 atmospheres. Ion-exchange chromatography uses stationary phases with acidic or basic functional groups to remove charged compounds; it can be used for water softening.
Distillation
Separates a mixture of liquids based on their boiling point by heating, causing the more volatile component to vaporize and condense in a different container while the other components remain in the original vessel. In the laboratory, the vapors are usually cooled with a water-based Liebig condenser or Vigreux condenser, and the product is collected in a round-bottomed receiving flask. In industry, multiple rounds of distillation are performed in a single "column" packed with trays, each of which can be modeled as a "theoretical plate." Oil refining relies on fractional distillation, in which different products are pulled out of the mixture at various trays along the column. Azeotropes are mixtures that cannot be separated because at the specified pressure and composition, both components boil at the same temperature.
Liquid-Liquid Extraction
Separates mixtures based on their relative solubilities in two immiscible solvents, such as oil and water. It is commonly performed by placing the mixture and solvents into a separatory funnel, shaking, and then using the stopcock to remove one of the two phases. The partition coefficient quantifies the desired compound's relative solubility in the two phases. The material left over after the desired material has been extracted is called the raffinate. A variant of liquid-liquid extraction that uses phenol and chloroform as the two solvents is used to isolate DNA from cells
Annelida (11,500 Species)
The annelids are segmented worms and represent the first lineage of truly acoelomate animals, meaning their body cavities are lined with tissue derived from the embryonic mesoderm. Annelid classes include the marine Polychaeta, as well as the mostly terrestrial Oligochaeta (including the earthworms, Lumbricus) and the mostly aquatic Hirudinea, or leeches. Characteristics of annelids include nephridia (kidney-like structures), blood vessels, and, in some classes, hermaphroditism.
Flower Parts
The calyx is composed of sepals, specialized green leaves that protect the flower as a bud and provide support for the fully bloomed flower. The stem supporting a flower is called the peduncle. If multiple flowers bloom from a peduncle, the stems supporting each flower are called pedicels. The torus is the swelling at the top of the pedicel or peduncle, just below the calyx. Petals are specialized leaves, often brightly colored to attract pollinating species. Collectively, they are called a corolla. The pollen-producing reproductive organ of the flower is called the stamen. The stamen consists of a thin filament topped by an anther, which actually contains the pollen. The pistil, or female reproductive organ of the flower, is composed of leaf-like carpels. The ovary-containing ovules are at the base of the pistil, while a tube called a style topped by a sticky, pollen-receptive stigma rises from the ovary. There may be one or many pistils in each flower.
Hydrogen (H, 1)
The first element on the periodic table and, by far, the most common element in the Universe. In addition to the main isotope (called protium), there are two other significant isotopes of hydrogen: deuterium (2H or D), which has one neutron, and tritium (3H or T), which has two neutrons. It naturally exists as a diatomic gas (H2), which was discovered by the British chemist Henry Cavendish. Hydrogen is highly flammable when exposed to high temperatures or electric current, a fact demonstrated by the Hindenburg disaster. It can react with nonmetals by losing an electron to form the H+ ion, or react with metals to form the hydride ion, H-.
Platyhelminthes (15,000 Species)
The flatworms are the most primitive phylum to develop from a triploblastic (three-layered) embryo. They have bilateral body symmetry, and are acoelomate (lacking a true body cavity), so that the space between the digestive tract and the body wall is filled with tissue. As the name implies, they are generally flat-bodied. They have a true head and brain, but the digestive system has only one opening, which functions as both mouth and anus. Most are hermaphroditic. This phylum includes parasites such as the tapeworms and flukes, as well as free-living (i.e., non-parasitic) organisms such as the planarians.
Gluons
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.
Io
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.
Inverse Trigonometric Functions
The inverse functions of the trigonometric functions. They are sometimes just called "inverse sine," etc, and are also given with the prefix "arc": arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent. 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.
Ganymede
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.
