A number of important technologies developed rapidly in the nineteenth and twentieth centuries, including electricity, electronics, and computers, but also biotechnology, nanotechnologies, and technologies derived from them. In fact, it is the exponential rate of development of these technologies, more than anything else, that has resulted in the scientific and economic developments of recent decades. In those technologies and in others, that rate is likely to increase.
Four Equations That Changed the World
Phenomena related to electricity—for example, magnetism, lightning, and electrical discharges in fish—remained scientific curiosities until the seventeenth century. Although other scientists had studied electrical phenomena before him, the first extensive and systematic research on electricity was done by Benjamin Franklin in the eighteenth century. In one experiment, reputedly conducted in 1752, Franklin used a kite to capture the electrical energy from a storm.
After the invention of the battery (by Alessandro Volta, in 1800), it was practicable to conduct a variety of experiments with electrical currents. Hans Christian Ørsted and André-Marie Ampere discovered various aspects of the relationship between electric currents and magnetic fields, but it was Michael Faraday’s results that made it possible to understand and harness electricity as a useful technology. Faraday performed extensive research on the magnetic fields that appear around an electrical current and established the basis for the concept of the electromagnetic field. He became familiar with basic elements of electrical circuits, such as resistors, capacitors, and inductors, and investigated how electrical circuits work. He also invented the first electrical motors and generators, and it was largely through his efforts that electrical technology came into practical use in the nineteenth century. Faraday envisioned some sort of field (he called it an electrotonic state) surrounding electrical and magnetic devices and imagined that electromagnetic phenomena are caused by changes in it.
However, Faraday had little formal mathematical training. It fell to James Clerk Maxwell to formulate the equations that control the behavior of electromagnetic fields—equations that now bear his name. In 1864 and 1865, Maxwell (who was aware of the results Faraday had obtained) published his results suggesting that electric and magnetic fields were the bases of electricity and magnetism and that they could move through space in waves. Furthermore, he suggested that light itself is an electromagnetic wave, since it propagates at the same speed as electromagnetic fields. The mathematical formulation of his results resulted in a set of equations that are among the most famous and influential in the history of science (Maxwell 1865, 1873).
In its original form, presented in A Treatise in Electricity and Magnetism, Maxwell’s formulation involved twenty different equations. (Maxwell lacked the tools of modern mathematics, which enable us to describe complex mathematical operations in simple form.) At the time, only a few physicists and engineers understood the full meaning of Maxwell’s equations. They were met with great skepticism by many members of the scientific community, among them William Thomson (later known as Lord Kelvin). A number of physicists became heavily involved in understanding and developing Maxwell’s work. The historian Bruce Hunt dubbed them the Maxwellians in a book of the same name (Hunt 1991).
One person who soon understood the potential importance of Maxwell’s work was Oliver Heaviside. Heaviside dedicated a significant part of his life to the task of reformulating Maxwell’s equations and eventually arrived at the four equations now familiar to physicists and engineers. You can see them on T shirts and on the bumpers of cars, even though most people don’t recognize them or have forgotten what they mean. On Telegraph Avenue in Berkeley, at the Massachusetts Institute of Technology, and in many other places one can buy a T shirt bearing the image shown in figure 3.1. Such shirts somehow echo Ludwig Boltzmann’s question “War es ein Gott der diese Zeichen schrieb?” (“Was it a god who wrote these signs?”), referring to Maxwell’s equations while quoting Goethe’s Faust.
Figure 3.1 The origin of the world according to Oliver Heaviside’s reformulation of Maxwell’s equations and a popular T shirt.
Hidden in the elegant, and somewhat impenetrable, mathematical formalism of Maxwell’s equations are the laws that control the behavior of the electromagnetic fields that are present in almost every electric or electronic device we use today. Maxwell’s equations describe how the electric field (E) and the magnetic field (B) interact, and how they are related to other physical entities, including charge density (ρ) and current density (J). The parameters ε0 and μ0 are physical constants called the permittivity and permeability of free space, respectively, and they are related by the equation
,
where c is the speed of light.
A field is a mathematical construction that has a specific value for each point in space. This value may be a number, in the case of a scalar field, or a vector, with a direction and intensity, in the case of a vector field. The electric and magnetic fields and the gravitational field are vector fields. At each point in space, they are defined by their direction and their amplitude. One reason Maxwell’s work was difficult for his contemporaries to understand was that it is was hard for them to visualize or understand fields, and even harder to understand how field waves could propagate in empty space.
The symbol ∇ represents a mathematical operator that has two different meanings. When applied to a field with the intervening operator ⋅ , it represents the divergence of the field. The divergence of a field represents the volume density of the outward flux of a vector field from an arbitrarily small volume centered on a certain point. When there is positive divergence in a point, an outward flow of field is created there. The divergence of the gravitational field is mass; the divergence of the electrical field is charge. Charges can be positive or negative, but there is no negative mass (or none was ever found). Positive charges at one point create an outgoing field; negative charges create an incoming field.
When ∇ is applied to a field with the operator ×, it represents the curl of the field, which corresponds to an indication of the way the field curls at a specific point. If you imagine a very small ball inside a field that represents the movement of a fluid, such as water, the curl of a field can be visualized as the vector that characterizes the rotating movement of the small ball, because the fluid passes by at slight different speeds on the different sides. The curl is represented by a vector aligned with the axis of rotation of the small ball and has a length proportional to the rotation speed.
