The Handy Math Answer Book, Second Edition (2012)
MATH IN THE PHYSICAL SCIENCES
PHYSICS AND MATHEMATICS
What science fields use an abundance of mathematics?
The sciences that are “heavy users” of mathematics are the so-called physical sciences, including physics, chemistry, geology, and astronomy. These scientific fields are often contrasted with the natural or biological sciences. The physical sciences analyze the nature and properties of energy and non-living matter, and they often need the help of mathematics to determine the complex relationships between their interactions.
What is physics?
Physics is often described as the science of the interactions between matter and energy. It includes the subfields of atomic structure, heat, electricity, magnetism, optics, and many other phenomena.
Traditionally speaking, physics is divided into classical and modern—although many subdivisions of the two overlap—and both are ruled by mathematics. Classical physics includes Newtonian mechanics, thermodynamics, acoustics, optics, electricity, and magnetism. Modern physics includes such fields as quantum field theory and relativistic mechanics.
Other common divisions of physics are experimental and theoretical physics. Theoretical physicists use mathematics to describe the physical world and predict how it will behave; they depend on experimental results to check, understand, change, or eliminate theories. Experimental physicists test their predictions with practical experiments, often using mathematics to conduct the experiments.
What is mathematical physics?
Mathematical physics uses the concepts of statistical mechanics and quantum field theory. But it is not the same as theoretical physics. Mathematical physics studies physics on a more abstract and meticulous level. Theoretical physics entails less mathematics than mathematical physics and has more to do with experimental physics. But like many fields of science, defining mathematical physics is not easy. For example, still another definition of modern mathematical physics states that it entails all areas of mathematics other than classical mathematical physics.
A diagram of a bloch sphere, or diagram of a qubit, which is the basic building block of quantum computers. Named after Swiss physicist Felix Bloch (1905–1983), the bloch sphere has two possible states, which can represent the 0s and 1s used in computing. It took highly advanced mathematics to develop this concept!
What are some physics fields that rely heavily on mathematics?
Almost every field of physics—especially modern physics—relies heavily on mathematics. For example, mathematics is needed to understand the concepts of acceleration, velocity, and gravitational forces. Statistical mechanics also uses an intense amount of mathematics. And it is impossible to comprehend quantum mechanics without a good knowledge of mathematics. In fact, the subject of quantum field theory is one of the most mathematically rigorous and abstract areas of the physical sciences.
CLASSICAL PHYSICS AND MATHEMATICS
How is mathematics used in physics to describe motion?
Everything in the universe is in motion, from the rotating Earth to subatomic particles. Motion in physics is described mainly through mathematics, including speed, velocity, acceleration, momentum, force (something that changes the state of rest or motion of an object), torque (when a force causes rotation or twisting around a pivot point), and inertia (a body at rest remains at rest, and a body in motion remains in motion, until acted upon by an outside force).
Unlike what most people think, speed and velocity are not the same. Speed is the rate at which something moves; velocity is speed in a certain direction. Speed is also called a scalar quantity, described by the following formula: speed = distance/time. For example, if you drive 200 miles in 2 hours, and your speed is constant, your average speed is 200/2, or 100 miles per hour. On the other hand, velocity is known as a vector quantity (for more about vectors, see “Mathematical Analysis”). That gives velocity both speed and direction—and that leads directly to acceleration.
When an object’s velocity changes, we say that it accelerates. Acceleration—also a vector like velocity—is represented as the change in velocity divided by the time it takes for the change to occur. We define the formula for acceleration—or the change in velocity per unit time—as a = Δv/Δt. In this equation, a is acceleration, Δv is change in velocity of an object (the delta symbol stands for change), and Δt is the change in time needed to reach the velocity. For example, if acceleration is constant, and a person drove from a standing point to 60 miles per hour in 5 seconds, the equation becomes: 60 miles per hour / 5 seconds (or final speed minus the initial speed, all divided by the elapsed time). This means that the acceleration is equal to 17.6 feet per second squared (you have to change the miles per hours to feet and seconds, respectively).
Momentum relates to the amount of energy maintained by a moving object; it is also defined as the force necessary to stop an object from moving. It depends on the mass and velocity of an object, and is represented as: M = mv, in which M is momentum, m is the mass of the object, and v is the object’s velocity.
What is the mathematical distinction between weight and mass?
There is a definite difference between weight and mass, which is easily represented in the formula W = mg, in which W is weight (or the gravitational pull on an object), m is mass (or the quantity of matter in an object), and g is the gravitational pull. For example, if you wanted to lose weight, you could move further from the Earth or live on the Moon (in both cases, the gravitational pull would be less). But remember, no matter where you travel in the universe, you will always have the same mass—unless you diet!
Whether you are standing on the Moon or on Earth, your mass will always be the same; however, you will weigh less on the Moon because the gravitational pull on your body is less. Taxi/Getty Images.
Can Newton’s three laws be expressed mathematically?
Yes, Newton’s three laws are all based on mathematical formulas, but the equations would be too complex for this text. They are defined (some with applicable mathematical notation) as follows: Newton’s First Law (Law of Inertia)states that without any forces acting on an object, it will maintain a constant velocity. In other words, an object will stay still or keep moving in a straight line until something pushes it to change its speed or direction. The one force that stops almost everything from standing completely still or moving in a straight line is gravity. This is easily seen when you throw a baseball in the air: The ball does not move in one, continuous, straight direction because the Earth’s gravity pulls it downward toward the surface.
Is mathematics used in physics to describe work and energy?
