 # Unit 2. More of the Basics

In this unit we'll cover an important mathematical topic: scientific notation and engineering notation. (Actually, we already started this topic in Unit 1 when we mentioned the prefixes, such as k and M, used to indicate large or small numbers.) With that topic under our belts, we'll be able to make sense of the color codes that appear on resistors and the numeric codes that appear on capacitors and inductors. We'll also talk about the solderless breadboard, and we'll see how to use the breadboard to connect components together in two different ways, called series connections and parallel connections.

##### Powers of Ten
• Often we express quantities as numbers multiplied by powers of ten.
• Example: Instead of writing 370,000, we may write 37 × 104.
• Another example: instead of writing 0.000019, we may write 1.9 × 10-5.
• The main reason for doing this is to reduce the number of zeroes we have to read when dealing with very large or very small numbers.
• ##### Two Standard Conventions
• Suppose I have a quantity—say it's 12,100 Ω—that I want to write as a power of ten. There are many ways I could do this, such as
1210 × 10 Ω
or 121× 102 Ω
or 12.1 × 103 Ω
or 1.21 × 104 Ω
or 0.121 × 105 Ω
• Question: Which one of those is the right way to write 12,100 Ω as a power of ten?
• Answer: there is no one single right way; all of the ways shown above are correct, since they are all equal to 12,100 Ω.
• However, physicists and other scientists usually prefer one particular way of doing this, while engineers and engineering technicians (that's us!) usually prefer another way of doing it. These two ways are called scientific notation and engineering notation.
##### Scientific Notation
• In scientific notation, a quantity is expressed as the product of a number between 1 and 10 times a power of ten.
• Another way of saying the same thing is that in scientific notation, a number is expressed as the product of a number with exactly one non-zero digit to the left of the decimal point, times a power of ten.
• Example: In scientific notation 12,100 Ω would be written as 1.21 × 104 Ω, since the number that we're multiplying by a power of ten must be between 1 and 10.
• Using either definition given above, convince yourself that none of the following expressions are in scientific notation:
1210 × 10 Ω
121× 102 Ω
12.1 × 103 Ω
0.121 × 105 Ω
• Note 1: When we say "between 1 and 10" in the definition, what we really mean is that the number must be greater than or equal to 1 and less than (but not equal to) 10.
• Example: 1000 V in scientific notation should be written as 1 × 103 V, not as 10 × 102 V.
• Note 2: Here's a special case: if you wish to express a quantity in scientific notation and the number is already between 1 and 10, then you just leave it alone.
• For example, 1.37 A expressed in scientific notation is just 1.37 A.
• ##### Engineering Notation
• In engineering notation, a quantity is expressed as the product of a number between 1 and 1000, times a power of ten whose exponent is a multiple of 3.
• Another way of saying the same thing is that in engineering notation, a number is expressed as the product of a number with one to three non-zero digits to the left of the decimal point, times a power of ten whose exponent is a multiple of 3.
• Example: In engineering notation 12,100 Ω would be written as 12.1 × 103 Ω.
• Using either definition given above, convince yourself that none of the following expressions are in engineering notation:
1210 × 10 Ω
121× 102 Ω
1.21 × 104 Ω
0.121 × 105 Ω
• Note 1: When we say "between 1 and 1000" in the definition, what we really mean is that the number must be greater than or equal to 1 and less than (but not equal to) 1000.
• Example: 1,000,000 V in scientific notation should be written as 1 × 106 V, not as 1000 × 103 V.
• Note 2: Here's a special case: if you wish to express a quantity in engineering notation and the number is already between 1 and 1000, then you just leave it alone.
• For example, 25.4 V expressed in engineering notation is just 25.4 V.
• ##### Metric Prefixes
• Metric prefixes are symbols that represent the powers of ten used in engineering notation. The most common metric prefixes are listed below.  Name Symbol Power of Ten Value pico p 10-12 0.000000000001 nano n 10-9 0.000000001 micro µ 10-6 0.000001 milli m 10-3 0.001 kilo k 103 1000 mega M 106 1,000,000 giga G 109 1,000,000,000 tera T 1012 1,000,000,000,000
• Do these symbols looks familiar? They should! In Unit 1 we already started using many of them (p, n, µ, m, k, and M) to express very large or very small numbers. Now you should have a better understanding of what these prefixes mean. For example:
• The prefix M stands for 1,000,000 (a million), so 1 MΩ is the same thing as 1,000,000 Ω.
• The prefix k stands for 1,000 (a thousand), so 22 kΩ = 22,000 Ω.
• The prefix m stands for 0.001 (one thousandth), so 1.5 mH = 0.0015 H.
• The prefix µ stands for 0.000001 (one millionth), so 6 µH = 0.000006 H.
• The prefix n stands for 0.000000001 (one billionth), so 4.7 nF = 0.0000000047 F.
• The prefix p stands for 0.000000000001 (one trillionth), so 33 pF = 0.000000000033 F.
• Using metric prefixes, 12,100 Ω would be written as 12.1 kΩ.
• Throughout this course and other electronics courses at Sinclair, you should almost always express quantities in engineering notation using metric prefixes. On a homework or lab or test, if you find an answer of 12,100 Ω, you should express it as 12.1 kΩ, not as 12,100 Ω or as 1.21 × 104 Ω or as 12.1 × 103 Ω.
• ##### Metric-Prefix Matching Game
• You need to memorize which value each metric prefix stands for.
• To work on this skill, be sure to play the Metric-Prefix Matching Game. Like all of the games on the Games page, this game has a Study mode, a Practice mode, and a Challenge mode.