The first of Maxwell’s equations is equivalent to the statement that the total amount of electric field leaving a volume equals the total sum of charge inside the volume. This is the mathematical formulation of the fact that electric fields are created by charged particles.
The second equation states that the total amount of magnetic field leaving a volume must equal the magnetic field entering the volume. In this respect, the electric field and the magnetic field differ because, whereas there are electrically charged particles that create electric fields, there are no magnetic monopoles, which would create magnetic fields. Magnetic fields are, therefore, always closed loops, because the amount of field entering a volume must equal the amount of field leaving it.
The third equation states that the difference in electrical potential accumulated around a closed loop, which translates into a voltage difference, is equal to the change in time of the magnetic flux through the area enclosed by the loop.
The fourth equation states that electric currents and changes in the electric field flowing through an area are proportional to the magnetic field that circulates around the area.
The understanding of the relationship between electric fields and magnetic fields led, in time, to the development of electricity as the most useful technology of the twentieth century. Heinrich Hertz, in 1888, at what is now Karlsruhe University, was the first to demonstrate experimentally the existence of the electromagnetic waves Maxwell had predicted, and that they could be used to transmit information over a distance.
We now depend on electrical motors and generators to power appliances, to produce the consumer goods we buy, and to process and preserve most of the foods we eat. Televisions, telephones, and computers depend on electric and magnetic fields to send, receive, and store information, and many medical imaging technologies are based on aspects of Maxwell’s equations.
Electrical engineers and physicists use Maxwell’s equations every day, although sometimes in different or simplified forms. It is interesting to understand, in a simple case, how Maxwell’s equations define the behavior of electrical circuits. Let us consider the third equation, which states that the total sum of voltages measured around a closed circuit is zero when there is no change in the magnetic field crossing the area surrounded by the circuit. In the electrical circuit illustrated in figure 3.2, which includes one battery as a voltage source, one capacitor, and one resistor, the application of the third equation leads directly to
VR + VK – VC = 0.
Figure 3.2 An electrical circuit consisting of a capacitor, a resistor, and a voltage source.
This expression results from computing the sum of the voltage drops around the circuit, in a clockwise circulation. Voltage drop VC is added with a negative sign because it is defined against the direction of the circulation; voltages VK (in the battery) and VR (in the resistor) are defined in the direction of the circulation. It turns out that the current through the capacitor, IC, is the same (and in the opposite direction) as the current through the resistor, IR, because in this circuit all current flows through the wires.
Maxwell’s fourth equation implies that the current through the capacitor is proportional to the variation in time of the electric field between the capacitor plates, leading to the expression , since the electric field (, in this case) is proportional to the voltage difference between the capacitor plates, VC. In this equation, the dot above VC represents the variation in time of this quantity. Mathematically, it is called the derivative with respect to time, also represented as dVC//dt.
The linear relation between the current IR and the voltage VR across a resistor was described in 1827 by another German physicist, Georg Ohm. Ohm’s Law states that there is a linear relation VR = RIR between the two variables, given by the value of the resistance R. Putting together these expressions, we obtain
.
This expression, which is a direct result of the application of Maxwell’s equations and Ohm’s Law, is a differential equation with a single unknown, VC (the voltage across the capacitor, which varies with time). All other parameters in this expression are known, fixed physical quantities. Differential equations of this type relate variables and their variations with respect to time. This particular differential equation can be solved, either by analytical or numerical methods, to yield the value of the voltage across the capacitor as it changes with time. Similar analyses can be used to derive the behaviors of much more complex circuits with thousands or even millions of electrical elements.
We know now that neurons in the brain use the very same electromagnetic fields described by Maxwell’s equations to perform their magic. (This will be discussed in chapter 8.) Nowadays, very fast computers are used to perform brain simulation and electrical-circuit simulation by solving Maxwell’s equations.
The Century of Physics
The twentieth century has been called the century of physics. Although knowledge of electromagnetism developed rapidly in the nineteenth century, and significant insights about gravity were in place before 1900, physics was the driving force of the major advances in technology that shaped the twentieth century. A comprehensive view of the relationship between matter and energy was first developed in that century, when the strong and weak nuclear forces joined the electromagnetic and gravitational forces to constitute what we see now as the complete set of four interactions that govern the universe.
Albert Einstein’s “annus mirabilis papers” of 1905 started the century on a positive note. Einstein’s seminal contributions on special relativity, matter, and energy equivalence (1905c), on the photoelectric effect (1905b), and on Brownian motion (1905a) changed physics. Those publications, together with our ever-growing understanding of electromagnetism, started a series of developments that led to today’s computer and communications technologies.
The 1920s brought quantum mechanics, a counter-intuitive theory of light and matter that resulted from the work of many physicists. One of the first contributions came from Louis de Broglie, who asserted that particles can behave as waves and that electromagnetic waves sometimes behave like particles. Other essential contributions were made by Erwin Schrödinger (who established, for the first time, the probabilistic base for quantum mechanics) and by Werner Heisenberg (who established the impossibility of precisely and simultaneously measuring the position and the momentum of a particle). The philosophical questions raised by these revolutionary theories remain open today, even though quantum mechanics has proved to be one of the most solid and one of the most precisely tested physical theories of all time.
