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## Electrical engineering

### Course: Electrical engineering>Unit 1

Lesson 1: Getting started

# Numbers in electrical engineering

An overview of engineering notation, large and small numbers, number prefixes, and the grammar for units. Written by Willy McAllister.
Electrical engineers come across very large and very small numbers compared to everyday experience. This article gives you an initial exposure to large and small numbers and has examples of how they show up in engineering applications.
Engineering numbers are written in engineering notation, similar to scientific notation. It helps to get comfortable with engineering notation and the wide, dynamic range of numbers that engineers deal with on a regular basis.

## Scientific notation

If you've studied math or science, you have probably run across scientific notation. You can brush up on scientific notation with this video. To express a number in scientific notation, you rewrite it as a number is greater than or equal to, 1 and \lt, 10, multiplied by a power of 10. It might make more sense if we look at some examples:
• Avogadro's Number looks like this in scientific notation: 6, point, 02214082, times, 10, start superscript, 23, end superscript. You may see the same number in computer syntax like this: 6, point, 02214082, start text, space, E, end text, 23, where "E" for "Exponent" stands in for "times, 10, start text, space, t, o, space, t, h, e, space, point, point, point, end text".
• The speed of light is 299792458 meters per second. This is expressed in scientific notation as 2, point, 99792458, times, 10, start superscript, 8, end superscript, start text, m, slash, s, end text and may be rounded to fewer digits like this: 3, point, 00, times, 10, start superscript, 8, end superscript, start text, m, slash, s, end text.
• The charge on an electron is a tiny, unwieldy number. 0, point, 00000000000000000016021766208 coulombs. Rather than writing all those zeros—and getting it wrong most of the time—we can use scientific notation to write the number more simply: 1, point, 6021766208, times, 10, start superscript, −, 19, end superscript coulombs.

## Engineering notation

The habit in engineering is to use a slightly modified scientific notation. Engineers like exponents in multiples of three. This means the digits to the left of the decimal point fall in the range of one to 999. Our minds do a pretty good job relating to numbers in this range.
Engineering notation is only slightly different than scientific notation.
It takes light 0.0000333564095 seconds to travel 10 kilometers in a vacuum. Let's convert this small number into engineering notation:
• Find the decimal point.
• Hop over three digits at a time, going right, until you hop over one, two, or three nonzero digits. In this case, take two hops to the right, until you hop over 33.
• Write down 33.
• Add a decimal point: 33, point
• Write down the remaining digits: 33, point, 3564095.
• Because we hopped right, finish by writing 10 raised to the negative number of hops times three: minus, 2, start text, h, o, p, s, end text, times, 3, equals, minus, 6.
33, point, 3564095, times, 10, start superscript, minus, 6, end superscript seconds is the time it takes for light to travel 10 kilometers in a vacuum, in engineering notation.
A few more examples of engineering notation:
• Speed of light: 300, times, 10, start superscript, 6, end superscript, start text, m, slash, s, end text
• A blink of an eye: \sim, 350, times, 10, start superscript, minus, 3, end superscript, start text, s, end text
The number format rules are not rigid. As long as the point you are trying to make is clear and unambiguous, you may make exceptions. The blink of an eye may be clearly written 0.350 seconds if you intend the reader to compare the value to one second.
One flaw in engineering notation is that it can mislead about the number of significant figures. Engineers generally deal with wide tolerances of manufactured components, so the number of significant figures in circuit designs is usually small: two to three. If the tolerance is important, it is common to write it next to the number, as shown in this example:
A large resistance value: 33, point, 3, times, 10, start superscript, 6, end superscript, \Omega, plus minus, 1, percent.
Over time, you will develop a feel for numerical accuracy and rounding in different situations. When done appropriately, rounding to a few digits is not a sign of laziness, but a realization that real-world components are not all the same—and yet your design still has to work every time. There are other instances, such as during long calculations using computer arithmetic, where even tiny rounding errors are important to anticipate and control. It all depends on the situation. This is the engineering art.

