Base 5 Calculation

Table of Contents PREFACE iii
1. Introduction 1
2. What is base 5 number 1
3. How to read a Multiplication TABLE 2
4. How to read an Addition TABLE 3
5. Process of multiplying two base 5 number 3
APPENDIX A: LIST OF FIGURES 6
APPENDIX B: REFERENCE PAGE 6

ii PREFACE
Thanks the help of my professor Nancy Acemian and the students who contribute their ideas when I write this instruction.
Wenjun zhu
February 2008

iii

1. Introduction

In reality, people use base 10 numbers quite often in different fields, such as shopping, calculation. However, a few people hear about base 5 number which is quite similar to base 10 number in the calculation. This set of instruction is to explain what is base 5 number and how to multiply a 3-digit positive base 5 number by a 2-digit positive base 5 number.

1
2
2. What is base 5 number
Base 5 numbers are the representation of quantity with symbols which are written using only 0, 1, 2, 3, 4 digits. 3. How to read a Multiplication Table
Let’s understand how to read the Base 5 multiplication TABLE. This table is quite similar to the regular decimals multiplication table. Grey shade cells of the first row and first column are operands .The blue shade cell is the operation. In this case, the operation is multiplication. White shade cells are results which store the answer of two operands. Light red shade cell is the sample. Please see in Figure 3-1. x 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 11 13
3 0 3 11 14 22
4 0 4 13 22 31

Figure 3-1 Multiplication Table
For example:
2 × 4 = 13 Here you look at 2 and go across the row, then look at four and go down the column. Where they intersect, that is the answer.

3
4. How to read an Addition Table
+ 0 1 2 3 4
0 0 1 2 3 4
1 1 2 3 4 10
2 2 3 4 10 11
3 3 4 10 11 12
4 4 10 11 12 13

Figure 4-1 Addition Table
The rule to read figure 4-1 is the same as to read the figure 3-1.
For example:
2 + 4 = 11. Here you look at 2 and go across the row, then look at 4 and go down the column. Where they intersect, that is the answer.

5. Process of multiplying two base 5 numbers
Now, let’s learn multiply a 3-digit base 5 number by a 2-digit base 5 number following these steps with the help of the Addition and Multiplication Table.
Step 1: write the 43 under the 342 aligning vertically for each digit, and draw a horizontal line under them. 3 4 2
× 4 3
---------
Step 2: Start with the ones-column, the multiplicand is 342 and ones-multiplier is 3.Perform the multiplication under the line. At this point, use the multiplication TABLE to do the operation.3 × 2 = 11.Eleven has two digits. Write its last digit,1, in the ones-column under the line, and write the “carry digit” which is 1 above the top digit of the next column: in this case the next column is the tens-column. 4 1 3 4 2
× 4 3
---------
1

Step 3: Perform 3×4 = 22.Suppose to write down the 2 under the line in the tens-column. However, there is still one carry digit,1, which added to 2 is equals to 3.So write down the 3 under the line in the tens-column, write the “carry digit” which is 2 on top of the hundreds-column. 2 1 3 4 2
× 4 3
---------
3 1

Step 4: Perform 3 × 3 =14. Note this is a special situation. Since 14 is a two digits number, write down the 1 in the thousands-column under the line temporarily. Add 4 with the previous carry digit,2, which equals to 11 with the help of the Addition Table. Write down 1 in the hundreds-column under the line, and write down the carry digit,1, on top of the thousands-column. Next, write down the sum of the thousands-column under the line. The sum is the leftmost digit,1, plus the carry digit,1. Thus, the result is 2. 1 2 1 1 2 1 3 4 2 3 4 2 × 4 3 × 4 3 --------- --------- 1 1 3 1 2 1 3 1

5
Step 5: Erase all the carry digits on top of the multiplicand. 3 4 2
× 4 3
---------
2 1 3 1
Step 6: Perform the multiplication which the multiplicand is 342 and the tens-multiplier is 4.Repeat the multiplication steps as above in the tens-row, under the ones-row, but shifted one column to the left. Then draw a horizontal line under the tens-row. Remember to write down the carry digit in correct columns correspond to the number in the tens-row. 1 3 1 1 3 1 3 4 2 3 4 2 × 4 3 × 4 3 ------------ ------------ 2 1 3 1 2 1 3 1 2 2 2 3 3 0 2 3 ------------ ------------
Step 7: Add 2131 and 3023 between the first and second lines. Write down the 1 in ones-column under the line. Then 3+3= 11, seen from figure 4-1.So,write down the 1 in the tens-column under the line, and carry digit,1, on top of the hundreds-column. After that, add all the digits in the hundreds-column including the carry digit, and write down the result 4 under the line. Finally, just write down 3,2 in their own column under the line, respectively. 1 2 1 3 1
+ 3 0 2 3 ------------ 3 2 4 1 1

6 Here is the final result. 3 4 2 × 4 3 ------------ 2 1 3 1 3 0 2 3 ------------ 3 2 4 1 1

APPENDIX A: LIST OF FIGURES
List of Figures
Fig. 3-1 Multiplication Table...................................................…........................... 2
Fig. 4-1 Addition Table................................................….......................................3

APPENDIX B: REFERENCE PAGE
REFERENCES
[1] Wikipedia, “Elementary arithmetic,” http://en.wikipedia.org/wiki/Elementary_arithmetic#Addition_algorithm
, (current February 13 ,2008).

References: [1] Wikipedia, “Elementary arithmetic,” http://en.wikipedia.org/wiki/Elementary_arithmetic#Addition_algorithm , (current February 13 ,2008).