Yes, mathematical equations can be used to describe work and energy. Energy comes in many forms, but its basic definition is in terms of work. Work is done when a force moves a body a certain distance. It is expressed in the simple equation: W = Fd, in which W is work, F is the force, and d is the distance. In this definition, only force in the direction of the object’s motion counts.
Newton’s Second Law (Law of Constant Acceleration) states that if a force acts on an object, the object accelerates in the direction of the force; the force creates an acceleration proportional to the force (and inversely proportional to the mass). This is written in the following notation: F = ma, in which F is the force, m is the mass, and a is the acceleration. Newton actually expressed this in terms of the calculus—a form of mathematics he created to explain these physical laws (for more about the calculus and Newton, see “History of Mathematics” and “Mathematical Analysis”). He wrote the equation as follows:
in which m is mass, Δv is the change in velocity, and Δt is the change in time. This is because instantaneous acceleration is equal to the instantaneous change in velocity in an instance in time (or change in time).
Newton’s Third Law (or Law of Conservation of Momentum) states that forces on an object are always mutual. To put it another way, if a force is exerted on an object, the object reacts with an equal and opposite force on the phenomenon that initially exerted the force. Simply stated, objects exert equal but opposite forces on each other. This is often phrased, “For every action, there is an equal and opposite reaction.” The mathematical equations for these forces are complex and beyond the scope of this text.
What is Newton’s Law of Universal Gravitation?
As hard as it is to comprehend, (almost) everything in the universe is attracted to everything else. This physical law is not only one of the most well-known but also one of the most important. Newton’s law states that the gravitational force between two masses, m and M, is proportional to the product of the masses and inversely proportional to the square of the distance (r) between them. In formula form, this is written as follows (note: in some texts, the masses of the two objects are written as m1 and m2):
G is a constant in nature (also called a universal constant), indicating how strong a gravitational force exists. In other words, the farther away the objects, the less the attraction between the objects.
What is statistical mechanics?
Statistical mechanics applies statistics to the field of mechanics—the motion of particles or objects when subjected to a force. It is used to understand the properties of single atoms and molecules of liquids, solids, gases—even the individual quanta of light that make up electromagnetic radiation—to the bulk properties of everyday materials. Because statistical mechanics mathematically helps to understand the interactions between a large number of microscopic elements, it is used in a wide range of fields. In a way, it is also the “opposite” of thermodynamics, which approaches the same types of systems from a macroscopic, or large-scale, point of view.
What is Ohm’s Law?
Ohm’s Law is important to the field of electrical studies. It states that direct current flowing in a conductor is directly proportional to the potential difference between its ends. First summarized by German physicist Georg Simon Ohm (1789–1854), it is usually seen in formula form as: V = IR (or I = V/R), in which V is the potential difference (voltage), I is the current (also written in some texts as i), and R is the resistance of the conductor. This can also be written in terms of electric quantities (voltage = current × resistance) and with units of measure (volts = amps × ohms).
Georg Ohm discovered the law about how electrical currents flow through a conductor.
Who developed the mathematical equations that explain electricity and magnetism?
One of the major early works about electricity and magnetism was written by Scottish physicist James Clerk Maxwell (1831–1879), who in 1873 published A Treatise on Electricity and Magnetism. It contained his mathematically based theory of the electromagnetic field. These equations, now known as Maxwell’s equations, include four partial differential equations that provided a basis for the unification of electric and magnetic fields, the electromagnetic description of light, and, ultimately, Albert Einstein’s theory of relativity. Although most people recognize Isaac Newton’s work on mechanics, few remember Maxwell’s electromagnetic theories (including the idea of the electromagnetic wave) when it comes to classical physics. But his theories eventually led to many things we take for granted today, including radio waves and microwaves.
MODERN PHYSICS AND MATHEMATICS
What is modern physics and how is mathematics used?
In contrast to classical physics—although there are overlapping topics—modern physics includes relativistic mechanics, atomic, nuclear and particle physics, and quantum physics. Mathematics is very relevant to all these topics, with physicists using mathematics as their toolbox to find out the physical answers to the universe. For example—and these are only generalizations—the study of quantum mechanics often means a person uses calculus, algebraic techniques, and to some extent, probability and statistics; general relativity uses differential geometry; and quantum field theory uses matrices and group theory. In fact, according to most physicists, you can never know enough mathematics when you study physics because it seems one type of math often leads to another in the natural, physical world.
How did Albert Einstein use mathematics?
German-born American theoretical physicist Albert Einstein (1879–1955) is recognized as one of the greatest physicists of all time, but he was also a notable mathematician. In 1905 he developed the special theory of relativity, mathematically demonstrating that two observers moving at great speeds with respect to one another will experience different time intervals and measure lengths differently, that the speed of light is the “speed limit” for all objects having mass, and that mass and energy are equivalent.
By around 1915, Einstein completed a mathematical formulation of a general theory of relativity, this time adding gravitational effects to determine curvature of a time-space continuum. He further tried to discover a unified field theory, which would combine gravity, electromagnetism, and subatomic phenomena under one set of rules, but he, and no one since, has ever found such a theory. (For more about Albert Einstein, see “History of Mathematics”)
What does Einstein’s famous equation E = mc2 signify?
Among his other accomplishments, Albert Einstein showed mathematically that there was a connection between mass and energy: energy has mass, and mass represents energy. This equation, also called the energy-mass relation, is expressed as E = mc2, in which E is energy, m is mass, and c is the speed of light. Because c is a very large number, even a very small amount of mass represents an enormous amount of energy.