After World War II, Julian Schwinger, Richard Feynman, and Sin-Itiro Tomonaga independently proposed techniques that solved numerical difficulties with existing quantum theories, opening the way for the establishment of a robust theory of quantum electrodynamics. Feynman was also a talented storyteller and one of the most influential popular science writers of all time. In making complicated things easy to understand, few books match his QED: The Strange Theory of Light and Matter (1985).
High-energy physics led to a range of new discoveries, and a whole zoo of sub-atomic particles enriched the universe of particle physics. The idea that fundamental forces are mediated by particles (photons for the electromagnetic force, mesons for the nuclear forces) was verified experimentally. A number of other exotic particles, including positrons (the antimatter version of the electron), anti-protons, anti-neutrons, pions, kaons, muons, taus, neutrinos, and many others—joined the well-known electrons, neutrons, protons, and photons as constituents of matter and energy. At first these particles were only postulated or were found primarily by the ionized trails left by cosmic rays, but with particle accelerators such as those at CERN and Fermilab they were increasingly produced. And even more exotic particles, including the weakly interacting massive particle (WIMP), the W particle, the Z0 particle, and the elusive Higgs boson, joined the party, and will keep theoretical physicists busy for many years to come in their quest for a unifying theory of physics.
A number of more or less exotic theories were proposed to try to unify gravity with the other three forces (strong interaction, electromagnetic interaction, and weak interaction), using as few free parameters as is possible, in a great unifying theory. These theories attempt to replace the standard model, which unifies the electromagnetic, strong, and weak interactions and which comprises quantum electrodynamics and quantum chromodynamics. Among the most popular are string theories, according to which elementary particles are viewed as oscillating strings in some higher-dimensional space. For example, the popular M-theory (Witten 1995) requires space-time to have eleven dimensions—hardly a parsimonious solution. So far, no theory has been very successful at predicting observed phenomena without careful tuning of parameters after the fact.
At the time of this writing, the Large Hadron Collider at CERN, in Switzerland, represents the latest effort to understand the world of high-energy physics. In 2012, scientists working at CERN were able to detect the elusive Higgs boson, the only particle predicted by the standard model that had never been observed until then. The fact that such a particle had never been observed before is made even more curious by the fact that it should be responsible for the different characteristics of photons (which mediate electromagnetic force, and are massless) and the massive W and Z bosons (which mediate the weak force).
The existing physical theories are highly non-intuitive, even for experts. One of the basic tenets of quantum physics is that the actual outcome of an observation cannot be predicted, although its probability can be computed. This simple fact leads to a number of highly counter-intuitive results, such as the well-known paradox of Schrödinger’s cat (Schrödinger 1935) and the Einstein-Podolsky-Rosen paradox (Einstein, Podolsky, and Rosen 1935).
Schrödinger proposed a thought experiment in which the life of a cat that has been placed in a sealed box depends on the state of a radioactive atom that controls, through some mechanism, the release of a poisonous substance. Schrödinger proposed that, until an observation by an external observer took place, the cat would be simultaneously alive and dead, in a quantum superposition of macroscopic states linked to a random subatomic event with a probability of occurrence that depends on the laws of quantum mechanics.
Schrödinger’s thought experiment, conceived to illustrate the paradoxes that result from the standard interpretation of quantum theory, was, in part, a response to the Einstein-Podolsky-Rosen (EPR) paradox, in which the authors imagine two particles that are entangled in their quantum states, creating a situation such that measuring a characteristic of one particle instantaneously makes a related characteristic take a specific value for the other particle. The conceptual problem arises because pairs of entangled particles are created in many physical experiments, and the particles can travel far from each other after being created. Measuring the state of one of the particles forces the state of the other to take a specific value, even if the other particle is light-years away. For example, if a particle with zero spin, such as a photon, decays into an electron and a positron (as happens in PET imaging, which we will discuss in chapter 9), each of the products of the decay must have a spin, since the spins must be opposite on account of the conservation of spin. But only when one of the particles is measured can its spin be known. At that exact moment, the spin of the other particle will take the opposite value, no matter how far apart they are in space. Such “spooky” instant action at a distance was deemed impossible, according to the theory of relativity, because it could be used to transmit information at a speed exceeding that of light. Posterior interpretations have shown that the entanglement mechanism cannot in fact be used to transmit information, but the “spooky action-at-a-distance” mechanism (Einstein’s expression) remains as obscure as ever.
The discussions and excitement that accompanied the aforementioned paradoxes and other paradoxes created by quantum mechanics would probably have gone mostly unnoticed by the general public had it not been for their effects on the real world. Many things we use in our daily lives would not be very different if physics had not developed the way it did. We are still raising cattle and cultivating crops much as our forebears did, and we go around in cars, trains, and planes that, to a first approximation, have resulted from the first industrial revolutions and could have been developed without the advantages of modern physics.
There are, however, two notable exceptions. The first was noticed by the whole world on August 6, 1945, when an atomic bomb with a power equivalent to that of 12–15 kilotons of TNT was detonated over Hiroshima. The secret Manhattan Project, headed by the physicist J. Robert Oppenheimer, had been commissioned, some years before, to explicitly explore the possibility of using Einstein’s equation E = mc2 to produce a bomb far more powerful than any that existed before. The further developments of fission-based and fusion-based bombs are well known.
The second exception, less noticeable at first, was the development of an apparently simple device called the transistor. Forty years later, it would launch humanity into the most profound revolution it has ever witnessed.
