How do computers do math?


Before we can answer the question “How do computers do math?“, we need to get our toes wet with a short introduction to electrical circuits. What are the electrical properties and what does Ohm’s law have to do with it?\(\)

Electrical circuits

Turbulence in thunderclouds builds up static electricity because of the friction between water droplets. Your physics teacher might run a van der Graaf generator causing her hair to stand up, or you can simply rub a balloon over your sweater (demo). Objects that rub against each other create static charge. This charge caused by an excess or lack of electrons. The electrons can be easily transferred by rubbing objects, because they only have a loose bond with their atomic nucleus (protons and neutrons).

Static electricity may suddenly discharge, such as when a bolt of lightning flashes through the sky. Other times, objects with opposite charges attract like when a balloon clings to a sweater. In between these two extremes are experiments with a modest flow of electrons, such as the van der Graaf generator discharging through a fluorescent light tube.

Electrical charge is expressed in Coulomb and corresponds to a charge of \(-6.24151\times 10^{18}\) electrons. Electric current is the movement of charge (electrons). Your light bulb, computer and phone all harness the movement of electrons.

Current

Electric current flows when there is a closed loop, a circuit, around which electrons flow. For example, in a flashlight, once you close the switch the current flows from the battery through the bulb, wires, switch and back through the battery. The current through the thin wire of the light bulb causes it to heat and light up. When you open the switch, it breaks the current and the light turns off.

Electrons jumping through a semiconductor
Electron traversing a copper wire

The illustration above visualizes an electron traversing a wire from atom to atom. In real life, the scale if very different. The atoms are so small that they are invisible to the naked eye. To get an idea of the scale, imagine a current of about 1,000,000,000,000,000,000 electrons per second to power a light bulb.

Current is the movement of charge, with a unit of 1 Ampere corresponding to moving a charge of one Coulomb a second.

The unit name for current “ampere” is in remembrance of André-Marie Ampère, a French physicist and mathematician and early pioneer of electromagnetism. He introduced the symbol \(I\) from the French term “intensité de courant”. Electrical current is the flow of positive charge opposite to the movement of electrons. This is an historic artifact because he discovered current 60 years before the discovery of the electron.

Analogy

Let us take a step back and compare the electrical current to a water system as shown below. When you open the valve in the pipe, water starts flowing down the pipe, similar to closing the switch in flashlight.

The amount of water flow depends on the pipe diameter and the height difference of the water. Similarly, in the flashlight the current
depends on the voltage of the battery and the resistance of the light bulb.

Voltage

In the water model above, the voltage compares to the difference in water level. Voltage is the difference in charge between two points. For example, in the flashlight that is between the \(+\) and \(-\) terminals of the batteries. Instead of charge, people also use the term electric potential, the amount of force available to move electrons from one specific point in a circuit to another point.

The unit “volt” honors of the Italian physicist Alessandro Volta, who discovered the first chemical battery. The symbol for voltage depends on your location. Europe uses symbol \(U\) and the US uses the symbol \(V\). In either case, the unit Volt (\(V\)) is defined as the electric potential over a wire when an electric current of 1 Ampere dissipates 1 Joule of energy.

Resistance

The third electric property is resistance, a material’s tendency to resist the flow of charge (current). Free electrons move through a conductor with some degree of friction or resistance to motion.

A thin carbon wire has a high resistance, and a thick copper wire has a low resistance. In the water analogy, it compares with the narrowness of the water pipe. The narrower the pipe, the smaller the flow of water.

The symbol for resistance is \(R\). Resistance is measured in the unit Ohm (\(\Omega\)) after the German physicist Georg Ohm.

Ohm’s law

Ohm’s law describes the relationship between voltage, current and resistance. The amount of current is proportional with the voltage that motivates the electrons to move, and inversely proportional to the resistance that oppose the flow of electrons. Georg Ohm formulated this relationship as:

$$U=I \times R\nonumber$$
Ohm’s law

The illustration below shows the circuit and the popular Ohm’s law triangle. Here \(U\) represents the voltage [Volt], \(I\) the current [A] and \(R\) the resistance in Ohms [Ω].

Using Ohm’s law, we can determine the voltage, current or resistance if the other two properties are known. For example, if we have a circuit with the potential of 12 Volt, a resistance of 120 Ω, then the current follows as
$$
\frac{12\ \mathrm{V}}{120\ \Omega}=0.1\ \mathrm{A}\nonumber
$$

Another example is a 9 volt battery powering a light emitting diode in series with a resistor. If we want a current of 10 mA, and at that current, the LED has a potential drop of 2.2 volts then we can calculate the required current limiting resistor value as
$$\def\e#1{\times 10^{#1}}
\def\num#1{\numx#1}\def\numx#1e#2{{#1}\mathrm{e}{#2}}
R=\frac{U_{bat}-U_{led}}{I}=\frac{9 – 2.2}{10\e{-3}}=680\ \Omega\nonumber
$$

Armed with this knowledge about electrical circuits, let’s continue with Electronic Circuits.

Embedded software developer
Passionately curious and stubbornly persistent. Enjoys to inspire and consult with others to exchange the poetry of logical ideas.

2 Replies to “How do computers do math?”

  1. Coert, I have tried to make a square root in excel but it would be work. The multiply and divide work in excel and the CSM works. I can not send you a picture of my work in this system. Can you help me?
    Ronald lokker from the Netherlands

  2. Thanks. I think the sqrt schematic might use the q* instead of the q to build the new subtrahend.
    [in parallel to email conversation]

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