What is relativity?
Relativity refers to the idea that the velocity of an object can be determined only relative to the observer. For example, if a fly moves around the inside of a car at about 1 mile per hour, inside the car’s frame of reference, the fly is moving at 1 mile per hour. But if the car goes past you at 65 miles per hour, it will appear as if the fly is traveling at 66 miles per hour, not 1 mile per hour. In other words, it’s all a matter of reference, and it’s “relative” to your viewpoint.
What new ideas came out of Einstein’s (and others’) study of relativity? He showed that space and time could no longer be viewed as separate, independent entities, forming a four-dimensional continuum called space-time (also written as spacetime). It is not easy to verbally explain the intricacies of Einstein’s theory. The best way to interpret his works is with the use of formulations from certain mathematical branches, such as tensor calculus (for more about tensors, see “Mathematical Analysis”). But such complex equations are beyond the scope of this book.
Did Einstein’s theories change our concept of dimensions?
Yes, our ideas about dimensions changed dramatically thanks to Einstein’s theories, not to mention the mathematics involved in producing those theories. In particular, one can’t distinguish space and time as elements in the description of events. Instead, they are joined in what is called the fourth dimension—also called a four-dimensional manifold known as space-time (see above).
Although it sounds like something out of the television show Star Trek, spacetime events in the universe are described in terms of the four-dimensional continuum. Simply put, each observer locates an event by three spacelike coordinates (positions) and one timelike coordinate. The choice of the timelike coordinate in space-time is not unique; hence, time is not absolute but is relative to the observer. The strange effects go on: In general, events at different locations that are simultaneous for one observer will not be simultaneous for another observer. (For more information about dimensions, see “Geometry and Trigonometry.”)
Why was Max Planck important to quantum theory?
Quantum theory (or physics) entails the emission and absorption of energy by matter and the motion of material particles; it is a special situation in which very small quantities are involved. When added to the theory of relativity—in which great speeds are involved—both form the theoretical basis of modern physics.
One of the most important aspects of quantum theory is the quanta. In 1900 German physicist Max Karl Ernst Ludwig Planck (1858–1947) proposed that all forms of radiation—such as light and heat—come in bundles called quanta. These bundles are further emitted and absorbed in small, discrete amounts, thus behaving in some situations like particles of matter. For example, a bundle of light energy is known as light quanta or photons. Planck devised the equation: E = hv, in which E is the amount of energy in a single particle, v is the frequency of the wave, and h is the constant now known as Planck’s constant.
It is interesting to note that some people divide physics using Planck’s discovery: The term classical physics is often referred to as “before Planck”; while the term for modern physics is often referred to as “after Planck.”
What is quantum mechanics?
Quantum mechanics is a branch of quantum theory that simply determines the probability of an event happening, although the mathematical calculations to prove such things are very rigorous and not simple. In fact, quantum mechanics is often called the “final mathematical formulation of the quantum theory.” Developed during the 1920s, it accounts for matter at the atomic level, and is considered an extension of statistical mechanics, but based on quantum theory.
Physicist Max Planck was the first to propose the idea that energy existed in bundles called “quanta.” His theories later led to the development of quantum mechanics and modern physics.
What is Schrödinger’s equation?
Schrödinger’s equation was, of course, developed by Austrian physicist Erwin Schrödinger (1887–1961) in 1926. Called a wave equation, it could be applied to any physical system. The actual equation has more than one solution, and each solution (also called a quantum state, matter waves, or probability waves) to the equation is a probability wave that describes one of the possible outcomes, or behaviors, of that system. For example, looking at a hydrogen atom, each wave solution to the equations describes one of the allowed electron orbits of the atom.
What does wave mechanics mean in quantum studies?
One important part of quantum mechanics is wave mechanics, which is an extension of quantum mechanics based on Schrödinger’s equation. This idea states that atomic events can be explained as interactions between particle waves.
Austrian physicist Erwin Schrödinger developed the wave equation, the solution of which is a probability wave expressing the possible outcomes of a system.
What is the Pauli’s exclusion principle?
Quantum theory also relies on the Pauli exclusion principle. This principle, developed by Austrian-born Swiss physicist Wolfgang Pauli (1900–1958) in 1925, states that two particles of a certain class—called femions, and otherwise known as electrons, neutrons, and protons—can never be in the same energy state. For example, two electrons with the same quantum number can’t occupy the same atom.
How do quantum physicists regard light waves?
Added to the mix of the mathematically rich quantum theory was an idea developed by French physicist Prince Louis Victor Pierre Raymond de Broglie (1892–1987), who discovered the wave nature of electrons and of particles in general (and also devised a mathematical explanation of the kinetic theory of heat). He determined that not only do light waves often exhibit particle-like properties, but particles also often exhibit wave-like properties.
This opened a can of quantum worms. From there, two different formulations of quantum mechanics developed: first, the wave mechanics of Austrian physicist Erwin Schrödinger (1887–1961), who used a mathematical entity (the wave function) related to the probability of finding a particle in space at a given point. (Schrödinger also developed a model of the atom that differed from the traditional Niels Bohr model); second and mathematically equal to Schrödinger’s theory was the matrix mechanics of German physicist Werner Karl Heisenberg (1901-1976; for more about Heisenberg, see below).
German physicist Werner Karl Heisenberg developed the uncertainty principal, which states that it is impossible to note both the exact position and exact motion of an object at the same time.
What is the Heisenberg uncertainty principle?
