Analog: An analog computer (spelled analogue in British English) is a form of computer that uses the continuously-changeable aspects of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved. In contrast, digital computers represent varying quantities incrementally, as their numerical values change. In an analog computer, quantities represent – they are analogous to – quantities in the problem being solved. In operation, analog computers contain a mathematical model of the problem. In popular usage, analog has come to refer to devices and media that represent and store images, sound, motion pictures, etc. as analogs of what they represent, while digital devices basically work with numerical representations, using analog to digital conversion for input, and often digital to analog conversion for output. Mechanical analog computers were very important in gun fire control in World War II and the Korean War; they were made in significant numbers. In particular, development of transistors made electronic analog computers practical, and before digital computers had developed sufficiently, they were commonly used in science and industry. Analog computers can have a very wide range of complexity. Slide rules and nomographs are the simplest, while naval gun fire control computers and large hybrid digital/analogue computers were among the most complicated. Digital computers have a certain minimum (and relatively great) degree of complexity that is far greater than that of the simpler analog computers. This complexity is required to execute their stored programs, and in many instances for creating output that is directly suited to human use. Setting up an analog computer required scale factors to be chosen, along with initial conditions – that is, starting values. Another essential was creating the required network of interconnections between computing elements. Sometimes it was necessary to re-think the structure of the problem so that the computer would function satisfactorily. No variables could be allowed to exceed the computer's limits, and differentiation was to be avoided, typically by rearranging the "network" of interconnects, using integrators in a different sense. Running an electronic analog computer, assuming a satisfactory setup, started with the computer held with some variables fixed at their initial values. Moving a switch released the holds and permitted the problem to run. In some instances, the computer could, after a certain running time interval, repeatedly return to the initial-conditions state to reset the problem, and run it again. Digital: A computer is a machine that manipulates data according to a set of instructions. Although mechanical examples of computers have existed through much of recorded human history, the first electronic computers were developed in the mid-20th century (1940–1945). These were the size of a large room, consuming as much power as several hundred modern personal computers (PCs). Modern computers based on integrated circuits are millions to billions of times more capable than the early machines, and occupy a fraction of the space. Simple computers are small enough to fit into a wristwatch, and can be powered by a watch battery. Personal computers in their various forms are icons of the Information Age and are what most people think of as "computers". The embedded computers found in many devices from MP3 players to fighter aircraft and from toys to industrial robots are however the most numerous. The ability to store and execute lists of instructions called programs makes computers extremely versatile, distinguishing them from calculators. The Church–Turing thesis is a mathematical statement of this versatility: any computer with a certain minimum capability is, in principle, capable of performing the same tasks that any other computer can perform. Therefore computers ranging from a mobile phone to a supercomputer are all able to perform the same computational tasks, given enough time and storage capacity. Hybrid: Hybrid computers are computers that exhibit features of analog computers and digital computers. The digital component normally serves as the controller and provides logical operations, while the analog component normally serves as a solver of differential equations. In general, analog computers are extraordinarily fast, since they can solve most complex equations at the rate at which a signal traverses the circuit, which is generally an appreciable fraction of the speed of light. On the other hand, the precision of analog computers is not good; they are limited to three, or at most, four digits of precision. Digital computers can be built to take the solution of equations to almost unlimited precision, but quite slowly compared to analog computers. Generally, complex equations are approximated using iterative numerical methods which take huge numbers of iterations, depending on how good the initial "guess" at the final value is and how much precision is desired. (This initial guess is known as the numerical seed for the iterative process.) For many real-time operations, the speed of such digital calculations is too slow to be of much use (e.g., for very high frequency phased array radars or for weather calculations), but the precision of an analog computer is insufficient. Hybrid computers can be used to obtain a very good but relatively imprecise 'seed' value, using an analog computer front-end, which is then fed into a digital computer iterative process to achieve the final desired degree of precision. With a three or four digit, highly accurate numerical seed, the total digital computation time necessary to reach the desired precision is dramatically reduced, since many fewer iterations are required. Consider that the nervous system in animals is a form of hybrid computer. Signals pass across the synapses from one nerve cell to the next as discrete (digital) packets of chemicals, which are then summed within the nerve cell in an analog fashion by building an electro-chemical potential until its threshold is reached, whereupon it discharges and sends out a series of digital packets to the next nerve cell. The advantages are at least threefold: noise within the system is minimized (and tends not to be additive), no common grounding system is required, and there is minimal degradation of the signal even if there are substantial differences in activity of the cells along a path (only the signal delays tend to vary). The individual nerve cells are analogous to analog computers; the synapses are analogous to digital computers. Note that hybrid computers should be distinguished from hybrid systems. The latter may be no more than a digital computer equipped with an analog-to-digital converter at the input and/or a digital-to-analog converter at the output, to convert analog signals for ordinary digital signal processing, and conversely, e.g., for driving physical control systems, such as servomechanisms. Sometimes seen is another usage of the term "hybrid computer" meaning a mix of different digital technologies to achieve overall accelerated processing, often application specific using different processor technologies.