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Contemporary Mathematics

4.5 Multiplication and Division in Base Systems

Contemporary Mathematics4.5 Multiplication and Division in Base Systems

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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
A woman is solving an equation on a white board.
Figure 4.6 The processes for multiplication and division are the same for arithmetic in any bases. (credit: modification of work “NCTR Intern Claire Boyle” by Danny Tucker/U.S. Food and Drug Administration, Public Domain)

Learning Objectives

After completing this section, you should be able to:

  1. Multiply and divide in bases other than 10.
  2. Identify errors in multiplying and dividing in bases other than 10.

Just as in Addition and Subtraction in Base Systems, once we decide on a system for counting, we need to establish rules for combining the numbers we’re using. This includes the rules for multiplication and division. We are familiar with those operations in base 10. How do they change if we instead use a different base? A larger base? A smaller one?

In this section, we use multiplication and division in bases other than 10 by referencing the processes of base 10, but applied to a new base system.

Multiplication in Bases Other Than 10

Multiplication is a way of representing repeated additions, regardless of what base is being used. However, different bases have different addition rules. In order to create the multiplication tables for a base other than 10, we need to rely on addition and the addition table for the base. So let’s look at multiplication in base 6.

Multiplication still has the same meaning as it does in base 10, in that 4×64×6 is 4 added to itself six times, 4×6=4+4+4+4+4+44×6=4+4+4+4+4+4.

So, let’s apply that to base 6. It should be clear that 0 multiplied by anything, regardless of base, will give 0, and that 1 multiplied by anything, regardless of base, will be the value of “anything.”

Step 1: So, we start with the table below:

* 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4
3 0 3
4 0 4
5 0 5

Step 2: Notice 2×2=42×2=4 is there. But we didn’t hit a problematic number there (4 works fine in both base 10 and base 6). But what is 2×32×3? If we use the repeated addition concept, 2×3=2+2+2=4+22×3=2+2+2=4+2. According to the base 6 addition table (Table 4.4), 4+2=104+2=10. So, we add that to our table:

* 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 10
3 0 3 10
4 0 4
5 0 5

Step 3: Next, we need to fill in 2×42×4. Using repeated addition, 2×4=2+2+2+2=10+2=122×4=2+2+2+2=10+2=12 (if we use our base 6 addition rules). So, we add that to our table:

* 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 10 12
3 0 3 10
4 0 4 12
5 0 5

Step 4: Finally, 2×5 =2+2+2+2+2=12+2=142×5 =2+2+2+2+2=12+2=14. And so we add that to our table:

* 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 10 12 14
3 0 3 10
4 0 4 12
5 0 5 14

Step 5: A similar analysis will give us the remainder of the entries. Here is 4×54×5 demonstrated: 4×5 =4+4+4+4+4=12+12+4=24+4=324×5 =4+4+4+4+4=12+12+4=24+4=32.

This is done by using the addition rules from Addition and Subtraction in Base Systems, namely that 4+4=124+4=12, and then applying the addition processes we’ve always known, but with the base 6 table (Table 4.4). In the end, our multiplication table is as follows:

* 0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 2 3 4 5
2 0 2 4 10 12 14
3 0 3 10 13 20 23
4 0 4 12 20 24 32
5 0 5 14 23 32 41
Table 4.8 Base 6 Multiplication Table

Notice anything about that bottom line? Is that similar to what happens in base 10?

To summarize the creation of a multiplication in a base other than base 10, you need the addition table of the base with which you are working. Create the table, and calculate the entries of the multiplication table by performing repeated addition in that base. The table needs to be drawn only the one time.

Example 4.38

Creating a Multiplication Table for a Base Lower Than 10

Create the multiplication table for base 7.

Your Turn 4.38

1.
Create the multiplication table for base 4.

Example 4.39

Creating a Multiplication Table for a Base Higher Than 10

Create the multiplication table for base 12.

Your Turn 4.39

1.
Create the multiplication table for base 14.

The multiplication table in base 2 below is as minimal as the addition table in the solution for Table 4.6. Since the product of 1 with anything is itself, the following multiplication table is found.

* 0 1
0 0 0
1 0 1
Table 4.10 Base 2 Multiplication Table

As with the addition table, we can use the multiplication tables and the addition tables to perform multiplication of two numbers in bases other than base 10. The process is the same, with the same carry rules and placeholder rules.

Example 4.40

Multiplying in a Base Lower Than 10

  1. Calculate 456×246456×246.
  2. Calculate 1012×11021012×1102.

Your Turn 4.40

1.
Calculate 43 6 × 52 6 .
2.
Calculate 11101 2 × 11 2 .

Summarizing the process of multiplying two numbers in different bases, the multiplication table is referenced. Using that table, the multiplication is carried out in the same manner as it is in base 10. The addition rules for the base will also be referenced when carrying a 1 or when adding the results for each digit’s multiplication line.