German physicist Werner Karl Heisenberg (1901–1976) not only helped with quantum theory of light waves, he also developed the Heisenberg uncertainty principle, an idea used extensively in the modern electronics field. The principle states that it is impossible to determine, at the same time, both the energy and velocity of a particle. Or, in other words, it’s impossible to predict, measure, or know both the exact position of an object and its exact motion at the same time. This principle made a huge dent in Newtonian physics, which relied on a deterministic, “predictable” view of the universe. In simple terms, it showed that the universe does what it wants.
How else is mathematics used in physics?
There are hundreds of other applications of mathematics in physics. The following lists only a few:
Fluid mechanics—This is the study of air, water, and other fluids in motion. It includes the mathematics of turbulence, wave propagation, and so on.
Geophysics—This is a geological study with a physics basis. Much of the field is dominated by the mathematics of large scale movement of materials, such as earthquakes, volcanic activity, and fluid mechanics (for example, underground molten volcanic material).
Optics—This is the mostly mathematical study of the propagation and evolution of electromagnetic waves, such as diffraction and the path of light rays. This requires a great knowledge of geometry and trigonometry, not to mention complex equations.
What are superstrings?
Amazing as it sounds, one of the most fascinating discoveries in our universe started with an instrument’s vibrating strings. These structures are called superstrings and are often compared to a string on a musical instrument, such as a violin. The musical notes made by the string are said to be excitation modes of the string under tension. In string theory, the observed elementary particles in a particle accelerator could be considered the excitation modes of elementary strings. But unlike the violin, super-strings are floating in space-time; they also have tension like the violin without being tied to anything.
Greek mathematician and philosopher Pythagoras of Samos (c. 582-c. 507 B.C.E.), an expert on the lyre, was probably the first “string theorist,” as he figured out the (mathematical) harmonic relationship of the strings. He determined that vibrating lyre strings with equal tension but different lengths would produce harmonious notes (such as a middle C and a high C) if the ratio of the lengths of the two strings was a whole number. Eventually, mathematicians took Pythagoras’s ideas to new heights, more precisely encoding the harmonics with complex mathematics.
And, of course, there are other mathematical calculations and measurements used when talking about superstrings. For example, the tension in string theory is denoted by the quantity /(2 p a'), in which a' is said to be “alpha prime” and equal to the square of the string length scale. There is also the size of the string (equal to the Planck length, or about 10-33 centimeters) and supersymmetry (for every particle that transmits a force, there is a corresponding particle that makes up matter).
CHEMISTRY AND MATH
What is chemistry?
Chemistry is the science of matter. It studies the composition, structure, and properties of substances (matter) and its reactions and changes. Because chemistry includes all materials in the universe, it is useful for studying many things—from the chemical composition of gases in galaxies to the chemical reactions within living cells. It also includes mathematics in many forms, such as when determining chemical compositions and understanding relationships between certain chemicals.
What are the atomic number and mass of an element?
The atomic number is the number of protons in an atomic nucleus. The atomic mass of an atom—usually measured in atomic mass units—is the total mass of the atom, or the combined mass of its protons and neutrons (the mass of the electrons is negligible). The importance of atomic numbers and mass is simple: The atoms of each element has a specific atomic number and mass—each determined by “adding” or “subtracting” protons and neutrons within the atom.
What is an electron?
Chemically speaking, matter is composed of minute particles called atoms; in turn, atoms contain elementary (or subatomic) particles: protons (positively charged particles), neutrons (particles with no charge), and electrons (negatively charged found around the nucleus of the atom). When an atom gains or loses electrons, it acquires a net electric charge.
Three naturally occurring isotopes of oxygen are oxygen-16, oxygen-17, and oxygen-18. They differ by the number of neutrons in their nuclei. Oxygen-16 is the most common form of oxygen in our atmosphere at 99.76% occurrence.
There is a strange part of an electron’s definition, especially in the field of physics: is an electron a particle or a wave? In reality, everything is a wave or particle; and the more mass an object has, the smaller the waves. Thus, electrons, which have less mass, have larger waves. In quantum mechanics, electron waves are actually thought of as probability waves (and so are matter waves in general). But again, there are some different characteristics of these waves. In particular, the wave does not indicate where the electron is found, only where it might be found—or the probability of the electron being in that place.
What is an ion?
Scientists know that atoms can gain or lose electrons, thus acquiring a negative or positive electrical charge (determined by the number of protons minus the number of electrons). For example, if there are 4 protons and 6 electrons, the net charge is -2 (often called the valence). An ion is an atom—or group of atoms—that takes on the net electric charge, and can be positive (cation) or negative (anion). Based on the “mathematics” of losing or gaining electrons, ions can be formed or destroyed. This is one example of where math comes in handy in chemistry.
What is an angstrom?
An angstrom (Å) is a unit of measurement often used in chemistry, most often in reference to molecules; for example, the average molecule diameter is between 0.5 to 2.5 angstroms. An angstrom is equal to about 3.937 × 10-9 inch or one hundredth of a millionth of a centimeter (10-8 centimeter). (For more information about measurement, see “Mathematics throughout History.”)
What is Avogadro’s number?