##### Converting between Kilohms and Ohms
• Below we'll look at the general problem of how to convert a quantity from one metric prefix to another. But before we look at this problem in its general form, let's "get our feet wet" by looking at a couple of specific cases that you'll run into pretty often.
• To convert from kilohms to ohms, move the decimal point 3 places to the right.
• Example: 4.7 kΩ = 4,700 Ω
• Going in the opposite direction, to convert from ohms to kilohms, move the decimal point 3 places to the left.
• Example: 390,000 Ω = 390 kΩ
• ##### Converting between Megohms and Ohms
• To convert from megohms to ohms, move the decimal point 6 places to the right.
• Example: 6.8 MΩ = 6,800,000 Ω
• Going in the opposite direction, to convert from ohms to megohms, move the decimal point 6 places to the left.
• Example: 10,000,000 Ω = 10 MΩ
• ##### Metric Prefix Conversions
• Often we need to convert a quantity from one metric prefix to another.
• For example, perhaps you know that a particular resistance is equal to 3.3 MΩ, but you want to express it in kilohms (kΩ) instead of megohms (MΩ).
• Or perhaps you know that a capacitance is equal to 1000 pF, and you want to express it in microfarads (µF) instead of picofarads (pF).
• To do these conversions, you'll need to adjust the size of the number by moving the decimal point to the right or to the left. This requires you to figure out which way to move the decimal point (to the right or to the left), and how far to move the decimal point.
• Let's look at two cases in more detail.
##### Converting from a Larger Prefix to a Smaller Prefix
• If you're converting from a larger prefix to a smaller prefix, the number must get larger, so you must move the decimal point to the right.
• For the example of converting 3.3 MΩ into kilohms, we're converting from a larger prefix (MΩ) to a smaller prefix (kΩ), so we know that the number that we end up with must be larger than 3.3. We'll move the decimal point to the right.
• Once you know which direction to move the decimal point, how many places do you move the decimal point? You figure that out by finding the difference in the powers of ten between the two prefixes.
• Returning the example of converting 3.3 MΩ into kilohms, mega- (M) is equal to 106, while kilo- (k) is equal to 103, so we must move the decimal point by three places. Since we decided above that we must move the decimal point the to right, our final answer is:

3.3 MΩ = 3,300 kΩ

• ##### Converting from a Smaller Prefix to a Larger Prefix
• If you're converting from a smaller prefix to a larger prefix, the number must get smaller, so you must move the decimal point to the left.
• For the example of converting 1000 pF into microfarads, we're converting from a smaller prefix (pF) to a larger prefix (µF), so we know that the number that we end up with must be smaller than 1000. We'll move the decimal point to the left.
• Once you know which direction to move the decimal point, how many places do you move the decimal point? Again, you figure that out by finding the difference in the powers of ten between the two prefixes.
• Returning the example of converting 1000 pF into microfarads, pico- (p) is equal to 10-12, while micro- (µ) is equal to 10-6, so we must move the decimal point by six places. Since we decided above that we must move the decimal point the to left, our final answer is:

1000 pF = 0.001 µF

• ##### Calculator's Exponent Key
• Scientific calculators have an exponent key (usually labeled EE, EXP, or E) that lets the user enter a number multiplied by a power of 10.
• On the Casio fx-115, the key is labeled EXP, and is located in the position shown here. • On the Texas Instruments TI-86, the key is labeled EE, and is located in the position shown below. • Examples:
• To enter 25.7 × 103, press 25.7 E 3. Don't press 25.7 × 10 E 3.
• To enter 1.49 × 10-3, press 1.49 E -3.
• Learn how to use this feature of your calculator. It will save you from making mistakes. For instance, if you're doing a calculation that involves 1.75 μA, you're much less likely to make a mistake if you enter it as 1.75 E -6 than if you try to enter it as 0.00000175.
• ##### Calculator Modes
• Most scientific calculators also have an engineering mode and a scientific mode, which cause the answer to be displayed in engineering notation or in scientific notation.
• If you don't know how to put your calculator into engineering mode or scientific mode, ask me or one of your classmates to help you figure out how.
• Examples:
• Put your calculator in engineering mode, and then use it to calculate 1 ÷ 2500. You should get an answer of 400 E -6, which means 400×10-6.
• With your calculator still in engineering mode, use it to calculate 330 × 1540. You should get an answer of 508.2 E 3, which means 508.2×103.
• Learn how to use this feature of your calculator. It will save you from making mistakes.
• ##### Calculator Manuals Online
• Many calculator manufacturers have websites where you can find online versions of the user's manuals for their calculators. So if you've misplaced your calculator's manual and need help figuring out how to use the calculator, the following links might come in handy:
• If you have a calculator made by another manufacturer, please let me know and I'll try to find a link for their website.