Transistors, Chips, and Microprocessors
The first patent for a transistor-like device (Lilienfeld 1930) dates from 1925. Experimental work alone led to the discovery of the effect that makes transistors possible. However, the understanding of quantum mechanics and the resulting field of solid-state physics were instrumental in the realization that the electrical properties of semiconductors could be used to obtain behaviors that could not be obtained using simpler electrical components, such as resistors and capacitors.
In 1947, researchers at Bell Telephone Laboratories observed that when electrical contacts were applied to a germanium crystal (a semiconductor) the output signal had more power than the input signal. For this discovery, which may have been the most important invention ever, William Shockley, John Bardeen, and Walter Brattain received the Nobel Prize in Physics in 1956. Shockley, foreseeing that such devices could be used for many important applications, set up the Shockley Semiconductor Laboratory in Mountain View, California. He was at the origin of the dramatic transformations that would eventually lead to the emergence of modern computer technology.
A transistor is a simple device with three terminals, one of them a controlling input. By varying the voltage at this input, a large change in electrical current through the two other terminals of the device can be obtained. This can be used to amplify a sound captured by a microphone or to create a powerful radio wave. A transistor can also be used as a controlled switch.
The first transistors were bipolar junction transistors. In such transistors, a small current also flows through the controlling input, called the base. Other types of transistors eventually came to dominate the technology. The metal-oxide-semiconductor field-effect transistor (MOSFET) is based on different physical principles, but the basic result is the same: A change in voltage in the controlling input (in this case called the gate) creates a significant change in current through the two other terminals of the device (in this case called the source and the drain). From the simple description just given, it may be a little difficult to understand why such a device would lead to the enormous changes that have occurred in society in the last thirty years, and to the even larger changes that will take place in coming decades. Vacuum tubes, first built in the early twentieth century, exhibit a behavior similar to that of transistors, and can indeed be used for many of the same purposes. Even today, vacuum tubes are still used in some audio amplifiers and in other niche applications. Even though transistors are much more reliable and break down much less frequently than vacuum tubes, the critical difference, which took several decades to exploit to its full potential, is that, unlike vacuum tubes, transistors can be made very small, in very large numbers, and at a very small cost per unit. Although the British engineer Geoffrey Dummer was the first to propose the idea that many transistors could be packed into an integrated circuit, Jack Kilby and Robert Noyce deserve the credit for realizing the first such circuits, which required no connecting wires between the transistors.
Transistors, tightly packed into integrated circuits, have many uses. They can be used to amplify, manipulate, and generate analog signals, and indeed many devices, such as radio and television receivers, sound amplifiers, cellular phones, and GPS receivers, use them for that purpose. Transistors have enabled circuit designers to pack into very small volumes amplifiers and other signal-processing elements that could not have been built with vacuum tubes or that, if built with them, would have occupied a lot of space and weighed several tons. The development of personal mobile communications was made possible, in large measure, by this particular application of transistors.
However, the huge effect transistor technology has in our lives is due even more to the fact that integrated circuits with large numbers of transistors can be easily mass produced and to the fact that transistors can be used to process digital information by behaving as controlled on-off switches. Digital computers manipulate data in binary form. Numbers, text, images, sounds, and all other types of information are stored in the form of very long strings of bits (binary digits—zeroes and ones). These binary digits are manipulated by digital circuits and stored in digital memories. Digital circuits and memories are built out of logic gates, all of them made of transistors. For example, a nand gate has two inputs and one output. Its output is 0 if and only if both inputs are 1. This gate, one of the simplest gates possible, is built using four transistors, as shown in figure 3.3a, which shows four MOSFETs: two of type N at the bottom and two of type P at the top. A type N MOSFET behaves like a closed switch if the gate is held at a high voltage, and like an open switch if the gate is held at a low voltage. A P MOSFET, which can be recognized because it has a small circle on the gate, behaves in the opposite way. It behaves like a closed switch when the gate is held at a low voltage, and like an open switch when the gate is held at a high voltage.
Figure 3.3 (a) A nand gate made of MOSFET transistors. (b) The logic symbol for a nand gate.
If both X and Y (the controlling inputs of the transistors) are at logical value 1 (typically the supply voltage, VDD), the two transistors shown at the bottom of figure 3.3a work as closed switches and connect the output Z to the ground, which corresponds to the logical value 0. If either X or Y (or both) is at the logical value 0 (typically ground, or 0 volt), or if both of them are, then at least one of the two transistors at the bottom of figure 3.3a works as an open switch and one (or both) or the top transistors works as a closed switch, pulling the value of Z up to VDD, which corresponds to logical value 1. This corresponds in effect to computing , where the bar denotes negation and ∧ denotes the logic operation and. This is the function with the so-called truth table shown here as table 3.1.
Besides nand gates there are many other types of logic gates, used to compute different logic functions. An inverter outputs the opposite logic value of its input and is, in practice, a simplified nand gate with only two transistors, one of type P and one of type N. An and gate, which computes the conjunction of the logical values on the inputs and outputs 1 only when both inputs are 1, can be obtained by using a nand gate followed by an inverter. Other types of logic gates, including or gates, exclusive-or gates, and nor gates, can be built using different arrangements of transistors and basic gates. More complex digital circuits are built from these simple logic gates.