Helium (He, 2)
The lightest noble gas and the second most abundant element in the Universe, after hydrogen. Discovered by Sir William Ramsey, Pierre Janssen, and Norman Lockyer, it has two stable isotopes, helium-3 and helium-4, with helium-4 by far the more common. Because of their different quantum properties (the helium-3 nucleus is a fermion, while the helium-4 nucleus is a boson), the isotopes of helium actually have significantly different physical properties. Helium-4 can exist in a zero-viscosity state known as superfluidity when its temperature drops below the lambda point. Helium has the lowest boiling point of any element; liquid helium is used for devices that need intense cooling, such as MRI machines. Most helium on Earth results from radioactive decay, since the helium nucleus is equivalent to an alpha particle.
Mollusca (50,000 Species)
The molluscs are second in diversity only to the arthropods. Body plans within this phylum are diverse, but general characteristics include a soft body covered by a thin mantle, with a muscular foot and an internal visceral mass. There are two fluid-filled body cavities derived from mesodermal tissue: a small coelom and a large hemocoel that functions as an open circulatory system. Many molluscs have a shell composed of calcium carbonate and proteins, secreted by the mantle. Familiar groups within the Mollusca include the classes Gastropoda (slugs, snails), Bivalvia (clams, oysters, scallops), and Cephalopoda (nautilus, squids, octopi).
Earth's Moon
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.
Nitrogen (N, 7)
The most abundant element in Earth's atmosphere. Nitrogen, which was first isolated as "noxious air" by Daniel Rutherford, exists primarily as a diatomic molecule containing two triple-bonded nitrogen atoms (N2). Because nitrogen gas is extremely stable, N2 is unusable for many biological and chemical purposes. To make it useful, it often undergoes fixation to convert it into usable nitrogen species such as the ammonium ion (NH4+) — as it is by bacteria in the root nodules of legume plants—or ammonia gas (NH3), as is done industrially in the Haber-Bosch process. Conversely, its stability makes it useful in preventing unwanted combustion reactions. It also has a relatively low boiling point (-196°C), which makes liquid nitrogen useful as a refrigerant.
Aluminum (Al, 13)
The most common metal in Earth's crust, and the first metal in the p block of elements. First isolated by Hans Christian Ørsted, its primary ore is bauxite, from which it is refined using large amounts of electric current, via electrolysis, through the Bayer and Hall-Héroult processes. (Because aluminum exists only in a +3 oxidation state, it takes three moles of electrons to produce one mole of aluminum; as a result, it has been estimated that 5% of all electricity in the U.S. goes to purifying aluminum.) It is found in the mineral corundum, which is found in many gems, including sapphires and rubies; the specific impurities found in a gem determine its color. It is also found in aluminosilicates, such as feldspar.
Iron (Fe, 26)
The most common metal in the Earth, and one of the major components of the Earth's core. Iron was known to the ancients; its atomic symbol comes from the Latin name ferrum. Iron is the namesake of ferromagnetism; one of its ores is magnetite, Fe3O4, which contains iron in both of its most common oxidation states, 2+ and 3+. Iron(II) sulfide, FeS2, is formally known as pyrite, but because of its appearance has long been known as fool's gold. Iron can react with oxygen in the air to form iron(III) oxide, Fe2O3, in a relatively slow but exothermic process; this process is used in "all-day" heat patches. Hydrated iron(III) oxide is better known as rust; rust only forms when iron is exposed to both oxygen and water. Its isotope 56 is one of the most strongly bound of all nuclei, with the lowest mass per nucleon. Iron is one of the heaviest elements that is normally produced by stellar nucleosynthesis. The largest use of iron is in steel.
Arthropoda (Over 800,000 Species discovered)
The most diverse and successful animal phylum on earth (incorporating about 75% of all described animal species), the Arthropods are characterized by jointed legs and a chitinous exoskeleton. Like annelids, they are segmented, but unlike annelids, their segments are usually fused into larger body parts with specialized functions (such as the head, thorax, and abdomen of an insect). Arthropods are often divided into four subphyla: Uniramia (insects, centipedes, millipedes); Chelicerata (arachnids, sea spiders, horseshoe crabs); Crustacea (shrimps, lobsters, crabs, crayfish, barnacles, pillbugs), and Trilobitomorpha (the trilobites, now extinct).
