EMAT 6690: Using Computers in Mathematics Instruction
Article #2: Using Microsoft Excel¨ for the Arithmetic of Different Number Base
By Ryan Fox
My motivation for creating this came from my experience as my schoolÕs Math Team coach. A seemingly frequent question that was asked was to add or subtract two numbers in a different base: 431_{8} – 375_{8} being one such example. I was surprised how attached to the typical decimal number base my very bright and motivated high school students were that they could not think in different bases. My hope for this article is to use the computer software program to motivate ways to think of a decimal conversion of a number in a different base, so that the students would connect the number in a new number base to its decimal equivalent, and to get the students to begin arithmetic operations within the new number base.
The Excel spreadsheet program already does some work with different number bases. The program can convert numbers within three additional number bases: binary (base 2), octonary (base 8), and hexadecimal (base 16). My goal will be to try to increase the number bases that are available. My current intentions would be to do as much work as possible with bases between 3 and 9, with possible extensions to bases that are greater than 10.
Part I: Understanding the New Number Base
It was my impression that my students thought of place value very rigidly. The terminology ÒonesÓ, ÒtensÓ, ÒhundredsÓ, ÒthousandsÓ, and so on were so strictly constructed that the students did not necessarily make the connection between one and 10^{0}, tens and 10^{1}, hundreds and 10^{2}, thousands and 10^{3}, and so on. To me, making this connection is critical; once the student understands that the place value is really 10 raised to these powers, place value almost becomes arbitrary. Why is it that it is 10 raised to every power? Why not 8, or 3, or 12? Once I could get the students to think this way, I think I could explain new number bases this way.
It is at this point I want to introduce to the Microsoft Excel program. I am using Excel, although any spreadsheet program would suffice. The first connection I want make is to show how a different number base would make a decimal place value number. One could follow the link to the spreadsheet, while I provide a rationale to my constructed worksheet below:



Base
System= 

2 












Eighth 
Seventh 
Sixth 
Fifth 
Fourth 
Third 
Second 
First 
Decimal
# 
0 
1 
1 
1 
0 
0 
1 
1 
115 
In the top cell listed after ÒBase System =Ó, I wanted to give the students an opportunity to pick any base system that wanted to work with. (Currently my plan only works for number bases less than 10. The addition of letters to represent 11, 12, 13, etc. in a single place value is something I need to work on.) The students could then put the number from the different number system in the cells on the bottom row. I put the generic label ÒFirstÓ, ÒSecondÓ, ÒThirdÓ, etc., instead ÒOnesÓ, ÒTensÓ, ÒHundredsÓ, etc., not to confound the studentsÕ investigation of the new base with their firm understanding of the decimal place value system. Students could see what value a number in the new number base system has in relation to the decimal system. Students could type in the first eight digits of the new number into these cells. The spreadsheet program would generate the decimal equivalent to that number. In the example listed above, 1110011_{2} = 115 in the typical decimal form.
What is happening in this table is the spreadsheet program is using the number entered at the top of the table as the number base. The ÒFirstÓ digit used in this table is the ÒonesÓ place value, just like it is in the decimal place value system. However, looking at the ÒSecondÓ digit is where the spreadsheet takes the value entered by the student and raises it to the first power. In the example above, the ÒSecondÓ place value, represents the ÒtwosÓ place value. To convert to decimal values, having a 1 in the ÒtwosÓ column is the decimal equivalent of 2, which is found by multiplying the 1 the student entered by 2^{1} that is represented by the ÒSecondÓ column. The ÒThirdÓ digit represents the value of the base raised to the second power. Once again, in the above example, the ÒThirdÓ place value would be the ÒfoursÓ place, since 2^{2} = 4. This pattern continues for the remaining 5 place values in our spreadsheet. For example, in the example above, 1 in the ÒSixthÓ place corresponds to a decimal value of 32, since the ÒSixthÓ place refers to the base number being raised to the 5^{th} power, 2^{5} = 32 (in decimal form). Just like in the decimal place value of natural numbers, the number in the n^{th} position from the right represents the base number being raised to the (n – 1)^{st} power.
For clarification, and maybe to strengthen the understanding of place value, immediately underneath the ÒFirstÓ, ÒSecond, ÒThirdÓ, etc., columns, I placed the new place value that the students selected. In the example above, we could say a little more clearly now that the ÒFirstÓ column represents the Ones place, the ÒSecondÓ column represents the Two places, the ÒThirdÓ column represents the Fours place, and so down the line.
Part II: Converting Back to Decimal Numbers
What I am hoping to do in this portion of the spreadsheet is to strengthen the connection back and forth between the two number bases. As a result, the second worksheet is to take a number in the decimal system and give its equivalent form in the new number base.
An example of the spreadsheet printout is given below.



