2.1 SI Units

The scientific community has largely agreed upon a single standard of units known as the International System of Units, abbreviated as SI Units. This is the modern form of the metric system. Because of the large number of unit systems available, the goal of creating the SI standard was to create a single set of units that had a basis in physics and had a standard way of expressing larger and smaller quantities.

The imperial system of volumes illustrates the problem they were trying to solve. In the imperial system, there were gallons. If you divided a gallon into four parts, you would get quarts. If you divided quarts in half, you’d get pints. If you divided a pint into twentieths, you’d get ounces.

The imperial system was very confusing. Not only were there an enormous number of units but they all were divisible by differing amounts. The case was similar for length—12 inches in a foot, but 3 feet in a yard and 1,760 yards in a mile. This was a lot to memorize, and doing the calculations was not easy.

The imperial system does have some benefits (the quantities used in the imperial system match the sizes normally used in human activities—few people order drinks in milliliters), but for doing work which requires a lot of calculations and units, the SI system has largely won out. Scientific quantities are almost always expressed in SI units. In engineering it is more of a mix, just as engineering itself is a mix between scientific inquiry and human usefulness. However, the more technical fields have generally moved to SI units and stayed with them.

There are only seven base units in the SI system. Other units are available as well, but they all can be measured in terms of these base units. The base units for the SI system are shown in Figure 2-1.

Many other units are derived from these and are known as SI derived units. For instance, for measuring volume, liters are often used.¹ A liter, however, is not defined on its own, but in terms of meters. A liter is a thousandth of a cubic meter. Thus, we can take the unit of length and use it to describe a unit of volume.

A more complicated example is the newton, which is a unit of force. In the SI system, the newton is defined as being a “kilogram-meter per second squared.” This is another way of saying that a newton is the amount of force which accelerates 1 kilogram 1 meter per second, per second.

All of the things of interest to us in this book are ultimately defined in terms of SI base units. For the purposes of this book, it is not important to know which units are base units or derived units, and it is especially unnecessary to know how they are derived. The important thing to keep in mind is that you will be using a well-thought-out, standardized system of units. If the units seem to fit together well, it’s because they were designed to do so.

2.2 Scaling Units

Now, sometimes you are measuring really big quantities, and sometimes you are measuring very small quantities. In the imperial system, there are different units altogether to reach a different scale of a quantity. For instance, there are inches for small distances, yards for medium-sized distances, and miles for large distances. There are ounces for small volumes and gallons for larger volumes.

In the SI system, however, there is a uniform standard way of expressing larger and smaller quantities. There are a set of modifiers, known as unit prefixes, which can be added to any unit to work at a different scale. For example, the prefix kilo- means thousand. So, while a meter is a unit of length, a kilometer is a unit of length that is 1,000 times as large as a meter. While a gram is a unit of mass, a kilogram is a unit of mass that is 1,000 times the mass of a gram.

It works the other way as well. The prefix milli- means thousandth, as in . So, while a meter is a unit of length, a millimeter is a unit of length that is of a meter. While a gram is a unit of mass, a milligram is a unit of mass that is the mass of a gram.

Therefore, by memorizing one single set of prefixes, you can know how to modify all of the units in the SI system. The common prefixes occur at every power of 1,000, as you can see in Figure 2-2.

To convert between a prefixed unit (i.e., kilometer) and a base unit (i.e., meter), we just apply the conversion factor. So, if something weighs 24.32 kilograms, then I could convert that into grams by multiplying by 1,000. 24.32 ∗ 1000 = 24,320. In other words, 24.32 kilograms is the same as 24,320 grams.

To move from the base unit to a prefixed unit, you divide by the conversion factor. So, if something weighs 35.2 grams, then I could convert that into kilograms by dividing it by 1,000. 35.2/1000 = 0.0352. In other words, 35.2 grams is the same as 0.0352 kilogram.

You can also convert between two prefixed units. You simply multiply by the starting prefix and divide by the target prefix. So, if something weighs 220 kilograms and I want to know how many micrograms that is, then I will multiply using the kilo- prefix (1,000) and divide by the micro- prefix (0.000001):

In other words, 220 kilograms is the same as 220,000,000,000 micrograms.

You can do all of the unit scaling that you need just by knowing the multipliers. However, what usually helps me deal with these multipliers intuitively is to simply visualize where each one lands in a single number. Figure 2-3 shows all of the prefixes laid out in a single number.

So let’s say that I was dealing with fractions of a meter and I had something that was 0.000000030 meter. If you line this number up with the chart in Figure 2-3, there are only zeros in the unit, the milli-, and the micro- areas. The first nonzero digits appear in the “nano-” group. When lined up with the chart, the number in the nano- area is 030. Therefore, the number under consideration is 30 nanometers.

2.3 Using Abbreviations

Typing or writing words like kilogram, microsecond, and micrometer isn’t terribly difficult, but, when it occurs a lot (as what can happen in equations), it can get overwhelming. Therefore, every prefix and every unit has an abbreviation. Since the abbreviation for gram is g, and the abbreviation for kilo- is k, we can abbreviate kilogram as kg. Occasionally the abbreviation for the unit and the scaling prefix are identical, as in the case of meter (m) and milli- (also m). That’s fine, as, when you put them together, you get millimeter, which is abbreviated as mm.