Chordata (44,000 Species)
The phylum that contains humans, Chordata is divided into three subphyla: Urochordata, the sea squirts; Cephalochordata, the lancelets, and Vertebrata, the true vertebrates, which is the most diverse subphylum. Defining traits of chordates include pharyngeal gill slits, a notochord, a post-anal tail, and a dorsal hollow nerve cord. In vertebrates, some of these structures are found only in embryonic stages. The lancelet Amphioxus (Branchiostoma) is often used as a demonstration organism in biology labs.
Nematoda (15,000 Species)
The roundworms are unsegmented worms that live in a variety of habitats. They are pseudocoelomate; the three tissue layers are concentric, but the body cavity is not lined with tissue derived from the mesoderm (middle embryonic layer). They include both free-living and parasitic species; human parasites include hookworms and the causative agents of elephantiasis, trichinosis, and river blindness. Soil nematodes may be crop pests, while others are beneficial predators on other plant pests. The nematode species Caenorhabdis elegans is a common subject in genetics and developmental-biology labs.
Porifera (5000 Species)
The sponges are all water-dwellers (98% marine, 2% freshwater), and are sometimes classified separately from other animals because of their asymmetric bodies and lack of distinct tissues. They are sessile (immobile) except in early dispersing stages, and collect food particles via the sweeping motions of flagellated cells called choanocytes [koh-AN-oh-"sites"].
Xylem vs Phloem
There are two types of vascular tissue in plants. Xylem transports water and soluble nutrients from the roots to the leaves. Phloem, on the other hand, carries nutrients like sucrose from their origin of synthesis or absorption to all parts of the plant. Both tissues originate in the procambium of the apical meristems of both the stems and roots. In woody plants, secondary vascular tissues arise in the vascular cambium. Xylem contains distinct elongated cells called tracheids that have lignified cell walls and help provide structural support. Vessel elements are also reinforced by lignin, but they are open at each end at perforation plates and connect to form long tubes for water transport. Xylem functions via transpirational pull and osmosis. Cell types in phloem include companion cells, fibers, and sclereids. In trees, it is usually the innermost layer of the bark.
Periodic Functions
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.
Nuclear Magnetic Resonance
Uses magnetic fields to determine the arrangement of nuclei in a molecule. Typical NMR methods only work on nuclei that have nonzero spin, such as 1H and 13C—the nonzero spin means that the nuclei oscillate between two spin states in a phenomenon called Larmor precession. NMR measures the frequency at which each nucleus oscillates, whose deviation from a reference nucleus (such as tetramethylsilane) depends on the local electron density and is called the chemical shift. Nuclei that have more electron density (due to proximity to electron-donating groups like alkyl groups) are said to be "shielded" and have lower chemical shifts, while those that have less electron density (due to proximity to electron-withdrawing groups like halogens) are said to be "deshielded" and have higher chemical shifts. Peaks in NMR can be "split" into several peaks due to J-coupling if they are adjacent to identical nuclei, like the three hydrogens of a methyl group. NMR is the theoretical basis for magnetic resonance imaging (MRI) in medicine.
Fuses and Circuit Breakers
When excess current flows through a fuse, the fuse deforms or disconnects so that no current can flow in the circuit. This protects the equipment in the circuit from being damaged by power surges and also protects the surroundings (as excess current can cause wires to overheat, starting a fire). Fuses typically melt and thus must be completely replaced after tripping; a circuit breaker is a form of fuse that can be reset and restored after the circuit has been verified to be safe.
Sulfur (S, 16)
Widely known in the ancient world, and referred to in the Bible as brimstone. Its nature as an element was first recognized by Antoine Lavoisier. Its most stable allotrope is an eight-membered ring that exists as a yellow solid. It is most often isolated by injecting superheated steam into the ground in the Frasch process. As an element, it is used in the vulcanization process to cross-link the polymer strands of rubber to increase strength; similarly, sulfur-sulfur bonds hold many proteins together. Industrially, though, the majority of sulfur is used to make sulfuric acid, H2SO4 (in fact, sulfuric acid is the most widely produced chemical in the chemical industry). Sulfur compounds are noted for their strong and unpleasant odors; small quantities of hydrogen sulfide, H2S, are frequently added to natural gas — which is normally odorless — to help people notice gas leaks.