Avogadro’s number (also called Avogadro’s constant or Avogadro’s figure) was determined by Italian physicist Lorenzo Romano Amedeo Carlo Avogadro, Count of Quarengna and Cerreto (1776–1856), who was also the first one to use the term “molecule” in chemistry. It represents the number of elementary entities, such as atoms, molecules, or formula units, in a mole of any chemical substance (a mole is approximately 6.02214199 × 1023 atoms, according to the most recent number from the National Institute of Standards and Technology). To translate even further, a mole is the molecular weight of a substance in grams; one mole is the amount of a substance that contains Avogadro’s number. For example, the number of carbon atoms in 12 grams of the substance carbon-12 is equal to one mole.
What is density?
Density (usually abbreviated as d or r) is a mathematical concept used to describe the ratio between the mass of an object and its volume. The actual formula is the density times the volume is equal to an object’s mass, or d × v = m.In the standard (American) measurement system, density is measured in pounds per cubic foot. But in the sciences, the metric system is usually used and density is measured in grams per cubic centimeter (or grams per milliliter). For example, the density of water is 1 gram per cubic centimeter, lead is 11.3 grams per cubic centimeter, and gold is 19.32 grams per cubic centimeter. (Note: In the majority of cases, the higher the density, the “heavier” it feels to us on Earth.)
What are formulas and equations in chemistry?
Formulas and equations in chemistry don’t always mean the same as in mathematics. Formulas in chemistry are representations of a chemical compound using symbols for the elements and subscripts for the number of atoms present. For example, the chemical formula for water is H2O, in which there are two atoms of hydrogen (H) bonded to an atom of oxygen (O). The subscript 2 indicates that there are two atoms of hydrogen in the molecule; if there is no subscript number, as with the oxygen (O), a subscript of 1 is implied. (Remember, not all compounds are molecular; for example, NaCl, or sodium chloride [regular table salt], is called an ionic compound. In these cases, the formula shows the proportion of the atoms of each element making up the compound.) There are other types of formulas in chemistry, but this is the most familiar.
Equations in chemistry also differ from mathematics. Chemical equations represent the reaction relationship between two or more chemical compounds—along with the products of the chemical reaction. For example, the chemical equation 2H2 + O2 → 2H2O is the reaction of hydrogen with oxygen to form water. The arrow indicates the direction of the reaction toward the product; the reactants (or the substances that react) are hydrogen and oxygen. There is also a methodology in writing chemical equations. Simply put, first there is the process to determine the reactants and outcome; next, determine the formula for each substance; and finally, balance the equation.
What is the pH scale?
The “pH” scale stands for p(otential of) H(ydrogen) scale, or the logarithm of the reciprocal of hydrogen-ion concentration in gram atoms per liter. In simpler terms, the pH is merely the measure of the hydrogen ion concentration of a solution. The pH numbers are based on a scale from 0 to 14, in which numbers less than 7 represent acidic solutions and numbers greater than 7 represent alkaline (base) solutions. A reading of 7 is considered neutral.
Mathematically speaking, once the concentration of hydrogen ions is determined chemically (based on moles per liter), the pH value is established by taking the exponent used in expressing this concentration and reversing its sign. It is most often expressed as the notation pH = -log 10[H+]. For example, if the hydrogen ion concentration of a solution is determined to be 10-4 (or 0.0001) moles per liter, the pH is 4.
A pH scale indicating pH levels for common liquids and other substances.
Many people are familiar with the pH scale from high school, especially the practice of using special whitish paper called litmus paper to check for pH. The paper contains a powder extracted from certain plants, allowing the user to determine acidity (the paper turns red), neutrality (the paper stays white), or alkalinity (the paper turns blue) of various solutions. The stronger the acid or base, the more intense the red or blue, respectively. And pH isn’t just for use in chemistry class. For example, it is also important to people who work the soil. All plants need a certain soil pH to grow and flourish, which is why most gardeners and farmers determine the acidity or alkalinity of their soil in order to grow better crops.
What is radioactive decay?
Mathematics can also be applied to radioactive substances found within certain rocks. Radioactive decay is the disintegration of a radioactive substance and the emission of certain ionizing radiation (such as alpha or beta particles—or even gamma rays). Simply put, when rocks form, the minerals within the rock often contain certain radioactive atoms that decay at a specific rate.
Radioactive decay is especially important in radioactive dating, in which the original and decayed radioactive elements are used to determine the age of the rock. This is because certain radioactive elements will decay to a mixture of half the original element and half another element (or isotope) in a specific timeframe; this is also called the half-life of the original element. For example, “half” of the Uranium-238 in a rock will decay into Lead-207 in 704 million years (thus, the half-life of Uranium-238 is said to be 704 million years). Statistically, this change follows a specific decay function for each isotope of an element. And in each of these exponential functions, the time for the function’s value to decrease to half is constant, making radioactive dating perfect in determining the age of certain rocks.
What is the Universal Gas Law?
The Universal Gas Law (also called the Universal Gas Constant or the Perfect Gas Law), is a chemical law that can also be looked at mathematically. It is represented by the equation PV = nRT, in which P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is the temperature in Kelvin. (For more about the Kelvin temperature scale, see “Mathematics throughout History.”)
What are calories?
To most people, calories are usually associated with a very large piece of chocolate cake. But in that case, they are called nutritionist’s calories, or the unit of energy-producing potential equal to the amount of heat that is contained in food and released upon oxidation by the body. The body needs the calories in the foods we eat to use as energy. This is why nutrition and weight-control texts often contain such entries as “a 140 pound person walking for one hour at a moderate pace burns off 222 calories.”
