Table of Contents
Why Units Matter 🧮
In electrical engineering, every quantity must have a unit. A number without a unit is almost meaningless. Saying “the voltage is 5” is incomplete. You must say “5 volts” and write it as “5 V.”
Units tell you what is being measured and how it is scaled. They allow you to compare results, check calculations, and communicate clearly with others. Using the wrong unit or mixing units carelessly can lead to large errors and sometimes dangerous situations.
In this chapter we focus on how to write and handle units properly, how to use prefixes to express very large and very small values, and how to write numbers in scientific notation so that calculations and comparisons become easier and less error prone.
You will see these ideas used everywhere in later chapters, so it is important to become comfortable with them early.
SI Units in Electrical Engineering ⚡
In electrical engineering, we normally use the International System of Units, called SI units. This system defines base units and derived units. For this chapter you only need a few that appear all the time in circuits and measurements.
The table below lists some of the most common SI units for electrical quantities.
| Quantity | Symbol (quantity) | SI unit name | Unit symbol |
|---|---|---|---|
| Charge | $Q$ | coulomb | C |
| Current | $I$ | ampere | A |
| Voltage | $V$ or $U$ | volt | V |
| Resistance | $R$ | ohm | $\Omega$ |
| Power | $P$ | watt | W |
| Energy | $E$ or $W$ | joule | J |
| Time | $t$ | second | s |
| Frequency | $f$ | hertz | Hz |
You will meet the detailed definitions and relationships of these quantities in other chapters. Here, you should notice how each unit has a precise symbol. Always write quantities with their unit symbols, for example “3.3 kΩ” or “12 V.”
Rules for Writing Units ✍️
There are simple conventions that help you read and write units consistently:
- Unit symbols are case sensitive. “mA” is milliamps, “MA” is megaamps, which is a billion times larger. Mixing upper and lower case can change the meaning completely.
- There is usually a space between the number and the unit symbol. Write “5 V,” not “5V,” and “10 kΩ,” not “10kΩ.” Some component labels on hardware may omit the space because of limited room, but in written work include it.
- Unit symbols are not pluralized. Write “5 V,” not “5 Vs,” and “3 A,” not “3 As.”
- Do not mix unit names and symbols in the same expression. Use either “volts” or “V,” not “volt V” together.
- Prefixes attach directly to the unit symbol. For example “kΩ,” “mA,” “µF,” and “nH.”
These small details make a big difference in avoiding confusion, especially when documents or diagrams are shared between people.
Metric Prefixes for Large and Small Values 📏
Electrical quantities can be extremely large or extremely small. Instead of writing many zeros, we use metric prefixes which represent powers of ten. These prefixes are combined with the base unit to form convenient values.
Common Prefixes in Electrical Work
Here are the prefixes you will see very often:
| Prefix | Symbol | Factor | Power of ten | Example |
|---|---|---|---|---|
| giga | G | $1\,000\,000\,000$ | $10^9$ | $1\ \text{GW} = 10^9\ \text{W}$ |
| mega | M | $1\,000\,000$ | $10^6$ | $1\ \text{MΩ} = 10^6\ \Omega$ |
| kilo | k | $1\,000$ | $10^3$ | $1\ \text{kV} = 10^3\ \text{V}$ |
| (none) | $1$ | $10^0$ | $1\ \text{A}$ | |
| milli | m | $0.001$ | $10^{-3}$ | $1\ \text{mA} = 10^{-3}\ \text{A}$ |
| micro | µ | $0.000\,001$ | $10^{-6}$ | $1\ \text{µF} = 10^{-6}\ \text{F}$ |
| nano | n | $10^{-9}$ | $10^{-9}$ | $1\ \text{nF} = 10^{-9}\ \text{F}$ |
| pico | p | $10^{-12}$ | $10^{-12}$ | $1\ \text{pF} = 10^{-12}\ \text{F}$ |
The micro symbol µ may be written as “u” in plain text environments where the Greek letter is hard to type, for example “uF” instead of “µF.” You will see this often on older schematics or simple calculators. However, technically µF is correct.
Understanding the Meaning of Prefixes
Each prefix is a shorthand for multiplying by a specific power of ten. For example:
- $1\ \text{kA} = 1 \times 10^3\ \text{A} = 1000\ \text{A}$
- $1\ \text{mA} = 1 \times 10^{-3}\ \text{A} = 0.001\ \text{A}$
- $47\ \text{nF} = 47 \times 10^{-9}\ \text{F}$
- $2.2\ \text{kΩ} = 2.2 \times 10^3\ \Omega = 2200\ \Omega$
When you read a value with a prefix, mentally replace the prefix with the corresponding factor. When you write a value, pick a prefix that keeps the number itself in a comfortable range, usually between about 0.1 and 1000.
Important rule: A metric prefix multiplies the unit by a power of ten.
Example: $1\ \text{mA} = 10^{-3}\ \text{A},\quad 1\ \text{kΩ} = 10^3\ \Omega.$
If you forget this rule, it is easy to confuse very large and very small values, which can cause serious design or measurement mistakes.