## Prefixes

Many numbers have names derived from Greek or Latin. Engineers and scientists use Système International d'Unités (SI) number prefixes.
Some of the most common prefixes in engineering are listed below. Notice that the exponents are multiples of three. These prefixes are shorter and easier to say or abbreviate than the numerical equivalent: "times, 10, start text, space, t, o, space, t, h, e, space, point, point, point, end text".
NumberPrefixSymbolNote
10, start superscript, plus, 12, end superscripttera-start text, space, T, end text
10, start superscript, plus, 9, end superscriptgiga-start text, space, G, end text
10, start superscript, plus, 6, end superscriptmega-start text, space, M, end text
10, start superscript, plus, 3, end superscriptkilo-start text, space, k, end textthe only > 1 prefix in lower case
10, start superscript, 0, end superscript
10, start superscript, minus, 3, end superscriptmilli-start text, space, m, end text
10, start superscript, minus, 6, end superscriptmicro-mube careful mu (mu) doesn't turn into "m"
10, start superscript, minus, 9, end superscriptnano-start text, space, n, end text
10, start superscript, minus, 12, end superscriptpico-start text, space, p, end text

## Do engineers really deal with numbers this large and small?

Yes! Below are examples of large, medium and small numbers used in real-world electrical systems. These examples are common occurrences, and you can always find greater extremes.
Frequency: Frequency counts the number of times something happens per second—or another unit of time. The SI unit for frequency is hertz (Hz), which is the same as 1, slash, s. You could also say "reciprocal second" or "per second". The internal clock of a modern personal computer runs at a frequency of around 3 GHz left parenthesis, 3, times, 10, start superscript, 9, end superscript, start text, H, z, end text, right parenthesis. This corresponds to a clock period—the amount of time between clock ticks—of 1, slash, left parenthesis, 3, times, 10, start superscript, 9, end superscript, right parenthesis or 333 ps left parenthesis, 333, times, 10, start superscript, minus, 12, end superscript, start text, s, end text, right parenthesis. A human heart beats about one time per second (1 Hz), as detected by an electrocardiogram (ECG) machine.
Resistance: Resistance is measured in units of ohms left parenthesis, \Omega, right parenthesis. The resistance of a wire is often much less than one ohm. Resistance up to tens of megohms left parenthesis, 10, times, 10, start superscript, 6, end superscript, \Omega, right parenthesis is not unusual.
Voltage: The unit of electrical potential is the volt (V). A flashlight battery is 1.5 volts. You can hold this battery in your hand without fear of electric shock. Inside a computer, the chips usually operate with 3 to 5 volts. A car battery is 12 volts. A wall socket is 110 or 220 volts, depending on where you live. This voltage can be fatal if you touch it with bare hands. High-tension power lines overhead are hundreds of thousands of volts—tension is the French word for voltage. As for tiny voltages, wireless signals are measured in microvolts left parenthesis, 10, start superscript, minus, 6, end superscript, start text, V, end text, right parenthesis when detected by a radio or mobile phone receiver.
Current: Currents are measured in amperes (A). One ampere is a large current. Car batteries momentarily supply 100 amperes or more to start a car. A house may consume 150 amperes if everything is turned on. Currents can also be absurdly small. There are situations where 1 femtoamp left parenthesis, 1, times, 10, start superscript, minus, 15, end superscript, start text, A, end text, right parenthesis matters.
Time: Electrical circuits are capable of working at very short time scales. Time intervals in electronics range from 1 second, for the heartbeat example above, down to 1 picosecond left parenthesis, 1, times, 10, start superscript, minus, 12, end superscript, start text, s, end text, right parenthesis.
Capacitance: Capacitance has units of farads (F). A farad is defined as a coulomb per volt. Since a coulomb is such a large amount of charge, a farad is a large unit of capacitance. As a result, capacitance values you come across are tiny numbers. 100 microfarads is a large capacitance. If you twist two 1-inch (2- centimeter) pieces of ordinary insulated hookup wire together, those wires have a capacitance around one picofarad left parenthesis, 1, times, 10, start superscript, minus, 12, end superscript, start text, F, end text, right parenthesis.
Distance and Length: Distance and length have units of meters. We deal with huge distances and tiny lengths on a regular basis. Nature gives us some staggering distances—light travels at 300, times, 10, start superscript, 6, end superscript meters (300 million meters) in one second. Modern microelectronics blesses us with astoundingly small dimensions inside integrated circuits. Today's most aggressive—and expensive—integrated-circuit processes have dimensions as small as 15 nanometers left parenthesis, 15, times, 10, start superscript, minus, 9, end superscript, start text, m, end text, right parenthesis. That's 15 billionths of a meter!