In chemistry, a calorie also refers to a unit of energy. But in terms of chemical experiments, a calorie is the amount of heat required to raise the temperature of 1 gram of water by 1 degree Celsius from a standard initial temperature at a pressure of 1 atmosphere (sea level). The unit measurement for energy is a joule, in which 1 calorie equals 4.184 joules; 1 joule is translated (most often in metric) as the energy needed to lift 2,000 grams a distance of 10 centimeters.
What is the Boltzmann equation?
In the late 1860s, British physicist James Clerk Maxwell (1831–1879) and Austrian theoretical physicist Ludwig Edward Boltzmann (1844–1906) developed the Boltzmann equation to predict how a gaseous material distributes itself in space, along with how it responds to various environmental factors, such as temperatures, pressure, or velocity—all combined into what is called the kinetic theory of gases.
For almost 140 years, the equation remained unsolved, but in 2010 researchers found solutions to the 7-dimesional equation using modern mathematical techniques. Overall, they used partial differential equations and harmonic analysis to prove that the equation does have solutions. This not only helps scientists understand the probability of where molecules exist, but also explains how molecules just glance off one another rather than collide head on, apparently a dominant way collisions work for the full equation.
What other contributions did Ludwig Boltzmann and James Maxwell make to science?
Boltzmann and Maxwell’s contributions to science didn’t stop at the Boltzmann equation. Boltzmann is considered the founder of statistical mechanics. He also developed a statistical interpretation of the second law of thermodynamics.
Maxwell contributed much more to the sciences. For example, he is well known for writing four equations that summarize the relationship between electricity and magnetism, collectively called Maxwell’s equations. He also showed mathematically that an oscillating electric and magnetic field would propagate through space at the speed of light—meaning that light was a form of electromagnetic radiation.
ASTRONOMY AND MATH
What are astronomy and astrophysics?
Astronomy is the study of matter in outer space. It is usually considered a branch of physics. But because it encompasses (literally) an astronomical number of subjects— everything from the study of a star’s surface to the end of the universe—it is often considered a field of its own. Astrophysics is a branch of astronomy dealing with the physics of celestial bodies and the universe as a whole. It deals with problems that range from the structure, distribution, evolution, and interaction of stars and galaxies to the orbital mechanics of a near-Earth asteroid.
Who first calculated the distance from the Earth to the Sun and Moon?
Around 290 B.C.E., astronomer and mathematician Aristarchus of Samos (c. 310 B.C.E.-c. 230 B.C.E.) used geometric methods to calculate the distances to and sizes of the Moon and Sun. Based on his observations and calculations, he suggested that the Sun was about 20 times as distant from the Earth as the Moon (it is actually 390 times); he also determined that the Moon’s radius was 0.5 times the radius of the Earth (it is actually 0.28 times). The numbers differ not because Aristarchus had no geometric knowledge, but because of the poor instruments used at that time.
These calculations were not the only contribution made by Aristarchus. He was also the first to propose that the Earth orbits the Sun—many centuries before Nicolaus Copernicus (see below). This concept was radical for his time, because it conflicted with geocentric religious beliefs and Aristotle’s principle that all objects move toward the center of the Earth.
Who was Hipparchus?
Hipparchus of Rhodes (also seen as Hipparchus of Nicaea, as he was born there; c. 190-c. 120 B.C.E.) was one of the greatest Greek astronomers. A partial list of his discoveries includes: being the first to discover the precession of the equinoxes, compiling an extensive star catalogue, assigning “magnitudes” as a measure of stellar brightness, and calculating the length of the year to within 6.5 minutes of the correct value. His planetary models were mathematical, not mechanical. And although Hipparchus did not invent it, he was probably the first person to systematically use trigonometry, which was a necessity for most of his discoveries.
The famous astronomer Nicolaus Copernicus (pictured here) is commonly thought of as the first person to propose a heliocentric (Sun-centered) model of the solar system, but actually Aristarchus surmised the truth centuries before him.
What was De revolutionibus orbium coelestium?
In the year of his death, astronomer Nicolaus Copernicus (1473-1543; in Polish, Mikolaj Kopérnik) published De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres). This manuscript gave a full account of his theory that the Sun, and not the Earth, was at the center of the solar system (or universe). Although this theory was not new, Copernicus offered the idea in all its mathematical detail. This heliocentric (versus geocentric) view of the heavens, now known as the Copernican system, is the foundation of modern astronomy.
What astronomical event with mathematical significance occurred on December 25, 1758?
On December 25, 1758, the appearance of a comet we now call “Halley’s Comet” (or Comet Halley) proved a famous astronomer’s predictions (unfortunately, it was sixteen years after his death). From around 1695, Edmond Halley (1656-1742; also seen as Edmund Halley) carefully studied comets, especially those with parabolic orbits. But he also believed that some comets had elliptical orbits, and he thus theorized that the comet of 1682 (now Comet Halley) was the same comet that appeared in 1305, 1380, 1456, 1531, and 1607. In 1705 he predicted that the comet would appear again 76 years later—in 1758—a prediction that came true.
Such a calculation was a great feat in those days, with Halley even taking into account the comet’s orbital perturbations produced by the planet Jupiter. Even today, the comet maintains its 76-year cycle. Its last appearance was in 1986; it will again appear in the year 2062.
What are Kepler’s Laws of Planetary Motion?
A great deal of mathematics went into the formulation of Kepler’s Laws of Planetary Motion. These laws were devised by German astronomer and mathematician (and Danish astronomer Tycho Brahe’s [1546-1601] assistant) Johannes Kepler (1571–1630). He presented the first and second laws in his work Astronomia nova (New Astronomy) in 1609; the third law was published in 1619 in Harmonice mundi. The three laws are as follows:
Kepler’s first law (or law of elliptic orbits)—Each planet moves about the Sun in an orbit that is an ellipse, with the Sun at one of the two foci of the ellipse.