Nand gates are somewhat special in the sense that any logic function can be built out of nand gates alone (Sheffer 1913). In fact, nand gates can be combined to compute any logic function or any arithmetic function over binary numbers. This property results from the fact that nand gates are complete, meaning that they can be used to create any logic function, no matter how complex. For instance, nand gates can be used to implement the two-bit exclusive-or function, a function that evaluates to 1 when exactly one of the input bits is at 1. They can also be used to implement the three-bit majority function, which evaluates to 1 when two or more bits are at 1. Figure 3.4 illustrates how the two-bit exclusive-or function (which evaluates to 1 when exactly one input is 1) and the three-bit majority function (which evaluates to 1 when at least two inputs are 1) can be implemented using nand gates.
Figure 3.4Exclusive-or and majority gates made of nand gates.
In fact, circuits built entirely of nand gates can compute additions, subtractions, multiplications, and divisions of numbers written in binary, as well as any other functions that can be computed by logic circuits. In practice, almost all complex digital circuits are built using nand gates, nor gates, and inverters, because these gates not only compute the logic function but also regenerate the level of the electrical signal, so it can be used again as input to other logic gates. Conceptually, a complete computer can be built of nand gates alone; in fact, a number of them have been built in that way.
Internally, computers manipulate only numbers written in binary form. Although we are accustomed to the decimal numbering system, which uses the digits 0 through 9, there is nothing special about base 10. That base probably is used because humans have ten fingers and so it seemed to be natural. A number written in base 10 is actually a compact way to describe a weighted sum of powers of 10. For instance, the number 121 represents
1 × 102 + 2 × 101 + 1 × 100,
because every position in a number corresponds to a specific power of 10.
When writing numbers in other bases, one replaces the number 10 with the value of the base used. If the base is smaller than 10, fewer than ten symbols are required to represent each digit. The same number, 121 in base 10, when written in base 4, becomes
1 × 43 + 3 × 42 + 2 × 41 + 1 × 40
(which also can be written as 13214, the subscript denoting the base). In base 2, powers of 2 are used and there are only two digits, 0 and 1, which can be conveniently represented by two electrical voltage levels—for example, 0 and 5 V. The same number, 12110 in base 10, becomes, in base 2, 11110012, which stands for
Arithmetic operations between numbers written in base 2 are performed using logic circuits that compute the desired functions. For instance, if one is using four-bit numbers and wishes to add 710 and 210, then one must add the equivalent representations in base 2, which are 01112 and 00102.
The addition algorithm, shown in figure 3.5, is the one we all learned in elementary school. The algorithm consists in adding the digits, column by column, starting from the right, and writing the carry bit from the previous column above the next column to the left. The only difference is that for each column there are only four possible combinations of inputs, since each digit can take only the value 0 or 1. However, since the carry in bit can also take two possible values (either there is a carry or there is not), there are a total of eight possible combinations for each column. Those eight combinations are listed in table 3.2, together with the desired values for the output, C, and the carry bit, Cout, which must be added to the bits in the next column to the left. The carry is simultaneously an output of a column (Cout) and an input in the next column to the left (Cin).
Figure 3.5 Adding together the numbers 01112 and 00102.
Table 3.2 Logic functions for the addition of two binary digits. C gives the value of the result bit; Cout gives the result of the carry bit.
It is easy to verify by inspection that the function C is given by the exclusive-or of the three input bits, a function that takes the value 1 when an odd number of bits are 1. The function Cout is given by the majority function of the same three input bits.
Therefore, the circuit on the left of figure 3.6 computes the output (C) and the carry out (Cout) of its inputs, A, B, and Cin. More interestingly, by wiring together four of these circuits one obtains a four-bit adder, like the one shown on the right of figure 3.6. In this four-bit adder, the topmost single-bit adder adds together the least significant bits of numbers A and B and the carry bit propagates through the chain of adders. This circuit performs the addition of two four-bit numbers, using the algorithm (and the values in the example) from figure 3.5.
Figure 3.6 (a) A single-bit adder. (b) A four-bit adder, shown adding the numbers 0111 and 0010 in binary..
More complex circuits (for example, multipliers, which compute the product of two binary numbers) can be built out of these basic blocks. Multiplications can be performed by specialized circuits or by adding the same number multiple times, using the same algorithm we learned in elementary school. Building blocks such as adders and multipliers can then be interconnected to form digital circuits that are general in the sense that they can execute any sequence of basic operations between binary numbers. The humble transistor thus became the workhorse of the computer industry, making it possible to build cheaply and effectively the adders and multipliers Thomas Hobbes imagined, in 1651, as the basis of all human reasoning and memory.
Transistors and logic gates, when arranged in circuits that store binary values over long periods of time, can also be used to build computer memories. Such memories, which can store billions or trillions of bits, are part of every computer in use today. Transistors can, therefore, be used to build general-purpose circuits that compute all possible logic operations quickly, cheaply, and effectively. A sufficiently complex digital circuit can be instructed to add the contents of one memory position to the contents of another memory position, and to store the result in a third memory position. Digital circuits flexible enough to perform these and other similar operations are called Central Processing Units (CPUs). A CPU is the brain of every computer and almost every advanced electronic device we use today. CPUs execute programs, which are simply long sequences of very simple operations. (In the next chapter, I will explain how CPUs became the brains of modern computers as the result of pioneering work by Alan Turing, John von Neumann, and many, many others.)