Base
System= 

8 






Decimal
Number = 

4000000 



Eighth 
Seventh 
Sixth 
Fifth 
Fourth 
Third 
Second 
First 
Decimal
# 
2097152 
262144 
32768 
4096 
512 
64 
8 
1 

1 
7 
2 
0 
4 
4 
0 
0 
4000000 
The first entry is to give the new number base system in which the students will be working. Currently, the only selections of numbers a student can enter are integer values between 2 and 9. (The number 10 could be a selection, however there would be nothing significant about this selection.) The entry below that is the decimal number that will be converted to the new number system.
Each of the columns underneath these two numbers represents the different place values for the new number base. The spreadsheet works for all decimal numbers that are less than the expression new number base raised to the ninth power. The ÒfirstÓ place value gives the ones place value of the decimal number converted to the new number base. For the rest of the place values, I had to work backwards. Each place value that listed as the n^{th} place value in the third row of the spreadsheet, is equal to the new base being raised to the (n – 1)^{st} power, which means for this particular example, the fifth place value represents the 4096s place, where 4096 = 8^{4}. Each corresponding entry in the fifth row of the spreadsheet represents the number of units from that place value. The decimal value 4,000,000 can be written in base eight as the following sum of the powers of 8:
Once we represented the decimal number four million in terms of the powers of 8, we can then write the number in terms of the new number base.
Part III: Learning to Add in the New Number Base
Now that the groundwork has been
laid for understanding the new numbers in terms of the familiar decimal numbers,
we can focus our attention to the arithmetic of the new number system. Before getting too involved in addition
of the two numbers, we might want to develop addition tables much in the same
manner addition tables were traditionally taught. Students have a familiarity that in the decimal system 5 + 4
= 9, but what does five plus four equal in base six (5 + 4 = ?_{6})? At this point, I have introduced the
addition table for the new number system.
An example of such is given below:





Base # 
8 















+ 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
0 
0 
1 
2 
3 
4 
5 
6 
7 
* 
* 
1 
1 
2 
3 
4 
5 
6 
7 
10 
* 
* 
2 
2 
3 
4 
5 
6 
7 
10 
11 
* 
* 
3 
3 
4 
5 
6 
7 
10 
11 
12 
* 
* 
4 
4 
5 
6 
7 
10 
11 
12 
13 
* 
* 
5 
5 
6 
7 
10 
11 
12 
13 
14 
* 
* 
6 
6 
7 
10 
11 
12 
13 
14 
15 
* 
* 
7 
7 
10 
11 
12 
13 
14 
15 
16 
* 
* 
8 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
9 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
This table was designed to look like the familiar addition table for adding two singledigit decimal numbers. Those numbers are represented in bold in the above table. If the number in the entry of the table is at least as big as the new number base, then an asterisk (*) is placed in the corresponding entry. For this particular table, any entry that is either the 8 or 9 row or 8 or 9 column has an asterisk since 8 = 8 and 9 > 8. At this point we will add the singledigit numbers of the new number base together. The sum is also given in the new number base. As a result, while most students would be able to recite the fact that 3 + 7 = 10 in the decimal number system the sum would appear to have a different result in the new number base of 8: 3_{8} + 7_{8} = 12_{8}. If a student were to have particular difficulties with the addition number system, a suggestion would be to have the student figure the sum in the familiar decimal form then use a preceding worksheet to convert that decimal number into the new number base. This particular worksheet will be useful when the student is able to determine the sum of multidigit numbers in the new number base.
Part IV: Learning to Multiply in the New Number Base
Once addition made sense to the students, I wanted to move to multiplication. I would believe, based on the level of the students I am intending for these spreadsheets, that subtraction would naturally follow from addition. Also, in my progression of spreadsheets, I wanted the multiplication table to follow the addition table so that the tables were presented before any calculations were performed.
I wanted to design a multiplication table that reminded the students of the multiplication of the decimal numbers they were familiar with. Like the addition spreadsheet in the previous part, we will indicate when the numbers in the table are too
large for the new number base. One example of a multiplication table is provided below.
The multiplication table provides an additional challenge compared to the addition table. In the addition table, the only possible value for the second digit was 1; in the multiplication table, all numbers in that base are possible for the second digit.





Base # 
9 















* 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
* 
1 
0 
1 
2 
3 
4 
5 
6 
7 
8 
* 
2 
0 
2 
4 
6 
8 
11 
13 
15 
17 
* 
3 
0 
3 
6 
10 
13 
16 
20 
23 
26 
* 
4 
0 
4 
8 
13 
17 
22 
26 
31 
35 
* 
5 
0 
5 
11 
16 
22 
27 
33 
38 
44 
* 
6 
0 
6 
13 
20 
26 
33 
40 
46 
53 
* 
7 
0 
7 
15 
23 
31 
38 
46 
54 
62 
* 
8 
0 
8 
17 
26 
35 
44 
53 
62 
71 
* 
9 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
Part V: Adding Multidigit Numbers
The next thing I wanted to focus on was the addition of any two numbers in the new number base, provided that the numbers are eight digits in length or less. I chose eight digits because of the size of the cells I selected to fit on one page. Additionally, I believe the concept could easily be extended to larger numbers without being addressed in this spreadsheet.