Example 4.41

Multiplying in a Base Higher Than 10

Calculate 3A12×74123A12×7412.

Your Turn 4.41

1.
Calculate B 3 12 × 47 12 .

Division in Bases Other Than 10

Just as with the other operations, division in a base other than 10, the process of division in a base other than 10 is the same as the process when working in base 10. For instance, 72÷9=872÷9=8 because, we know that 9×8=729×8=72. So, for many division problems, we are simply looking to the multiplication table to identify the appropriate multiplication rule.

Example 4.42

Dividing with a Base Other Than 10

  1. Calculate 146÷56146÷56.
  2. Calculate 5A12÷7125A12÷712

Your Turn 4.42

1.
Calculate 10 6 ÷ 3 6
2.
Calculate 50 12 ÷ A 12 .

Errors in Multiplying and Dividing in Bases Other Than Base 10

The types of errors encountered when multiplying and dividing in bases other than base 10 are the same as when adding and subtracting. They often involve applying base 10 rules or symbols to an arithmetic problem in a base other than base 10. The first type of error is using a symbol that is not in the symbol set for the base.

Example 4.43

Identifying an Illegal Symbol in a Base Other Than Base 10

Explain the error in the following calculation, and determine the correct answer:

46×26=8646×26=86

Your Turn 4.43

1.
Explain the error in the following calculation and determine the correct answer: 13 4 × 21 4 = 54 4

The second type of error is using a base 10 rule when the numbers are not in base 10. For instance, in base 17, 617×917=5417617×917=5417 would be incorrect, even though in base 10, 6×9=546×9=54. That rule doesn’t apply in base 17.

Example 4.44

Identifying an Error in Arithmetic in a Base Other Than Base 10

Explain the error in the following calculation. Determine the correct answer:

1812×712=126121812×712=12612

Your Turn 4.44

1.
Explain the error in the following calculation. Determine the correct answer: 49 14 × 9 14 = 441 14

Check Your Understanding

28.
To create the multiplication table for a given base, what should be used?
29.
What are the differences between multiplying in base 10 and multiplying in a different base?
30.
When dividing in a base other than base 10, what table is referenced?
31.
Compute 24 6 × 53 6 .
32.
Compute 32 14 ÷ 4 14 .
33.
How do you know an error has occurred in a base 5 multiplication question if the answer obtained was 285?
34.
What are two common ways to determine an error is committed when computing in s base other than base 10?

Section 4.5 Exercises

For the following exercises, create the multiplication table for the given base.
1 .
base 5
2 .
base 3
3 .
base 16 (Hint: Use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.)
4 .
base 2
For the following exercises, perform the indicated base 6 operation.
5 .
4 6 × 3 6
6 .
14 6 × 5 6
7 .
31 6 × 3 6
8 .
43 6 × 34 6
9 .
532 6 × 23 6
10 .
254 6 × 143 6
11 .
20 6 ÷ 3 6
12 .
23 5 ÷ 5 6
For the following exercises, perform the indicated base 12 operation.
13 .
5 12 × 6 12
14 .
3 12 × A 12
15 .
34 12 × 7 12
16 .
76 12 × B 12
17 .
59 12 × 1 A 12
18 .
A 1 12 × 36 12
19 .
53 12 ÷ 9 12
20 .
2 B 12 ÷ 7 12
21 .
Explain two ways to detect an error in arithmetic in bases other than base 10.
22 .
Explain the error in the following calculation: 15 12 × 7 12 = 105 12
23 .
Explain the error in the following calculation: 45 8 × 6 8 = 94 8 .
24 .
In base 10 multiplication, there are 100 multiplication rules plus a rule for carrying a number. How many multiplication rules are there for base 6?
25 .
In base 10 multiplication, there are 100 multiplication rules plus a rule for carrying a number. How many multiplication rules are there for base 14?
26 .
In base 10 multiplication, there are 100 multiplication rules plus a rule for carrying a number. How many multiplication rules are there for base 2?
27 .
Consider the answers from Exercise 24 and Exercise 26. Which base do you think would be more efficient: base 10, base 6, or base 2?
For the following exercises, use the multiplication table that you created from Exercise 4 to perform the indicated base 2 operations.
28 .
11 2 × 11 2
29 .
101 2 × 10 2
30 .
11011 2 × 1011 2
31 .
1011 2 × 1010101 2
32 .
Convert 1011 2 and 1010101 2 to base 10. Then multiply those base 10 numbers. Next, convert the answer you got for Exercise 31 to base 10. Do these numbers match?
For the following exercises, use the multiplication table that you created from Exercise 3 to perform the indicated base 16 operations.
33 .
19 16 × 5 16
34 .
3 B 16 × A 16
35 .
25 16 × 16 16
36 .
C 16 ÷ 4 16
For the following exercises, explain how you know an error was committed without performing the operation in the given base.
37 .
43 5 × 32 5 = 126 5
38 .
5 14 × 9 14 = 45 14
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