Background: Prefixes and Notation

01
CHAPTER

Imagine you are handed two facts on the same afternoon. A bolt of lightning carries an electric current of roughly thirty thousand amperes. The tiny light sensor that lets your phone wake up when you glance at it responds to a current of about one ten-millionth of an ampere. Both are measured in the very same unit—the ampere—yet the first is about three hundred billion times larger than the second.

Electronics lives across an enormous range of sizes like this. Voltages run from the few thousandths of a volt produced by a microphone up to the three-quarters of a million volts humming through long-distance transmission lines. Resistances span from the near-zero of a thick copper bar to the trillions of ohms of a good insulator. Writing all of those values out as ordinary decimal numbers—with their long trains of zeros—would be clumsy, hard to read, and dangerously easy to get wrong. Misplace a single zero in a current rating and you may design a circuit that quietly overheats.

This chapter introduces the small toolkit that engineers, technicians, and scientists use to tame these numbers: scientific notation, engineering notation, and the family of metric prefixes such as kilo, milli, and micro. None of the mathematics here is difficult; most of it is bookkeeping with powers of ten. But getting comfortable with it now will make every later chapter—Ohm’s law, power, circuits, signals—read much more smoothly.

1.1 Why Size Matters in Electronics

Every measurement is really two pieces of information glued together: a number and a unit. The unit tells you what kind of thing you measured (volts, amperes, ohms), and the number tells you how much of it there is. Trouble starts when the “how much” part stretches across many orders of magnitude.

Consider electric current, measured in amperes (A). Figure 1.1 places a few everyday examples on a scale where each tick is ten times larger than the one before it. Notice that the points are not crowded together near one value—they are spread across roughly sixteen powers of ten. That spread is typical of electrical work, and it is exactly why a special way of writing numbers is so useful.

Figure 1.1 Electric currents in everyday devices span about sixteen orders of magnitude.

The same story repeats for the other quantities you will meet in this book. The table below gives a feel for the territory.

1.2 Scientific Notation

Scientific notation writes any number as a single value between 1 and 10 multiplied by a power of ten:

a × 10n where 1 ≤ a < 10

The value a is called the coefficient (sometimes the mantissa), and n is the exponent. The coefficient carries the meaningful digits; the exponent records the scale. For example, 92,500,000 becomes 9.25 × 107, because moving the decimal point so that just one digit sits to its left takes seven steps to the left.

How to convert a number to scientific notation

There are only three steps, and a simple sign rule:

1. Place the decimal point just after the first non-zero digit, so the coefficient lands between 1 and 10.

2. Count how many places you moved the decimal point.

3. That count is the exponent. If you moved the point left (the original number was large), the exponent is positive. If you moved it right (the number was a small fraction), the exponent is negative.

1.3 Engineering Notation

Scientific notation allows any exponent at all. Engineering notation is a close cousin with one extra restriction: the exponent is always a multiple of three (…, −6, −3, 0, 3, 6, 9, …). To make that work, the coefficient is allowed to be anywhere from 1 up to (but not including) 1000—that is, one, two, or three digits in front of the decimal point.

Why bother with this rule? Because, as you are about to see, the metric prefixes also step in jumps of one thousand: kilo is a thousand, mega is a thousand kilos, milli is a thousandth, and so on. Engineering notation lines up perfectly with those prefixes, so a number in engineering notation can be read straight off as a prefixed quantity. For instance, 47,000 Ω = 47 × 103 Ω, and 103 is exactly what “kilo” means—so this is simply 47 kΩ.

Scientific vs. engineering notation, side by side

Look down the last column. Every exponent is a multiple of three, and every one of those exponents has a prefix name waiting for it. That is the whole point of engineering notation—and the reason this book uses it almost everywhere.

1.4 The Metric Prefixes

A metric prefix is a short name that stands in for a power of ten, so that you can attach it to a unit and avoid the exponent entirely. You already use several without thinking: a kilometer is a thousand meters, a milliliter is a thousandth of a liter. In electronics the same prefixes ride on volts, amperes, ohms, farads, hertz, and watts.

The prefixes that matter most step in factors of one thousand, which is why they pair so neatly with engineering notation. Figure 1.2 shows them as a ladder; Table 1.1 lists them with a real example of each.

Figure 1.2 The common metric prefixes climb and descend in steps of one thousand.

Table 1.1 The metric prefixes used throughout this book.