Base
System= 
5 













Eighth
Place 
Seventh
Pl. 
Sixth
Place 
Fifth
Place 
Fourth
Place 
Third
Place 
Second
Pl. 
First
Place 

78125 
15625 
3125 
625 
125 
25 
5 
1 
1st
Number 
2 
3 
4 
0 
1 
1 
3 
4 
2nd
Number 
3 
1 
2 
4 
3 
2 
1 
3 



























Sum
1st Pl. 
0 
4 
1 
4 
4 
3 
4 
2 
Sum
2nd Pl. 
1 
0 
1 
0 
0 
0 
0 
1 
After
CarryingÉ 








1 
0 
0 
1 
4 
4 
3 
0 
2 
0 
0 
1 
0 
0 
0 
0 
1 
0 
Final
Sum 








Ninth
Place 
Eighth
Place 
Seventh
Pl. 
Sixth
Place 
Fifth
Place 
Fourth
Place 
Third
Place 
Second
Pl. 
First
Place 
390625 
78125 
15625 
3125 
625 
125 
25 
5 
1 
1 
1 
0 
1 
4 
4 
4 
0 
2 
The first choice the student has to make is the value of the new number base. Right now, the spreadsheet is set up for number bases less than 10. I believe that bases greater than 10 could be created using logic statements. After the selection of the new number base, I give the student the opportunity to enter the two addends that they would like to compute. Right now, a student could insert a number larger than the base value and it would still compute the sum. Above the two addends, I have given the student a chance to see the new place value. This way, the student could see the new place value and hopefully visualize each place value in terms of the new number system, and not ÒonesÓ, ÒtensÓ, ÒhundredsÓ, and so forth. I made each column colorcoordinated so that the student may see the place values in the work that the spreadsheet is doing for them.
Once the student has entered the place value and the addends, the spreadsheet program pretty much does the rest. The resulting output shows the steps that are necessary to arrive at the final sum. The first calculation that is being performed is the sum of the two addends being represented in terms of the new number base. Referring back to the addition table in the earlier spreadsheet can check the sums. A problem for this spreadsheet that was not a problem in the addition table spreadsheet is how this particular spreadsheet handles ÒcarryingÓ. In the example from above, if we look at the ÒonesÓ 4_{5} + 3_{5} =12_{5}, while this is not a problem in the addition table, we must account for the additional five that this sum has created. We must move the one additional five to the next place value. In this spreadsheet, we have represented 12_{5} as 2 ÒonesÓ (in the yellow box) and 1 ÒfiveÓ (in the teal box immediately below the yellow box). The staggering of colors in the second group of boxes represent the ÒcarryingÓ that occurs when the sum of the two addends is greater than the number base. At this point, we must add a second time for the ÒcarryingÓ that took place in the previous step. It is possible that the ÒcarryingÓ in the previous step might yield a number larger than the base; once again, we have to perform a carrying step. In this example, if we look at the ÒonesÓ place we have 4_{5} + 3_{5} = 12_{5}, which means we have 2 in the ÒonesÓ place, and an additional 1 in the ÒfivesÓ place. When we go to add the ÒfivesÓ place, we are adding 3 and 1 from the addends, and then we will also add 1 from the ÒcarryingÓ in the ones place. We now have 3_{5} + 1_{5} +1_{5} = 10_{5}, meaning that our carrying caused a secondary carrying. This explains the third group of boxes we have. The method works exactly the previous set of boxes, but this group accounts for the additional carrying that might occur when the sum of the two addends is less than the value of the base and the carrying, but the carrying makes the sum at least as much as the value of the base.
The final answer is presented at the bottom of the spreadsheet outlined in a box. The place values for the digits in the sum are provided immediately above corresponding digits. I did this to help encourage the students who may have a difficult time understanding this concept a quicker way to verify the solution by converting back to the familiar decimal system. HereÕs what happens if a student were to make such a comparison:
All of the conversions could be verified by using earlier spreadsheets in this program.
Part VI: Subtracting Multidigit Numbers
Now we want to handle subtraction. While we were concerned with carrying in the addition spreadsheet, we must focus our attention to borrowing. Just like with addition, we start by declaring our number base and then creating the two numbers we will subtract. In this spreadsheet, the place values for each digit are given to help facilitate the transition from the decimal number system to the new number base.




