Chapel Field Christian Schools
Most human cultures count in what is known as base ten or decimal. Some count in base twelve. However, because electrical circuits can only be on or off, computers must work in base two, or binary. The only numbers a computer can truly understand are 0 (off) and 1 (on). We are going to learn how to use various counting systems and then master the use of binary.
Power: synonym for exponent meaning the number of times we will multiply the base number.
Base: the number we are going to raise to a certain power or the number of digits in a counting system.
Place: the value of the position a digit has in the number.
Western mathematics is based on the base ten system. This means that the first place has a value of 100 * d, the second place has a value of 101 * d, and so on. The first place is known as the “ones” place, the second is known as the “tens” place, the third is known as the “hundreds” place, and the fourth is known as the “thousands” place. So a base ten number such as 3491 represents (103 * 3) + (102 * 4) + (101 * 9) + (100 * 1). It also could be seen as three sets of one thousand, four sets of one hundred, nine sets of ten, and one set of one.
Why do you think base ten has been favored by most cultures?
Raise your hand and have Mr. Olinda verify this checkpoint. Alternatively, you may record your screen as you complete each step from this checkpoint and submit the videos for credit.
Answer the following questions.
Raise your hand and have Mr. Olinda verify this checkpoint. Alternatively, you may record your screen as you complete each step from this checkpoint and submit the videos for credit.
Base twelve was the counting system used by some cultures such as the early English. However, there are some who propose that we go back to this system because of the overall benefits it would offer in mathematics. Read this page explaining how the duodecimal system works. Note that the author has gone beyond simply modifying the counting system and has developed duodecimal systems of measurement, time, and more.
Right now, we are only interested in learning how to count in base twelve. In duodecimal the first place is 120 * d, the second place is 121 * d, and so on. When we write duodecimal numbers, we use a and b to represent 10 and 11 so that we do not have a multi-digit number in a place. So the duodecimal number 1b37 converted to decimal represents (120 * 7) + (12 1 * 3) + (122 * 11) + (123 * 1), or 7 + 36 + 1584 + 1728, or 3355.
| Decimal Digits | Duodecimal Digits |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | a |
| 11 | b |
How would you convert from decimal to duodecimal (or any other system)? If you aren't sure, take a look at this resource on converting from decimal to duodecimal. But be prepared to explain at least one method verbally.
Raise your hand and have Mr. Olinda verify this checkpoint. Alternatively, you may record your screen as you complete each step from this checkpoint and submit the videos for credit.
Answer the following questions.
Raise your hand and have Mr. Olinda verify this checkpoint. Alternatively, you may record your screen as you complete each step from this checkpoint and submit the videos for credit.
Hexadecimal is used by computers to encode information using sixteen different digits. Watch this video for an overview of how the system works and then continue.
As you saw in the video, the first ten digits are the same in decimal and hexadecimal, but 10, 11, 12, 13, 14, and 15 are represented with the letters a, b, c, d, e, and f. Any time you need to indicate a number is hexadecimal, you can write it as follows: D816.
| Decimal Digits | Hexadecimal Digits |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | a |
| 11 | b |
| 12 | c |
| 13 | d |
| 14 | e |
| 15 | f |
Raise your hand and have Mr. Olinda verify this checkpoint. Alternatively, you may record your screen as you complete each step from this checkpoint and submit the videos for credit.
Answer the following questions. Be prepared to show your work for any of these problems.
Raise your hand and have Mr. Olinda verify this checkpoint. Alternatively, you may record your screen as you complete each step from this checkpoint and submit the videos for credit.
At the most basic level, computers operate using switches. Some of the first computers used mechanical switches that could be on or off. Modern computers use digital switches, called bits. Data storage, calculations, and communication all rely on on (1) or off (0), which means each less information can be transmitted in the same amount of space. This is offset by the fact that computers are able to calculate so much faster than humans.
We want to understand how binary counting works. But we also want to study how to convert from decimal to binary, and from hexadecimal to binary. This is because most humans use decimal, while computers use binary and hexadecimal. A firm understanding of the relationship between these three systems will help you going forward in computer science.
In the binary system, there are only two digits: 1 and 0. This means that we are working with a base of two digits. The first place is 20, the second is 21, the third is 22, and so on. Duodecimal and hexadecimal numbers are usually smaller than decimal because they contain more information per digit. Binary numbers will be much larger because each digit contains less information.
| Decimal Digits | Binary Digits |
|---|---|
| 0 | 0 |
| 1 | 1 |
The number 2810, which contains two sets of ten and eight sets of one, converts to 111002, which contains one set of 16, one set of eight, one set of four, zero sets of two, and zero sets of one. Watch the video below to see his method for doing decimal to binary conversions.
However, it's also important to learn how to convert directly from hexadecimal to binary. What we tend to do is convert from one alternative base to decimal, and then to another alternative base. But it is possible to convert directly from one system to another. Watch the video below and see if you can learn the basic method for converting from hexadecimal to binary and from binary to hexadecimal. Show your work as you follow along.
As you can see, using the block method allows you to convert very efficiently. Now, see if you can convert a three-digit number from hexadecimal to binary using this method. Show your work.
Raise your hand and have Mr. Olinda verify this checkpoint. Alternatively, you may record your screen as you complete each step from this checkpoint and submit the videos for credit.
Answer the following questions. Be prepared to show your work for any of these problems.
Raise your hand and have Mr. Olinda verify this checkpoint. Alternatively, you may record your screen as you complete each step from this checkpoint and submit the videos for credit.