The hardest one to write is the one for micro-, μ. This is the Greek letter mu (pronounced “mew”). It’s essentially the Greek way of writing the letter m, and it is used because we already have a lowercase m (milli-) and an uppercase M (mega-) in use. Since micro- begins with an m, lacking any additional English/Latin way of writing an m, it is written with the Greek μ. However, this is sometimes hard to type. Therefore, since the squashed way that it is written makes it look kind of like a u, sometimes people will write u instead of μ if they don’t know how to type out μ with their computer/keyboard. In this book, we never use u for this purpose, but, if you are reading elsewhere something like 100 us, that means 100 microseconds.

2.4 Significant Figures

Significant figures are the bane of many science books. Nearly everything in science has to be rounded, and significant figures are basically the rounding rules for science. We need to talk about them simply so that you are aware of how I achieve the rounding that I do in my exercises.

These rules aren’t hard, but they can cause some newer students to stumble. If you just are wanting to play with electronics, you can skip this section; just be aware that I may have rounded the answers to problems differently than you do.

The goal of significant figures is to prevent us from thinking that we are being more precise than we really are. Let’s say that I measured a piece of wood to be 1 meter long, but I wanted to cut that wood into thirds. How should I report the distance, in decimal, of the length of each desired piece? Well, 1 divided by 3 is 0.33333333333… I can keep writing three’s until the cows come home. But do I really need the length to be that precise? Is my measurement of the initial length of the wood precise enough to warrant that sort of precise request? Significant figures allow us to answer that question and report numbers with a justifiable precision.

So, for any measured quantity, we need to count the number of significant figures. For the most part, the number of significant figures in a number is the same as the number of digits, with a few exceptions. First, significant figures ignore all leading zeros. So, if I measured something as being 102 feet, it has three signficant figures, even if I add leading zeros. So 102 feet and 0000000000102 feet both have three significant figures. Additionally, these leading zeros are still ignored even if they are after the decimal point. So the number 0.00042 has two significant figures. The second rule in counting is that trailing zeros aren’t counted as significant if the measuring device isn’t capable of measuring that accurately (or the quantity isn’t reported that accurately). For instance, the value 1 meter and the value 1.000 meter refer numerically to the same number. However, the second one is usually used to indicate that we can actually measure that precisely. We wouldn’t report 1.000 meter unless our scale can actually report accurately to a thousandth of a meter.

The situation is a little more complicated with zeros on the left side of the decimal. If I say, “1,000 people attended the event,” how precise is that number? Did you count individuals and get exactly 1,000? Is it possibly estimated to the tens or hundreds place? This gets murky. To simplify the issue for this book, you can assume that all digits on the right count as significant figures. So, if we say “1,000 people” attended the event, that is a number with four significant digits. However, if we say “1 kilopeople” attended the event, that is a number with one significant digit. If we say “ 1.03 kilopeople” attended the event, that is a number with three significant digits. We will use this convention when writing down problems, but results may just be a rounded number with trailing zeros (i.e., instead of writing “1.03 kilopeople,” we might write the result as 1,030 people).

One other small rule: If a number is exact, then it is essentially considered to have infinite significant digits. So, for instance, it takes two people to have a baby. This is an exact number. It isn’t 2.01 or 2.00003 people, it is exactly two. So, for the purposes of significant figures, this value has an infinite number of digits. Conversion factors are generally considered to be exact values.

So that is how to count significant digits. This is important because the significant digits affect how caclulations are rounded. There are two rules—one for multiplying and dividing and another for adding and subtracting.

For multiplying and dividing, you should find the input value with the fewest significant figures. The result should then be rounded to that many significant figures. For instance, if we had 103 ∗ 55, then the result should be rounded to two significant figures. So, even though the result is 5,665, we should report it as 5,700. Let’s say we have 55.0 ÷ 3.00. Since both of the input values have three significant figures, then the result should have three significant figures. Therefore, the result is 18.3.

For addition and subtraction, instead of using significant figures, the decimal points for the numbers are lined up, the operation is performed, and the result is rounded to the number of decimal places available in the input value that has the least precision (fewest numbers to the right of the decimal). So, for instance, if I have 1.054 + 0.06, the result is 1.104. However, this would be rounded to 1.10 because that is how many digits to the right the decimal 0.06 had.

If there are a series of operations, significant figures are usually applied at the end of the whole calculation, or when necessary to limit the complication of intermediate results. Calculators will round for you at some point anyway, so there is no getting around some amount of intermediate rounding. Therefore, you should recognize that if your answer differs from the book’s answer by the least significant digit, it is likely that you are correct, but that you rounded in different stages in your calculation.

In professional science and engineering data reporting, significant figures are important. In playing around with electronics, they are much less so. Additionally, even the rules for significant figures aren’t perfect—there are places where their usage leads to problematic results. Entire books have been written on the subject.² Significant figures are there not because they are perfect, but so that we all have a common, straightforward way of communicating the precision of our results. The most important thing to keep in mind is that the degree to which you are precise in your measurements affects the degree to which you can be precise in your calculations.

2.5 Apply What You Have Learned