The first digital computers were built by interconnecting logic gates made from vacuum tubes. They were bulky, slow, and unreliable. The ENIAC—the first fully electronic digital computer, announced in 1946—contained more than 17,000 vacuum tubes, weighted more than 27 tons, and occupied more than 600 square feet.
When computers based on discrete transistors became the norm, large savings in area occupied and in power consumed were achieved. But the real breakthrough came when designers working for the Intel Corporation recognized that they could use a single chip to implement a CPU. Such chips came to be called microprocessors. The first single-chip CPU—the 4004 processor, released in 1971—manipulated four-bit binary numbers, had 2,300 transistors, and weighted less than a gram. A present-day high-end microprocessor has more than 3 billion transistors packed in an area about the size of a postage stamp (Riedlinger et al. 2012).
Nowadays, transistors are mass produced at the rate of roughly 150 trillion (1.5 × 1014) per second. More than 3 × 1021 of them have been produced to date. This number compares well with some estimates of the total number of grains of sand on Earth. In only a few years, we will have produced more transistors than there are synapses in the brains of all human beings currently alive.
The number of transistors in microprocessors has grown rapidly since 1971, following an approximately exponential curve which is known as Moore’s Law. (In 1965, Intel’s co-founder, Gordon Moore, first noticed that the number of transistors that could be placed inexpensively on an integrated circuit increased exponentially over time, doubling approximately every two years.) Figure 3.7 depicts the increase in the number of transistors in Intel’s microprocessors since the advent of the 4004. Note that, for convenience, the number of transistors is shown in a logarithmic scale. Although the graph is relative to only a small number of microprocessors from one supplier, it illustrates a typical case of Moore’s Law. In this case, the number of transistors in microprocessors has increased by a factor of a little more than 220 in 41 years. This corresponds roughly to a factor of 2 every two years.
Figure 3.7 Evolution of the number of transistors of Intel microprocessors.
Many measures of the evolution of digital electronic devices have obeyed a law similar to Moore’s. Processing speed, memory capacity, and sensor sensitivity have all been improving at an exponential rate that approaches the rate predicted by Moore’s Law. This exponential increase is at the origin of the impact digital electronics had in nearly every aspect of our lives. In fact, Moore’s Law and the related exponential evolution of digital technologies are at the origin of many of the events that have changed society profoundly in recent decades.
Other digital technologies have also been improving at an exponential rate, though in ways that are somewhat independent of Moore’s Law. Kryder’s Law states that the number of bits that can be stored in a given area in a magnetic disk approximately doubles every 13 months. Larry Roberts has kept detailed data on the improvements of communication equipment and has observed that the cost per fixed communication capacity has decreased exponentially over a period of more than ten years.
For the case of Moore’s Law, the progress is even more dramatic than that shown in figure 3.7, since the speed of processors has also been increasing. In a very simplified view of processor performance, the computational power increases with both the number of transistors and the speed of the processor. Therefore, processors have increased in computational power by a factor of about 30 billion over the period 1971–2012, which corresponds to a doubling of computational power every 14 months.
The technological advances in digital technologies that led to this exponential growth are unparalleled in other fields of science, with a single exception (which I will address in chapter 7): DNA sequencing. The transportation, energy, and building industries have also seen significant advances in recent decades. None of those industries, however, was subject to the type of exponential growth that characterized semiconductor technology. To put things in perspective, consider the fuel efficiency of automobiles. In approximately the same period as was discussed above, the fuel efficiency of passenger cars went from approximately 20 miles per gallon to approximately 35. If cars had experienced the same improvement in efficiency over the last 40 years as computers, the average passenger car would be able to go around the Earth more than a million times on one gallon of fuel.
The exponential pace of progress in integrated circuits has fueled the development of information and communication technologies. Computers, interconnected by high-speed networks made possible by digital circuit technologies, became, in time, the World Wide Web—a gigantic network that interconnects a significant fraction of all the computers in existence.
There is significant evidence that, after 25 years, Moore’s Law is running out of steam—that the number of transistors that can be packed onto a chip is not increasing as rapidly as in the past. But it is likely that other technologies will come into play, resulting in a continuous (albeit slower) increase in the power of computers.
The Rise of the Internet
In the early days of digital computers, they were used mostly to replace human computers in scientific and military applications. Before digital computers, scientific and military tables were computed by large teams of people called human computers. In the early 1960s, mainframes (large computers that occupied entire rooms) began to be used in business applications. However, only in recent decades has it become clear that computers are bound to become the most pervasive appliance ever created.
History is replete with greatly understated evaluations of the future developments of computers. A probably apocryphal story has it that in 1943 Thomas Watson, the founder of IBM, suggested that there would be a worldwide market for perhaps five computers. As recently as 1977, Ken Olson, chairman and founder of the Digital Equipment Corporation, was quoted as saying “There is no reason anyone would want a computer in their home.” In 1981, Bill Gates, chairman of Microsoft, supposedly stated that 640 kilobytes of memory ought to be enough for anyone. All these predictions vastly underestimated the development of computer technology and the magnitude of its pervasiveness in the modern world.
However, it was not until the advent of the World Wide Web (which began almost unnoticeably in 1989 as a proposal to interlink documents in different computers so that readers of one document could easily access other related documents) that computers entered most people’s daily lives. The full power of the idea of the World Wide Web was unleashed by the Internet, a vast network of computers, interconnected by high-speed communications equipment, that spans the world. The Internet was born in the early 1970s when a group of researchers proposed a set of communication protocols known as TCP/IP and created the first experimental networks interconnecting different institutions.
