*© Fernando Caracena*, 2017

## Early "Digital" computers

Humans began computing in prehistory, perhaps as early as they could draw pictures of animals on cave walls. Normally, a person has five fingers on each hand for a total of ten. By association, we call the first ten Arabic numerals (0, 1, 2,...9) "digits". A computer that deals in discrete digits is therefore called a digital computer. A computer that outputs on a sliding scale that is proportional to the magnitude of a computed quantity is called an analog computer.

One can use fingers from both hands to represent numbers, and thereby to do arithmetic. You might think that that would limit the use of this computational device to small, single digit numbers, but that is not the case if the fingers are used to represent binary bits. If a person holds up both his hands flat and parallel to a table top without touching it, that can represent the number zero. Depressing the rightmost finger down to touch the table, leaving the other fingers in place, can then represent the number 1 in the following binary configuration: 0000000001, where the digits displayed are either 0 or 1 and represent which fingers are touching the table.

Binary arithmetic, which involves combinations of only the two digits, zeroes and ones, is similar to the arithmetic that we were taught in grammar school, except that in binary arithmetic we add 1 to 1 to get 0, and then carry a one to a higher position to the left . For example,

1010 +11 = 1101. (1)

A binary number represents an aggregate of powers of two, instead of one that is an aggregate of powers of ten,

2^{n}2^{n-1} ... 2^{0} 2^{1} . (2)

The binary numbers in (1) translate into the following numbers in decimal digits:

(1* 2^{3}+0* 2^{2} +1* 2^{1} +0* 2^{0})+(1* 2^{1} +1* 2^{0}) = 1* 2^{3}+1 * 2^{2} +0* 2^{1} +1* 2^{0},

or

(8+2)+(2+1) = (8+4+0+1)

10+3=13.

The binary system is but one example of a numbering system that has a base other than ten. For a discussion of numbering systems in general, see the Wikipedia article, "Numeral system".

At this point a big jump is taken, which is calculated to spur the reader into a further study of the subject. On ten fingers, one can represent all binary numbers beginning with zero, up to a number equal to 2^{10}-1, or 1023 in decimal notation! That means that one can develop proper skills to do some powerful binary arithmetic by using just using ten fingers. The hands and fingers then become a form of a digital computer.

## The Abacus, an early digital computer

The abacus is and ancient digital computer that is operated manually. The device features movable beads in two sections, one above the other (Fig.1). The device pictured below, is a fancy model, which can be

striped down to the simpler device pictured in the Wiki for how to use an abacus. The beads are place holders for numbers in the lower digital range (0 to 4 or 5, in the lower section) and 5s in the upper section.

## Curta, the hand-cranked, digital computer

The Curta hand-held mechanical calculator was a nifty instrument that my advisor owned in the early 1960s. On it he was able to do more precise numerical calculations than on a slide rule, to at least 8 digits, and very fast. He set up the numerical inputs using the side tabs, and cranked the small lever, and presto, out came the result. Although, electrical calculators were replacing mechanical ones, the Curta remained popular with technically minded people because it was portable, more precise than the slide rule and less expensive than an electric calculator.`

## Analog computers

Analog computers use some physical aspect of a mechanical or fluid device, or electrical circuit to portray the solution to a given problem. The Antikythera mechanism is an example of a ancient analog computer.

The slide-rule was the constant companion of engineering and science students when I went to college. Originally called Napier's bones, it was an elegant analog computer. It was a device designed to do rapid multiplications and divisions to about three digit accuracy. We sometimes called it the "slip stick," because the way calculations were performed. Two, identical logarithmic scales carved on separate, sliding pieces were the basic components of the slide rule. The answer was read off from a scale, and the last digit was estimated by mentally interpolating between scale markings. My engineering friends wore their slide rules like swords hanging from their belts. Bigger was better. I chose to have a circular slide rule, which achieved its length on the circumference of a circle. The scales moved next to each other on concentric disks.

Electronic Analog computers computers are machines built to solve a type of problem electronically. A coil is the analog of a spring in an oscillator, a capacitor is the analog of a mass, and a resistor is the analog of a frictional component. Using these elements plus external voltage drivers, on can construct electronic circuits that predict the response of complex, physical systems. When I was in college, analog computers were constructed on "bread-board" devices that allowed electrial chords to be connected between various plugs.

Early analog modelling of geofluid dynamics was conducted by various scientists before fast digital computers made it possible to treat such circulations realistically through animated outputs. Here is an example of an analog simulation of a cyclone on a soap bubble . The analog simulation of a tornado is obtained here in a combustion chamber. Here is another analog simulation of a tornado in a liquid tank.

Liquid, rotating tank experiments can beautifully model atmospheric flows through the injection of colored dyes and blobs of different densities.