Addition of Same and Different Numerical Bases
There are parts of Mathematics that sometimes seem to have little value. OK so some feel as though most of Mathematics has little or no value. I myself find the discipline to be intimately integrated in all parts of our daily life. I keep an open mind where other’s opinion matters. The study of numerical bases is one of those hard to appreciate topics.
As the world’s monetary currency becomes aligned with a Base 10 system we lose contact with this particular Mathematical study. The latest to incorporate this is the Euro. Money kept alive the relationship between different coins and their relative values. Different numerical bases were a natural part of our developing societies. Ancient Babylon had a Base 60 system. Other societies also had their own cultural preference too. Using a Base 10 system for financial transactions is a somewhat recent preference. We prefer it because it has a natural symbol counting system that clearly represents an easy to understand value.
Still, knowing how to use different base systems is an early learning technique that encourages Mathematical understanding. Being able to manipulate a number is what everyone believes Mathematics is about. So, let’s use some of our Mathematical operations and compute some numbers.
Adding Two Numbers With the Same Base
In order to sum two numbers together they need to have the same base. As we add the two numbers together we simplify our answer by shifting full place values with their appropriate higher level value. Here is an example where we add two Base 4 numbers together: 132 + 233 = 365.
Unfortunately this is not a correct answer in Base 4. We need to simplify our answer. To do this we reduce from RIGHT to LEFT. Remember each place value needs to move 1 in the next left hand box for each complete count of 4. So, since 5 = 4 + 1 we need to add 1 to the next place value to the left of the right hand place. Leave the 1 because it is not a complete 4. Our first simplification then is 371.
OK, so we fixed the far right hand place value. Now we need to fix the middle place value. Since 7 = 4 + 3 we will replace the complete 4 in the middle value with a 1 in the next left hand place value. The 3 stays right where it is because it is not yet a 4. Our second simplification then gives us 431.
Our next simplification is to see that there is a complete grouping of 4 in the third place value from the right. But, remember that 4 = 4 + 0. Just as in the other two simplification steps we will replace the 4 with a 1 in the next left hand place value. We still need to keep a 0 in the third place because that is how many there are now in the third spot. We now end up with the answer of 1031. This means that when we are adding 132 + 233 in Base 4 our correct answer is 1031.
Homework Set #1
Homework part 1: Both numbers in this section are the same base.
1) Base 5: 243 + 1223 = __________
2) Base 3: 111 + 121 + __________
3) Base 4: 223 + 111 = __________
4) Base 4: 3221 + 1003 = __________
5) Base 7: 531 + 444 = __________
6) Base 2: 1101 + 111 = __________
7) Base 2: 1001 + 1100 = __________
8) Base 8: 563 + 567 = __________
9) Base 3: 2210 + 1122 = __________
10) Base 6: 4421 + 4555 = __________
Now, I made some of the homework above on the hard side for someone just learning to use different place value systems. You may wish to construct some easier problems if you feel your students need more of a running start. There is just not a great deal of room here for a full homework set. If there were I would start with easier problems with fewer place values to ease into the work. It does help to start out slow and proceed to difficult.
Adding Two Numbers With Different Bases
The next step is to compute the sum of two numbers that have different bases. This is a type of problem man has been confronted with for thousands of years. Here is an example. A young man who raises sheep for a living has fallen in love with a beautiful young lady. The young lady comes from a family that makes its living by raising cows. Her father presents the young father with 4 cows as her dowry. Since the young man raises sheep he must first convert the new cows in to sheep. Then he will have a nice herd of sheep to manage that will provide for his new family.
Here is the process through an example. Compute the sum of the Base 4 number 321 added to the Base 6 number 235. The first step is to know which of the two numbers to convert to which base. Be sure to make sure that the directions tell you. For this example I want the answer to be in Base 6. That means our first step will be to change the Base 4 number 321 to the equivalent Base 6 number. Please visit my first blog on changing a number from one base to the next by clicking here. I have also computed the changing of a base in the images I included for this hub.
The Base 4 number 321 is equal to 133 in Base 6. Our problem now becomes in Base 6: 133 + 235 = 368. This reduces to the correct expression as 412. I had to take complete 6’s from a lower place value and add a 1 to the next higher place value when I reduced. We had to reduce because both the 6 and the 8 we greater than or equal to 6 in the first answer we computed.
Homework Set #2
Homework 2: The first number is Base 4 and the second number is Base 7.
1) 333 + 621 = __________ in Base 4
2) 1112 + 342 = __________ in Base 7
3) 2231 + 1132 = __________ in Base 4
4) 1100 + 554 = __________ in Base 7
5) 1213 + 1532 = __________ in Base 7
Conclusion and Answers to Homework Sets
Granted this homework set is more complex. While middle school students will be able to perform the first homework assignment they will have difficulty performing this second homework set. Some of the brighter students may be able to compute these problems with some difficulty. This should be a good lesson for high school students. The problems in this second homework set are considerably more difficult. They will also require the student to know several place values we had not yet presented.
Your students may need one more reason to understand this topic today. Here is one that is in the news frequently. This is the quantum computer computational speed is considerably faster than conventional systems. Do you know why? A conventional computer is based on a Base 2 processing system. A switch can be either on or off; 0 or 1. It is known as a binary system. A quantum computer is at least a Base 3. A quantum switch can be either on or off or it can be both on and off at the same time. Line up the place values for Base 2 underneath the same place values for Base 3. You can see that that the magnitude of the placement values in Base 3 very quickly outdistance those for Base 2.
Knowing how to compute with different bases is still an important computational skill. It is another way to utilize operational skills as well as how to reduce/rewrite a number into an equivalent expression that is correct for the system. We still have to learn the other operations such as subtraction as well as multiplication and division.
Homework set 1: 1) 2021, 2)1002, 3) 1000, 4) 10230, 5) 1305, 6) 10100, 7) 10101, 8) 1352, 9) 11102, 10) 13420
Homework set 2: 1) 11310, 2) 524, 3) 21030, 4) 1030, 5) 2040