The TCP/IP protocol soon enabled thousands of computers, and later millions, to become interconnected and to exchange files and documents. The World Wide Web was made easily accessible, even to non-expert users, by the development of Web browsers—programs that display documents and can be used to easily follow hypertext links. The first widely used Web browser—Mosaic, developed by a team at the University of Illinois at Urbana-Champaign, led by Marc Andreessen—was released in 1993, and represented a turning point for the World Wide Web. The growth of the Internet and the popularity of the World Wide Web took the world by surprise. Arguably, no phenomenon has changed so rapidly and so completely the culture and daily life around the world as the Internet.
Figure 3.8 plots the number of users since the beginning of the Internet. The number of users grew from just a few in 1993 to a significant fraction of the world population in twenty years. Probably no other technology (except some that were developed on top of the Web) has changed the world as rapidly as the World Wide Web has.
Figure 3.8 Evolution of the number of users of the Internet.
Initially, the World Wide Web gave users access to documents stored in servers (large computers, maintained by professionals, that serve many users at once). Development of the content stored in these servers was done mostly by professionals or by advanced users. However, with the development of more user-friendly applications and interfaces it became possible for almost any user to create content, be it text, a blog, a picture, or a movie. That led to what is known as Web 2.0 and to what is known in network theory as quadratic growth of the Web’s utility. For example, if the number of users doubles, and all of them contribute to enriching the Web, the amount of knowledge that can be used by the whole community grows by a factor of 4, since twice as many people have access to twice as much knowledge. This quadratic growth of the network utility has fueled development of new applications and uses of the World Wide Web, many of them unexpected only a few years ago.
The Digital Economy
Easy access to the enormous amounts of information available on the World Wide Web, by itself, would have been enough to change the world. Many of us can still remember the effort that was required to find information on a topic specialized enough not to have an entry in a standard encyclopedia. Today, a simple Internet search will return hundreds if not thousands of pages about even the most obscure topic. With the advent of Web 2.0, the amount of information stored in organized form exploded. At the time of this writing, the English version of the online encyclopedia Wikipedia includes more than 4.6 million articles containing more than a billion words. That is more than 30 times the number of words in the largest English-language encyclopedia ever published, the Encyclopaedia Britannica. The growth of the number of articles in Wikipedia (plotted in figure 3.9) has followed an accelerating curve, although it shows a tendency to decelerate as Wikipedia begins to cover a significant fraction of the world knowledge relevant to a large set of persons.
Figure 3.9 Evolution of the number of articles in Wikipedia.
Wikipedia is just one of the many examples of services in which a multitude of users adds value to an ever-growing community, thus leading to a quadratic growth of utility. Other well-known examples are YouTube (which makes available videos uploaded by users), Flickr and Instagram (photos), and an array of social networking sites, of which the most pervasive is Facebook. A large array of specialized sites cater to almost any taste or persuasion, from professional networking sites such ass LinkedIn to Twitter (which lets users post very small messages any time, from anywhere, using a computer or a cell phone).
Electronic commerce—that is, use of the Web to market, sell, and ship physical goods—is changing the world’s economy in ways that were unpredictable just a decade ago. It is now common to order books, music, food, and many other goods online and have them delivered to one’s residence. In 2015 Amazon became the world’s most valuable retailer, outpacing the biggest brick-and-mortar stores. Still, Amazon maintains physical facilities to store and ship the goods it sells. In a more radical change, Uber and Airbnb began to offer services (respectively transportation and lodging) using an entirely virtual infrastructure; they now pose serious threats to the companies that used to provide those services in the form of physical facilities (cabs and hotels).
Online games offer another example of the profound effects computers and the Internet can have on the way people live their daily lives. Some massively multiplayer online role-playing games (MMORPGs) have amassed very large numbers of subscribers, who interact in a virtual game world. The players, interconnected through the Internet, develop their game-playing activities over long periods of time, building long-term relationships and interacting in accordance with the rules of the virtual world. At the time of this writing, the most popular games have millions of subscribers, the size of the population of a medium-size country. Goods existing only in the virtual world of online games are commonly traded, sometimes at high prices, in the real world.
Another form of interaction that may be a harbinger of things to come involves virtual worlds in which people can virtually live, work, buy and sell properties, and pursue other activities in a way that mimics the real world as closely as technology permits. The best-known virtual-world simulator of this type may be Second Life, launched in 2003. Second Life has a parallel economy, with a virtual currency that can be exchanged in the same ways as conventional currency. Second Life citizens can develop a number of activities that parallel those in the real world. The terms of service ensure that users retain copyright for content they create, and the system provides simple facilities for managing digital rights. At present the user interface is still somewhat limited in its realism, since keyboard-based interfaces and relatively low-resolution computer-generated images are used to interact with the virtual world. Despite its relatively slow growth, Second Life now boasts about a million regular users.
This is a very brief and necessarily extremely incomplete overview of the impact of Internet technology on daily life. Much more information about these subjects is available in the World Wide Web, for instance, in Wikipedia. However, even this cursory description is enough to make it clear that there are millions of users of online services that, only a few years ago, simply didn’t exist.