Comparing Prefixes with a Number Line
It helps to see prefixes as steps on a scale of powers of ten. The following table arranges several common prefixes from large to small.
| Power of ten | Name | Symbol | Example using volts |
|---|---|---|---|
| $10^9$ | giga | G | $1\ \text{GV} = 10^9\ \text{V}$ |
| $10^6$ | mega | M | $1\ \text{MV} = 10^6\ \text{V}$ |
| $10^3$ | kilo | k | $1\ \text{kV} = 10^3\ \text{V}$ |
| $10^0$ | $1\ \text{V}$ | ||
| $10^{-3}$ | milli | m | $1\ \text{mV} = 10^{-3}\ \text{V}$ |
| $10^{-6}$ | micro | µ | $1\ \text{µV} = 10^{-6}\ \text{V}$ |
| $10^{-9}$ | nano | n | $1\ \text{nV} = 10^{-9}\ \text{V}$ |
| $10^{-12}$ | pico | p | $1\ \text{pV} = 10^{-12}\ \text{V}$ |
When converting between prefixes, you can move along this scale and adjust the power of ten accordingly.
Converting Between Units and Prefixes 🔁
Often you will need to convert a value from one prefix to another. For instance, changing milliamps to amps, or kilohms to ohms. There are two simple approaches: use the definition of each prefix, or use scientific notation and powers of ten.
Using Prefix Definitions Directly
You can use the basic definitions as conversion factors, for example:
- $1\ \text{kΩ} = 1000\ \Omega$
- $1\ \text{mA} = 0.001\ \text{A}$
So to convert $4.7\ \text{kΩ}$ to ohms:
$$4.7\ \text{kΩ} = 4.7 \times 1000\ \Omega = 4700\ \Omega$$
To convert $250\ \text{mA}$ to amps:
$$250\ \text{mA} = 250 \times 0.001\ \text{A} = 0.25\ \text{A}$$
You can reverse the direction as well. For example, to express $0.005\ \text{A}$ in milliamps, divide by $10^{-3}$ or multiply by $10^3$:
$$0.005\ \text{A} = 0.005 \times 10^3\ \text{mA} = 5\ \text{mA}$$
Using Scientific Notation for Conversions
Scientific notation, which you will learn in detail in a later section of this chapter, is very powerful for unit conversions. If you express both the number and the prefix as powers of ten, you can adjust exponents to find the new form quickly.
For example, convert $3.3\ \text{kΩ}$ to megaohms.
You know $1\ \text{kΩ} = 10^3\ \Omega$ and $1\ \text{MΩ} = 10^6\ \Omega$. So:
$$3.3\ \text{kΩ} = 3.3 \times 10^3\ \Omega$$
To express this in megaohms, divide by $10^6$:
$$3.3 \times 10^3\ \Omega = 3.3 \times 10^3 \times 10^{-6}\ \text{MΩ} = 3.3 \times 10^{-3}\ \text{MΩ} = 0.0033\ \text{MΩ}$$
So $3.3\ \text{kΩ} = 0.0033\ \text{MΩ}$.
The key point is that the base unit ohms stays the same, only the power of ten changes.
Conversion rule: Keep the base unit the same and adjust the power of ten when changing prefixes.
Example: $1\ \text{MΩ} = 10^6\ \Omega = 10^3\ \text{kΩ} = 10^9\ \text{mΩ}.$
Writing Numbers in Scientific Notation 🔬
Scientific notation is a way to write very large or very small numbers using a coefficient and a power of ten. This is essential in electrical engineering, where values often span many orders of magnitude.
A number in scientific notation has the general form
$$N = a \times 10^b$$
where:
- $a$ is a real number, called the coefficient,
- $b$ is an integer, called the exponent.
In standard scientific notation, the coefficient is chosen so that
$$1 \le |a| < 10.$$
For example:
- $4700 = 4.7 \times 10^3$
- $0.00047 = 4.7 \times 10^{-4}$
Scientific notation makes it much easier to see the approximate size of a number at a glance and to perform calculations with powers of ten.
Converting a Large Number
Suppose you want to write $5600000$ in scientific notation.
You move the decimal point so that you have a number between 1 and 10. Count how many places you move it. Here, you move the decimal 6 places to the left:
$$5600000 = 5.6 \times 10^6$$
The exponent is positive because the original number is greater than 1.
Converting a Small Number
Now consider $0.0000039$.
Move the decimal point 6 places to the right, to get $3.9$:
$$0.0000039 = 3.9 \times 10^{-6}$$
The exponent is negative because the original number is less than 1.
Scientific notation rule:
Write $N$ as $a \times 10^b$ with $1 \le |a| < 10$.
If $N > 1$ then $b$ is positive.
If $0 < N < 1$ then $b$ is negative.
With practice, you will start to see powers of ten naturally when you read or write measurement values.