Kepler’s second law (or the law of areas)—An imaginary straight line joining a planet to the Sun will sweep out equal areas of the ellipse in equal periods of time.
Kepler’s third law (or the harmonic law)—The square of the period of a planet’s revolution is directly proportionate to the cube of the semi-major axis of its orbit.
Halley’s Comet, named after Edmond Halley, last appeared near Earth in 1986 and will be seen again in the night sky in 2062.
How did Pierre-Simon de Laplace apply mathematics to astronomy?
French mathematician, astronomer, and physicist Marquis Pierre-Simon de Laplace (1749–1827) was one of the first to work out the gravitational mechanics of the solar system using mathematics. In his Mécanique Céleste (Celestial Mechanics), Laplace translated the geometrical study of mechanics used by Isaac Newton to one based on calculus (or physical mechanics). He also proved the stability of the solar system, but only on a short time scale. Laplace is also known for his theory about the formation of the planets. He believed they originated from the same primitive mass of material, a theory now known as Laplace’s nebular hypothesis. His other studies included major contributions to differential equations and to the theory of probability.
What are astronomical units and light years?
An astronomical unit is one of the more common measurements used in astronomy. It is a distance equal to the average distance from the Earth to the Sun, or 92,960,116 miles (149,597,870 kilometers); it is often seen rounded off to 93,000,000 miles (149,598,770 kilometers) and used in reference to great astronomical distances. For example, the Earth is 1 AU from the Sun; the planet Venus is 0.7 AU; Mars is 1.5 AU; and Saturn is 9.5 AU from the Sun.
A light year is an even larger unit. As the name implies, it is the distance light travels in one year, or about 5.88 trillion miles (9.46 trillion kilometers). In most cases, light year measurement is reserved for deep space objects. (For more about measurement, see “Mathematics throughout History.”)
What is the Hubble constant?
Astronomers have always been interested in the age of our universe and the speed of various objects in space. The Hubble constant was devised by American astronomer Edwin Hubble (1889–1953). It is the ratio of the recessional speed of a galaxy—because the universe is expanding—to its distance from the observer. In other words, the velocity at which a typical galaxy is receding from Earth, divided by its distance from Earth.
How long does it take for the Sun’s light to reach the Earth?
Because the Sun is an average of 93,000,000 miles (149,598,770 kilometers) from the Earth, and the speed of light is approximately 186,000 miles per second, it is easy to determine the approximate time (t) it takes for the Sun’s light to reach the Earth using mathematics:
t = 93,000,000 miles / 186,000 miles per second
= 500 seconds (miles cancel each other out)
= 8.3 minutes
The reciprocal of the Hubble constant is then thought to be the age of the universe, usually written in terms of kilometers per second per million light years. If the number is high, the universe would be very young; if the number is low, the universe would be much more ancient. Although there have been numerous theories, the true age of the universe is usually considered to be somewhere between 12 and 20 billion years old.
The most recent agreed-upon rate at which the universe is expanding is approximately 20 kilometers per second per 106 light years of distance. That makes the universe about 15 billion years old.
What is the Titius-Bode Law?
The Titius-Bode Law was developed by German astronomer Johann Daniel Titius (1729–1796); Titius’s idea was brought to the forefront by German astronomer Johann Elert Bode (1747–1826). The law actually represents a simple mathematical rule that allows one to determine the distances (also called the semi-major axis) of the planets in astronomical units. It is determined using the equation a = 0.4 + (0.3)2n, in which n is an integer and a is the astronomical unit. Interestingly enough, most of the planets—and even the asteroids in the Asteroid Belt—adhere to the law. The only exception is Neptune, the second-to-last planet in our solar system.
Distances of the Planets from the Sun in Astronomical Units
How does geometry work in terms of eclipses of the Sun and Moon?
Eclipses of the Sun (called a solar eclipse, when the Moon is between the Earth and the Sun) and the Moon (called a lunar eclipse, when the Earth is between the Moon and the Sun) are all a matter of angles. For example, in a solar eclipse, the shadow of the Moon follows a narrow path on the Earth, creating a shadow only in certain regions—not the entire planet (the Moon isn’t large enough to create a shadow that would cover the Earth). Depending on the angle and distance of the Moon from the Earth in its orbit, a solar eclipse can be an annular (the apparent size of the Moon is insufficient to cover the Sun completely, creating a ring of sunlight around the Moon); partial (the Moon only covers part of the Sun as seen from the Earth); or total (the Moon blocks out the entire face of the Sun in the eclipse path).
Neptune is the only planet in our solar system that does not fit the model set by the Titius-Bode Law.
What type of geometric path does the Sun seem to take in the Earth’s sky?
The Sun’s apparent path in the Earth’s sky is called an analemma, or the measurement of the Sun’s declination over the year. In fact, if one recorded the position of the Sun in the sky at the same time every day, the Sun would appear to follow a figure-eight path, the analemma—a feature often seen on globes of the Earth. Astronomers also use another way to note the path of the Sun: the difference in time between the clock and the position of the Sun (clock versus Sun times) is called the equation of time. For those living in the Northern Hemisphere, if one notes the Sun’s position is to the east, the equation of time is negative; if the sun is to the west, the equation of time is positive.