One defining aspect of present-day society is its extreme dependency on information and communication technologies. About sixty years ago, IBM was shipping its first electronic computer, the 701. At that time, only an irrelevant fraction of the economy was dependent on digital technologies. Telephone networks were important for the economy, but they were based on analog technologies. Only a vanishingly small fraction of economic output was dependent on digital computers.
Today, digital technologies are such an integral part of the economy that it is very difficult, if not impossible, to compute their contribution to economic output. True, it is possible to compute the total value created by makers of computer equipment, by creators of software, and, to a lesser extent, by producers of digital goods. However, digital technologies are so integrated in each and every activity of such a large fraction of the population that it isn’t possible to compute the indirect contribution of these technologies to the overall economy. A number of studies have addressed this question but have failed to assign concrete values to the contributions of digital technologies to economic output.
It is clear, however, that digital technologies represent an ever-increasing fraction of the economy. This fraction rose steadily from zero about sixty years ago to a significant fraction of the economic output today. In the United States, the direct contribution of digital technologies to the gross domestic product (GDP) is more than a trillion dollars (more than 7 percent of GDP), and this fraction has increased at a 4 percent rate in the past two decades (Schreyer 2000)—a growth unmatched by any other industry in history. This, however, doesn’t consider all the effects of digital technologies on everyday life that, if computed, would lead to a much higher fraction of GDP.
There is no reason to believe that the growth in the importance of digital technologies in the economy will come to a stop, or even that the rate of growth will reduce to a more reasonable value. On the contrary, there is ample evidence that these technologies will account for an even greater percentage of economic output in coming decades. It may seem that, at some point, the fraction of GDP due to digital technologies will stop growing. After all, some significant needs (e.g., those for food, housing, transportation, clothing, and energy) cannot be satisfied by digital technologies, and these needs will certainly account for some fixed minimum fraction of overall economic activity. For instance, one may assume, conservatively, that some fixed percentage (say, 50 percent) of overall economic output must be dedicated to satisfying actual physical needs, since, after all, there is only so much we can do with computers, cell phones and other digital devices. That, however, is an illusion based on the idea that overall economic output will, at some point, stagnate—something that has never happened and that isn’t likely to happen any time soon. Although basic needs will have to be satisfied (at least for quite a long time), the potential of new services and products based on digital technology is, essentially, unbounded. Since the contribution of digital technologies to economic growth is larger than the contribution of other technologies and products, one may expect that, at some point in the future, purely digital goods will represent the larger part of economic output. In reality, there is no obvious upper limit on the overall contribution of the digital economy. Unlike physical goods, digital goods are not limited by the availability of physical resources, such as raw materials, land, or water. The rapid development of computer technology made it possible to deploy new products and services without requiring additional resources, other than the computing platforms that already exist. Even additional energy requirements are likely to be marginal or even non-existent as computers become more and more energy efficient.
This is nothing new in historical terms. Only a few hundred years ago, almost all of a family’s income was used to satisfy basic needs, such as those for food and housing. With the technological revolutions, a fraction of this income was channeled to less basic but still quite essential things, such as transportation and clothing. The continued change toward goods and services that we deem less essential is simply the continuation of a trend that was begun long ago with the invention of agriculture.
One may think that, at some point, the fraction of income channeled into digital goods and services will cease to increase simply because people will have no more time or more resources to dedicate to the use of these technologies. After all, how many hours a day can one dedicate to watching digital TV, to browsing the Web, or to phone messaging? Certainly no more than 24, and in most cases much less. Some fraction of the day must be, after all, dedicated to eating and sleeping. However, this ignores the fact that the digital economy may create value without direct intervention of human beings. As we will see in chapter 11, digital intelligent agents may, on their behalf or on behalf of corporations, create digital goods and services that will be consumed by the rest of the world, including other digital entities. At some point, the fraction of the overall economic output actually attributable to direct human activity will be significantly less than 100 percent. To a large extent, this is already the case today. Digital services that already support a large part of our economy are, in fact, performed by computers without any significant human assistance. However, today these services are performed on behalf of some company that is, ultimately, controlled by human owners or shareholders. Standard computation of economic contributions ultimately attributes the valued added by a company to the company’s owners.
In this sense, all the economic output generated today is attributable to human activities. It is true that in many cases ownership is difficult to trace, because companies are owned by other companies. However, in the end, some person or group of persons will be the owner of a company and, therefore, the generator of the economic output that is, in reality, created by very autonomous and, in some cases, very intelligent systems. This situation will remain unchanged until the day when some computational agent is given personhood rights, comparable to those of humans or corporations, and can be considered the ultimate producer of the goods or services. At that time, and only at that time, we will have to change the way we view the world economy as a product of human activity.
However, before we get to that point, we have to understand better why computers have the potential to be so disruptive, and so totally different from any other technology developed in the past. The history of computers predates that of the transistor and parallels the history of the discovery of electricity. Whereas the construction of computers that actually worked had to await the existence of electronic devices, the theory of computation has its own parallel and independent history.
,
, since the electric field (
, in this case) is proportional to the voltage difference between the capacitor plates, VC. In this equation, the dot above VC represents the variation in time of this quantity. Mathematically, it is called the derivative with respect to time, also represented as dVC//dt.
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, where the bar denotes negation and ∧ denotes the logic operation and. This is the function with the so-called truth table shown here as 