Base
System= 
5 













Eighth
Place 
Seventh
Pl. 
Sixth
Place 
Fifth
Place 
Fourth
Place 
Third
Place 
Second
Pl. 
First
Place 

78125 
15625 
3125 
625 
125 
25 
5 
1 
1st
Number 
4 
1 
0 
1 
0 
1 
0 
1 
2nd
Number 
3 
4 
4 
4 
4 
4 
4 
4 



























Diff.
1st Pl. 
1 
2 
1 
2 
1 
2 
1 
2 
Diff.
2nd Pl. 
0 
1 
1 
1 
1 
1 
1 
1 



















0 
1 
0 
1 
0 
1 
0 
2 
In creating the spreadsheet for the operation of subtraction, we have to figure out a way to account for borrowing. In addition, we had to be mindful of carrying. In both of these instances, the operation involve one place value extends to a second place value. If the value of the minuend is less than the value of the Òfirst numberÓ, then we need to account for the ÒborrowingÓ from the next higher place value to create a value that is larger than the minuend, allowing us to make the subtraction of the two integers. To account for this in the Excel spreadsheet, we created a logical statement that said if our Òsecond numberÓ is larger than our Òfirst numberÓ then reduced the value of the next higher place value by 1. To find the difference between the two numbers, we will subtract the Òfirst numberÓ (including any borrowed value) from the Òsecond numberÓ, which is the row listed in the spreadsheet as ÒDiff. 1^{st} Pl.Ó. Notice that the place values in the difference have the corresponding colors as the place values in the subtraction problem. Had taking the difference required any ÒborrowingÓ from the next higher place value, a Ò1Ó was placed in the row marked ÒDiff. 2^{nd} Pl.Ó to signify as such. The colors of the boxes in that row are staggered to emphasize that. The final difference, represented in the final row of the spreadsheet, shows the difference after the ÒborrowingÓ had taken place. Once again, the colors of the place values in the difference correspond to the colors of the boxes in the original numbers.
If additional reinforcement were needed, we could convert the numbers back to the decimal number system and solve the subtraction problem as the student understands.
Part VII: Multiplying Multidigit Numbers
The selection of multiplying a threedigit number by another threedigit number was completely arbitrary. The process to create my desired result was rather involved.


Base
Number= 
7 






















Third 
Second 
First 




1st
Number 
1 
6 
2 




2nd
Number 
0 
5 
2 



















Partial
Products 
2 
15 
4 






5 
42 
13 





0 
0 
0 

















0 
4 





1 
5 





0 
2 







1 
3 





4 
2 





0 
5 







0 
0 





0 
0 





0 
0 














0 
0 
2 
6 
1 
4 


0 
0 
1 
0 
1 
0 


0 
1 
3 
0 
1 
4 








The first thing I wanted to start with was to find the product of each digit separately. I included this portion of the spreadsheet to make the connection back to the multiplication table from an earlier worksheet. These products are located in the 3 x 3 array next to ÒPartial ProductsÓ.
The key for my developing the product of multidigit numbers was to have the place values for the partial products in the correct location and to guarantee that any values that had to be ÒcarriedÓ were properly considered. In the ones values, the place value highlighted in yellow, the only value that must be checked is the product of the ones place of each digit. All that must be accounted for in the place value in the final product is the remainder when the base number divides the product. Once we move to the second place value, the sevens place in the above example, the process of determining the product becomes more challenging. There are values that contribute to the sevens place value: if the partial product was greater than the number base, the remainder when the base number divides the product of the second digit of the first number by the first digit of the second number, and the remainder when the base number divides the product of the first digit of the first number by the second digit of the second number. Those three values are added together, for which it is possible that the sum of these three numbers is greater than the value of the number base. As a result, the value of the second place value in the final product is equal to the remainder when the base value divides the sum of the three numbers. If the value of the sum is greater than the number base, then there will be some ÒcarryingÓ involved. This is represented in our spreadsheet in the second column of our final answer with the red box that represents the second place value and the maroon box underneath it to represent the value that will be added to the maroon box for the third place value. I continued this process until I arrived at my final result.
Part VIII: Potential Extensions: Bases Larger than 10
I believe that this spreadsheet could be extended to doubledigit bases. The Excel program already comes understanding the hexadecimal, or base 16, number system. Extending to doubledigits would involve, in my opinion, creating logic statements so that anytime a number larger than 10 were needed, it would be converted to a letter. For example, if we wanted to create work for base 11, we would reserve the letter ÒAÓ to represent the decimal number 10. We could create a spreadsheet page that could help facilitate a conversion from base 11 back to base 10. Likewise, if we wanted base 12, we would reserve ÒAÓ for the decimal number 10 and ÒBÓ for the decimal number 11. We can continue this pattern up to base numbers of 36. If we wanted larger base numbers, we might be tempted to include Greek letters. The possibilities for extensions are nearly limitless.