##### Resistor Color Code
• Usually, a resistor's value in ohms is indicated by several colored bands on the resistor's body.
• Each integer from 0 to 9 is represented by a color. The table below shows which colors represent these integers.
 Integer Color 0 Black 1 Brown 2 Red 3 Orange 4 Yellow 5 Green 6 Blue 7 Violet 8 Gray 9 White
• Two other colors (silver and gold) are also part of the resistor color code, but they have special meanings. Instead of simply representing numbers, these two colors represent tolerances ratings. In particular, gold represents a 5% tolerance, and silver represents a 10% tolerance. We often write this as ±5% or ±10%. (The expression ±5% is read as "plus or minus five percent." Similarly, ±10% is read as "plus or minus ten percent.") Finally, a tolerance of ±20% is represented by no colored band at all.
 Tolerance Color ±5% Gold ±10% Silver ±20% None
##### Color-Code Matching Game
• You need to memorize which colors stand for which numbers.
• To work on this skill, be sure to play my Color-Code Matching Game. Like all of the games on the Games page, this game has a Study mode, a Practice mode, and a Challenge mode.
##### Reading a Resistor's Value
• Now that you know what the individual colored bands stand for, let's see how to put them all together to find a resistor's value.
• Usually, a resistor's value in ohms is indicated by four colored bands on the resistor's body.
• The first three bands of the color code give the resistor's nominal value.
• The first two colored bands represent the first and second digits of the nominal value. The third band represents the number of zeroes following those first two digits.
• Example: If the first three bands are yellow-violet-red, then the nominal value is 4700 ohms, or 4.7 kilohms. That's because yellow stands for 4, violet stands for 7, and red stands for 2. (So the red band tells you to add two zeroes after the 4 and the 7.)
• The fourth band (or tolerance band), gives the percentage variation from the nominal value that the actual resistance may have.
• Example: If the four bands are yellow-violet-red-gold, then we saw above that the nominal value is 4700 Ω, or 4.7 kΩ. But you would not expect the resistor's actual value to be exactly 4700 Ω. It might actually be a bit higher or lower. The gold band is the manufacturer's way of assuring you that the actual value lies within 5% of the nominal value.
• ##### Resistor Identification Game
• You'll need to become an expert at reading resistor color codes. To work on this skill, play my Resistor Identification Game. Like all of the games on the Games page, this game has a Study mode, a Practice mode, and a Challenge mode.
##### More Practice from our Friends in Wisconsin
• Instructors in the Wisconsin Technical College System have created a library of short online animations and quizzes to help students learn electronics. I'll include links to some of these "learning objects." Whenever you see the icon below, click it to see a learning object on the material you're studying. The Wisconin learning object will open in a new window; close the winow when you're finished and want to return to this lesson.
• This first one will give you more practice with resistor color codes. Click the icon now.
• ##### Reading a Capacitor's Value
• Capacitor codes are similar to resistor codes. But with capacitors, the numbers are usually printed on the component, so you don't have to remember which colors stand for which numbers.
• Usually, a capacitor's nominal value in picofarads (pF) is indicated by three numbers on the capacitor's body.
• The first two numbers represent the first and second digits of the nominal value. The third number represents the number of zeroes following those first two digits.
• Example: If a capacitor's numeric code is 472, then the nominal value is 4700 pF, which is the same as 4.7 nF.
• ##### Capacitor Identification Game
• You'll need to become an expert at reading capacitor codes. To work on this skill, play the Capacitor Identification Game. Like all of the games on the Games page, this game has a Study mode, a Practice mode, and a Challenge mode.
##### Capacitor Tolerance Codes
• Often the numeric code on a capacitor will be followed by a letter that indicates the capacitor's tolerance rating. Here are the tolerance letters that you'll see most often:
 Letter Tolerance Rating F ±1% G ±2% J ±5% K ±10% M ±20%
• ##### Reading an Inductor's Value
• Inductor codes are similar to capacitor codes.
• Usually an inductor's nominal value in microhenries (µH) is indicated by three numbers on the inductor's body.
• The first two numbers represent the first and second digits of the nominal value. The third number represents the number of zeroes following those first two digits.
• Example: If an inductor's numeric code is 472, then the nominal value is 4700 µH, which is the same as 4.7 mH.
• ##### Inductor Tolerance Codes