1.5 Converting Between Prefixes

Changing a measurement from one prefix to another is the single most common piece of arithmetic in this book, so it is worth doing slowly until it becomes automatic. The key idea is reassuring:

the physical quantity never changes—only the label does. When you switch to a smaller unit, the number must grow larger to describe the same amount; when you switch to a larger unit, the number shrinks.

The decimal-point method

To convert between two prefixes:

1. Find the difference between their two exponents. That difference is how many places the decimal point moves.

2. Decide the direction. Moving to a smaller unit makes the number bigger, so slide the decimal point to the right. Moving to a larger unit makes the number smaller, so slide it to the left.

1.6 Arithmetic with Prefixed Quantities

Adding and subtracting

You can only add or subtract quantities that are expressed in the same prefix—just as you cannot add 3 apples and 2 dozen eggs without first agreeing on a common count. The recipe is simple: convert every term to one shared prefix, then add or subtract the plain numbers.

Multiplying and dividing

When you multiply or divide quantities, the powers of ten simply combine, and the prefixes often cancel in a satisfying way. This is a sneak preview of Ohm’s law from Chapter 2, where voltage equals current times resistance.

1.7 Significant Figures and Sensible Rounding

A calculator will happily report a current as 0.0473684 A, but that long tail of digits is usually a fiction. If your meter only resolves three digits, claiming seven of them implies a precision you do not actually have. Significant figures are the digits in a number that genuinely carry information.

Two quick rules cover almost every case you will meet:

• All non-zero digits count, and any zeros between them count. So 4070 has three or four significant figures depending on whether that last zero was measured.

• Leading zeros in a small decimal are just placeholders and do not count. In 0.00352 there are three significant figures (3, 5, 2); the zeros only locate the decimal point.

Scientific and engineering notation make precision unambiguous, which is one more reason to use them: writing 1.50 × 103 Ω clearly states three significant figures, whereas the bare number 1500 Ω leaves a reader guessing.

For the problems in this book, a good habit is to carry full precision through the middle of a calculation and round only the final answer—typically to three significant figures unless told otherwise. Rounding too early, and rounding repeatedly, lets small errors pile up.

Chapter Summary

  • Electrical quantities span a huge range—often sixteen or more orders of magnitude—so a compact way of writing numbers is essential.
  • Scientific notation writes a number as a × 10n, with the coefficient between 1 and 10. Large numbers take positive exponents; small fractions take negative ones.
  • Engineering notation restricts the exponent to multiples of three and lets the coefficient run from 1 to 1000, so it lines up directly with the metric prefixes.
  • Metric prefixes (peta, tera, giga, mega, kilo, milli, micro, nano, pico, femto) name the common powers of ten and step in factors of one thousand.
  • Prefix symbols are case-sensitive: M (mega) and m (milli) differ by a factor of a billion; kilo is a lowercase k; micro is µ.
  • To convert between prefixes, move the decimal point by the difference in exponents. A smaller unit always yields a larger number (slide right); a larger unit yields a smaller number (slide left).
  • Add or subtract only after converting all terms to a common prefix. When multiplying or dividing, the powers of ten combine and prefixes such as milli and kilo often cancel.
  • Round final answers to a sensible number of significant figures—three is a good default—and use notation to make that precision clear.

Key Terms

Review Problems

A. Write in scientific notation

A1. 68,000

A2. 0.00025

A3. 4,700,000

A4. 0.0903

A5. 12

B. Write in engineering notation

B1. 150,000

B2. 0.0000047

B3. 33,000

B4. 0.250

B5. 8,200,000

C. Convert between prefixes

C1. 4.7 kΩ to ohms

C2. 0.1 µF to nanofarads

C3. 2,500 mV to volts

C4. 3 GHz to megahertz

C5. 0.0033 A to milliamps

C6. 1,200,000 Ω to megohms

C7. 560 pF to nanofarads

C8. 0.75 MW to kilowatts

D. Add or subtract

D1. 220 Ω + 1.5 kΩ

D2. 470 mA + 0.8 A

D3. 100 nF + 0.22 µF

D4. 3.3 kΩ + 680 Ω + 1.0 kΩ

E. Apply what you know

E1. A power supply delivers a current of 0.5 A. Express this in milliamps.

E2. A resistor is printed with the marking 2M2. What is its resistance in ohms?

E3. A wireless router operates at 5 GHz. How many hertz is that? Write the answer in scientific notation.

E4. Two capacitors, 0.01 µF and 4700 pF, are connected so that their values add. Express the total in nanofarads.