## Unit grammar

These are the grammatical guidelines for writing unit names and symbols.
• Names of all units start with a lowercase letter, even if the unit is named after a person.
• Symbols for units are uppercase if the unit is named after a person, otherwise lowercase.
Symbol nameexample namesSymbolexample symbolsNamed after
second1 millisecondstart text, s, end text2, start text, space, n, s, end text
meter300 kilometerstart text, m, end text35, start text, space, n, m, end text
hertz10 kilohertzstart text, H, z, end text100, start text, space, M, H, z, end textHertz
ohm2 megohm\Omega47, start text, space, k, end text, \OmegaOhm
ampere35 microampstart text, A, end text65, start text, space, m, A, end textAmpère
volt11 kilovoltstart text, V, end text5, mu, start text, V, end textVolta
Isn't it cool. Ohm gets a Greek symbol: \Omega, "Ohm ega."
The short form "amp" is a perfectly acceptable way to abbreviate "ampere".
The numbers you encounter as you study electrical engineering span a huge range. You will come across these numbers on Khan Academy, in textbooks, and in real-world electronic systems.

## Want to join the conversation?

• how fast electricyti travel?
• The speed of electricity is not due to the individual speed of each electron but the combined movement of all the electrons in a circuit acting together. It's a bit like the links of a bicycle chain where each individual link does not travel that fast the but all of them move almost at once. An electron can only move a tiny distance called 'drift' but they all act as one field. The speed of electricity is really the speed of the electromagnetic field created in an electrical circuit, but this is something for study at a more advanced stage.
• what are the scopes in Electrical engineering?
• A scope is designed to measure varying voltage signals, which is important when you want to find out if, for example, a particular component in a circuit is working properly. A scope can measure very tiny quantities which would be impossible by other means.
• What is the difference between e-engineering and computer science?
• Engineering is building or inventing. Computer science is programing objects.
• Why do electrical engineers have to deal with such small numbers when they have to build/invent things? You can't build something that you can't see.
• Probably the best reason we have for building things really small is that small transistors work better than big transistors. They are faster, and take less power to do the same thing. And every time you make a transistor half as big as the previous generation, you can pack four times as many of them in the same space on an integrated circuit. It's all good. The silicon electronics was invented in the 1950's and has been getting smaller ever since. When we want to make something too small to see, we don't give up, we invent better microscopes to see with!
• How long would it take for an electrical pole's circuit to blow out when it has an overload of half of its usual input/output of electricity?
• Let's clarify your question. Do you mean that there is an electric circuit, like one on an outdoor pole, that is protected by a circuit breaker? And is the question how long the circuit breaker will take to trip? Also, when you say overload of half are you saying that there is 150% of the rated amps? Many circuit breakers have a piece of metal holding the circuit closed. The amount and type of metal is chose so that the metal strip will heat up if more amps of electricity pass through than the circuit breaker is rated for. The heating takes some amount of time and then the piece of metal will deflect and then an internal spring will cause the circuit breaker to flip - meaning open up and no more electricity will flow. The reason that this is important is that many electronic devices are sensitive to the type of voltage fluctuations that cause high current, but the heating is not instantaneous and so the high voltage can reach your computer (or stereo or TV) and destroy it well before the circuit breaker flips. Among electricians there is an adage that circuit breakers protect the electric company, not the customer. For your protection you need a surge suppressor. I'm not sure this is what you asked, but it's important to know ;)
• In engineering notation, what is the purpose of hopping 3 digits at a time? I noticed that the common prefixes are also 10³ apart from each other:

10⁻³ milli-
10⁻⁶ micro-
10⁻⁹ nano-
10⁻¹² pico-

Is this related, or just a coincidence?
• The names and the group-by-3 are the same idea, and the same as putting commas every 3 digits in big numbers. The thought is that we have a good quick sense of arithmetic and comparisons for numbers between 1 and 1000. You can visualize $1 and$1,000 pretty easy. The same goes for component values, voltages, currents, frequencies.
• I'm in fifth grade. Is it okay for me to be learning this type of material?
• This engineering topic is fairly challenging for a 5th grader. We use math ideas that you will learn about in Algebra in a few years. However, you will probably enjoy the first several videos that talk about the basic concepts of charge and current and voltage, and how we talk about circuits. In these videos it's perfectly ok to skip past any math parts you don't get just yet. You can learn anything.