Combining Scientific Notation and Prefixes 🔗
In electrical engineering, it is common to combine scientific notation with prefixes for clarity. The exponent in scientific notation tells you which prefix is convenient.
For example:
- $4.7 \times 10^3\ \Omega = 4.7\ \text{kΩ}$
- $3.3 \times 10^{-9}\ \text{F} = 3.3\ \text{nF}$
- $1.0 \times 10^{-6}\ \text{F} = 1.0\ \text{µF}$
Scientific notation and metric prefixes both express powers of ten. You can convert between them by matching exponents.
The table below connects some common exponents with typical prefixes in electrical work.
| Scientific form | Equivalent with prefix |
|---|---|
| $10^3$ | kilo (k) |
| $10^6$ | mega (M) |
| $10^9$ | giga (G) |
| $10^{-3}$ | milli (m) |
| $10^{-6}$ | micro (µ) |
| $10^{-9}$ | nano (n) |
| $10^{-12}$ | pico (p) |
When choosing how to write a value, try to keep the numeric part simple and use a prefix that makes it easy to read. For example, $4.7\ \text{kΩ}$ is usually clearer than $4700\ \Omega$ and also clearer than $0.0047\ \text{MΩ}$.
Operations with Powers of Ten ✖️➗
When values are written in scientific notation, calculations often involve multiplying or dividing powers of ten. There are two basic rules for exponents that you will use frequently.
Multiplying Numbers in Scientific Notation
If you have
$$(a \times 10^m) \times (b \times 10^n),$$
you can multiply the coefficients and add the exponents:
$$(a \times 10^m) \times (b \times 10^n) = (a b) \times 10^{m+n}.$$
For example, suppose a current is $2 \times 10^{-3}\ \text{A}$ and a resistance is $3 \times 10^3\ \Omega$. The voltage $V$ across the resistor is
$$V = I R = (2 \times 10^{-3}) \times (3 \times 10^3)\ \text{V}.$$
So
$$V = (2 \times 3) \times 10^{-3+3}\ \text{V} = 6 \times 10^0\ \text{V} = 6\ \text{V}.$$
Dividing Numbers in Scientific Notation
If you have
$$\frac{a \times 10^m}{b \times 10^n},$$
you divide the coefficients and subtract the exponents:
$$\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}.$$
For example, suppose power $P$ is $5 \times 10^{-3}\ \text{W}$ and voltage $V$ is $1 \times 10^{-1}\ \text{V}$. The current $I$ is
$$I = \frac{P}{V} = \frac{5 \times 10^{-3}}{1 \times 10^{-1}}\ \text{A} = 5 \times 10^{-3 - (-1)}\ \text{A} = 5 \times 10^{-2}\ \text{A} = 0.05\ \text{A}.$$
You will see these exponent rules appear repeatedly in formulas involving electrical quantities.
Exponent rules:
$10^m \times 10^n = 10^{m+n}$
$\dfrac{10^m}{10^n} = 10^{m-n}$
Typical Scales in Electrical Quantities 📊
It is helpful to build an intuitive sense of the ranges of values that appear in practical circuits. This will also guide your choice of prefixes.
The table below gives some common orders of magnitude for several quantities.
| Quantity | Typical small scale | Typical large scale |
|---|---|---|
| Voltage | millivolts ($\text{mV}$) | kilovolts ($\text{kV}$) |
| Current | microamps ($\text{µA}$) | hundreds of amps ($\text{A}$) |
| Resistance | milliohms ($\text{mΩ}$) | megaohms ($\text{MΩ}$) |
| Capacitance | picofarads ($\text{pF}$) | millifarads ($\text{mF}$) |
| Inductance | nanohenries ($\text{nH}$) | henries ($\text{H}$) |
| Power | milliwatts ($\text{mW}$) | megawatts ($\text{MW}$) |
Recognizing these ranges helps you notice when a value seems unrealistic. For example, a “5 MΩ resistor passing 1 A” should make you suspicious, because that would require a voltage of millions of volts.
Avoiding Common Mistakes 🚫
Beginners often make a few predictable mistakes with units, prefixes, and scientific notation. Being aware of them early will help you avoid them.
A very common error is confusing milli (m) and mega (M). These differ by a factor of $10^9$. Always check the case of letters carefully. Another mistake is forgetting to convert units before using a formula. For instance, using “10 mA” as if it were “10 A” will make the result 1000 times too small.
Sometimes people mix two different prefixes for the same quantity in one calculation without adjusting for the power of ten. For example, adding “1 kΩ” and “500 Ω” directly as if they had the same unit. The correct approach is to convert both to the same unit first, such as both to ohms or both to kilohms.
Finally, with scientific notation, it is easy to miscount decimal places when converting back to ordinary decimal form. Practice slowly at first, and always perform a quick reasonableness check. If you calculate a current of “$10^6\ \text{A}$” through a small resistor in a household circuit, you know something is wrong even before rechecking each step.
By handling units, prefixes, and scientific notation carefully, you build a solid foundation for every later topic in electrical engineering.