The illustration at the top shows what happens during a lunar eclipse, when the Moon passes through the shadow of the Earth (M-1 to M-2). Below that is what happens during a solar eclipse, when the Moon comes between the Sun and the Earth, casting a shadow on the Earth’s surface. A and B points show where a total eclipse would be, and the region between the Cs would experience a partial eclipse.
Why does the Sun take such a different path across the sky each year? There are two main reasons: first the Earth’s rotational axis (the Earth’s rotation creates the day and night) is tilted 23.5 degrees in relation to the plane of its orbit around the Sun. Second, the Earth does not orbit the Sun in a circle, but orbits in an ellipse (oval-shape). Because of the tilt and the orbit, over a year the Sun’s path seems to move in a figure-eight. (It’s interesting to note here that because of the tilt and orbit of the other planets in the solar system, they, too, have analemmas all their own, which are different from the Earth’s.)
What is the Hertzsprung-Russell diagram?
The Hertzsprung-Russell diagram is a “two-dimensional” graph of the mathematical relationship between the absolute magnitude, luminosity, stellar classification, and surface temperature of stars—all resulting in a diagram of the stellar life cycle. It was plotted by Danish astronomer Ejnar Hertzsprung (1873–1967) in 1911 and independently by American astronomer Henry Norris Russell (1877–1957) in 1913.
How do astronomers determine the distances to other planets?
Astronomers need mathematics, of course, to determine the distances to the planets and satellites of our solar system. One of the first astronomers to work this out was Nicolaus Copernicus (see above) using simple observations of planetary positions.
If you carefully chart the position of the Sun at the same time of day over the course of a year, at the mid-latitudes you will see that it appears to follow a figure eight pattern in the sky.
One of the earliest methods to determine such distances was to use the orbital period of a planet. This varies as the square root of the cube of the distance from the Sun: T = k × r(3/2), in which T is the time for one revolution, r is the distance between the centers of the Sun and the planet, and k is a constant.
How do astronomers determine the distances to the stars?
Relatively nearby objects beyond the solar system appear to shift position relative to more distant objects as the Earth moves from one side of the Sun to the other—a phenomenon called parallax. You can use parallax to determine the distance to stars, as long as these stellar objects are within a few dozen light years of Earth. (More distant objects in the sky do not change their position enough as the Earth orbits from one side of the Sun to the other.) First, measure the position of the star in the sky; then, measure it again in six months when the Earth is on the opposite side of its orbit. If the distance a and the angle ac are known (as seen in the diagram below), using trigonometry, c can be determined as a / cos (ac).
Is it easy to figure out the size of a distant object?
Yes, it is often possible to figure out the size of distant objects, as long as they aren’t too small. The key is in knowing the distance to the object. For example, if someone holds a nickel at arm’s length and then has someone hold the nickel 200 yards away, the coin may appear to be smaller, but its size really hasn’t changed. If a person knows how large an object appears to be, and how distant it is, they can work backward to determine the true size of the object. Simply put, this is also how astronomers work out the size of distant objects in outer space.
Using parallax to determine the distance to a star is done by calculating the apparent shift of a star as the Earth orbits the Sun.
What “mathematical measurement” error once occurred when a spacecraft reached Mars?
The Mars Climate Orbiter spacecraft, a joint effort between Lockheed Martin and NASA’s Jet Propulsion Laboratory (JPL), was supposed to go into orbit around the red planet on September 23, 1999. Instead, the Martian craft lost all contact with Earth. After much deliberation, a review panel for the incident came to a disconcerting conclusion: A thruster error developed when project teams used different measuring systems for the navigation commands—NASA used metric units; Lockheed Martin used English standard units, and no one caught the discrepancy (for more about metric and standard units, see “Mathematics throughout History”).
To this day, scientists can only speculate as to what happened to the orbiter, a craft sent to study the climate and weather patterns of the Martian atmosphere. Some say that because the orbiter dropped down to within 36 miles (60 kilometers) of the planet—about 62 miles (100 kilometers) closer than planned—atmospheric friction probably overheated the propulsion system and tore the vehicle apart. Others believe the craft was propelled through the atmosphere (or bounced off) and out into space again and is now perhaps circling the Sun like an artificial asteroid or comet.
How was math used to discover extrasolar planets?
Astronomers have always dreamed about detecting other planets outside our solar system—extrasolar planets. In 1994, Polish astronomer Alekzander Wolszczan (1946-) announced the discovery of the first extrasolar planet—actually, two planets with masses 3.4 and 2.8 times that of Earth’s mass—orbiting the pulsar PSR B1257+12. A pulsar star sends out a periodic pulse of light detected from Earth; Wolszczan found the planets by measuring the periodic variation in the pulse arrival time.
There are several major methods used to search for extrasolar planets, and all entail using mathematics. For example, the Doppler shift method measures the change in wavelength (color) of light coming from a star over the course of days, months, and years. The change in wavelength—or the Doppler shift of the light—is caused by the star orbiting a common center of mass with a companion planet. An example in our own solar system is the gas giant Jupiter. Its massive gravitational pull causes the Sun to wobble around a circle with a velocity of 39.4 feet (12 meters) per second.
Another detection method is called astrometry, which measures the periodic wobble that a planet causes in the position of its parent star. In this case, the minimum detectable planet mass gets smaller in inverse proportion to the planet’s distance from the star.
These methods work, and as of 2011 more than 700 such planets have been discovered. If you want to see an updated list of the extrasolar planets, try http://exoplanet.eu/catalog.php maintained by astronomers at the Paris Observatory in France.