• In the lab we often wish to build and test circuits by connecting components together. To connect components together permanently, you can solder them to a circuit board. But for temporarily building circuits that can be taken back apart after you've tested them, it's more convenient to use a solderless breadboard.
• A solderless breadboard is a plastic-covered board with many holes into which you can insert the leads of components. Underneath the plastic cover are strips of metal that connect some of the holes to each other. Therefore, by inserting component leads into the proper holes, you connect the components to each other without having to solder them together.
• Here is a photograph of a breadboard on the digital-analog trainers found in Sinclair's labs: • Here is a closer view of a small portion of the breadboard: • As you can see, the breadboard has many holes into which you can insert component leads. These holes are arranged in rows and columns, which are labeled with letters and numbers to let you identify individual holes. For instance, the hole in the upper left-hand corner of this photo is in row a and column 1, so we could refer to it as hole a-1. The hole in the lower right-hand corner is in row f and column 30, making it hole f-30.
• These holes are connected together in vertical groups of six. So, for example, holes a-1, b-1, c-1, d-1, e-1, and f-1 are all connected to each other by a metal strip beneath the breadboard's plastic cover, but those six holes are not connected to any of the other holes on the breadboard. We could connect two components together by inserting one component's lead into hole a-1 and inserting the other component's lead into hole b-1 (or c-1 or d-1 or e-1 or f-1).

##### Connecting Two Components in Series
• We can use the breadboard to connect components together in different ways. For example, the photograph below shows two resistors on a breadboard. Notice that each resistor has one lead (or "leg") inserted into a hole in column 14; therefore the two resistors are connected to each other at this point. But notice that the resistor's other leads are not connected together. • When two resistors are connected together in this particular way, they are said to be connected in series with each other.
• In the same way, we could connect two capacitors in series, or two inductors in series, or a capacitor and a resistor in series, and so forth.
##### Connecting More than Two Components in Series
• Shown below is a photo of three resistors connected in series on a breadboard. The first and second resistors are connected because they each have a lead inserted into column 14. The second and third resistors are connected because they each have a lead inserted into column 22. • If you kept adding more resistors to this "chain" of resistors, you would have four resistors in series, then five resistors in series, and so on.
• Of course, we could also connect three or more capacitors in series, three or more inductors in series, and so on.
##### Total Resistance (or Total Capacitance or Total Inductance)
• When we connect several components of the same kind together, we may be interested in measuring the total value of those components. For example, in the photographs above of resistors on the breadboard, we may wish to know the total resistance of the resistors.
• We do this by connecting our meter's test leads to the two free ends of the "chain" of series components.
• In other words, in the case of two resistors connected in series, we would connect one of the meter's test leads to R1's left-hand lead, and connect the meter's other test lead to R2's right-hand lead.
• For three resistors connected in series, we would connect one test lead to R1's left-hand lead and the other test lead to R3's right-hand lead.
• We'll use the symbol RT to stand for total resistance.
• In the same way, we could measure the total capacitance (CT) of capacitors connected to each other, or the total inductance (LT) of inductors connected to each other.

##### Connecting Two Components in Parallel
• We've been talking about connecting components together in series. But there are other ways to connect components to each other.
• When two resistors are connected together as shown in the photograph below, they are said to be connected in parallel with each other. In particular, notice that the left-hand leads of the resistors are connected to each other, and the right-hand leads of the resistors are connected to each other. • To measure the total resistance, RT, of these two resistors, we would touch one of our meter's test leads to either resistor's left-hand lead, and touch the meter's other test lead to either resistor's right-hand lead. (Since the left-hand leads of the two resistors are connected by the breadboard, it doesn't matter which one we touch our test lead to. The same thing is true for the right-hand leads: they're connected to each other, so it doesn't mater which one we touch with our meter's test lead.)
• In the same way, we could connect two capacitors in parallel, or two inductors in parallel, and so forth.
##### Connecting More than Two Components in Parallel
• Extending these ideas, we could connect three, four, or more resistors in parallel. For example, here is a photograph showing three resistors connected in parallel on a breadboard. • Can you picture where you would touch the meter's test leads to measure the total resistance of these three resistors?
• In the same way, we could connect three or more capacitors in parallel, or three or more inductors in parallel, and so on.
• ##### Unit 2 Review
• This e-Lesson has covered some important topics, including:
• scientific notation and engineering notation
• resistor color codes, capacitor codes, and inductor codes
• connecting components in series
• connecting components in parallel.
• To finish the e-Lesson, take this self-test to check your understanding of these topics.
• Congratulations! You've completed the e-Lesson for this unit.