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Contemporary Mathematics

4.2 Early Numeration Systems

Contemporary Mathematics4.2 Early Numeration Systems

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
Babylonian numerals are written on a clay tablet.
Figure 4.3 Babylonians used clay tablets for writing and record keeping. (credit: modification of work by Osama Shukir Muhammed Amin FRCP(Glasg), CC BY 4.0 International)

Learning Objectives

After completing this section, you should be able to:

  1. Understand and convert Babylonian numerals to Hindu-Arabic numerals.
  2. Understand and convert Mayan numerals to Hindu-Arabic numerals.
  3. Understand and convert between Roman numerals and Hindu-Arabic numerals.

Each culture throughout history had to develop its own method of counting and recording quantity. The system used in Australia would necessarily differ from the system developed in Babylon that would, in turn, differ from the system developed in sub-Saharan Africa. These differences arose due to cultural differences. In nearly all societies, knowing the difference between one and two would be useful. But it might not be useful to know the difference between 145 and 167, as those quantities never had a practical use. For example, a shepherd likely didn't manage more than 100 sheep, so quantities larger than 100 might never have been encountered. This can even be seen in our use of the term few, which is an inexact quantity that most would agree means more than two. However, as societies became more complex, as commerce arose, as military bodies developed, so did the need for a system to handle large numbers. No matter the system, the issues of representing multiple values and how many symbols to use had to be addressed. In this section, we explore how the Babylonians, Mayans, and Romans addressed these issues.

Understand and Convert Babylonian Numerals to Hindu-Arabic Numerals

The Babylonians used a mix of an additive system of numbers and a positional system of numbers. An additive system is a number system where the value of repeated instances of a symbol is added the number of times the symbol appears. A positional system is a system of numbers that multiplies a “digit” by a number raised to a power, based on the position of the “digit.”

The Babylonian place values didn’t use powers of 10, but instead powers of 60. They didn’t use 60 different symbols though. For the value 1, they used the following symbol:

Babylonian numeral 1 is displayed.

For values up to 9, that symbol would be repeated, so three would be written as

Babylonian numeral 3 is displayed.Babylonian numeral 3 is displayed.Babylonian numeral 3 is displayed.

To represent the quantity 10, they used

Babylonian numeral 10 is displayed.

For 20, 30, 40, and 50, they repeated the symbol for 10 however many times it was needed, so 40 would be written

Babylonian numeral 40 is displayed.Babylonian numeral 40 is displayed.Babylonian numeral 40 is displayed.Babylonian numeral 40 is displayed.

When they reached 60, they moved to the next place value. The complete list of the Babylonian numerals up to 59 is in Table 4.1.

Babylonian numerals 1 to 50 are displayed.
Table 4.1 Babylonian Numerals

You can see how Babylonians repeated the symbols to indicate multiples of a value. The number 6 is 6 of the symbol for 1 grouped together. The symbol for 30 is three of the symbols for 10 grouped together. However, their system doesn’t go past 59. To go past 59, they used place values. As opposed to the Hindu-Arabic system, which was based on powers of 10, the Babylonian positional system was based on powers of 60. You should also notice there is no symbol for 0, which has some impact on the number system. Since the Babylonian number system lacked a 0, they didn’t have a placeholder when a power of 60 was absent. Without a 0, 101, 110, and 11 all look the same. However, there is some evidence that the Babylonians left a small space between "digits" where we would use a 0, allowing them to represent the absence of that place value. To summarize, the Babylonian system of numbers used repeating a symbol to indicate more than one, used place values, and lacked a 0.

Who Knew?

Invention of 0

The idea of 0 is not a natural one. Most cultures failed to recognize the need for a 0. If someone asked a farmer in 300 B.C.E. how many cows they had, but they had none, they would not answer "zero." They’d say “I don’t have any” and be done with it. It wasn’t until roughly 3 B.C.E. that 0 appeared in Mesopotamia. It was independently discovered (or invented!) in the Mayan culture around 4 C.E. it made its appearance in India in the 400s C.E., and began to spread at that point. It wasn’t developed earlier mostly because positional systems were not yet fully developed. Once positional systems arose, the need to represent a missing power had to be addressed.

So how do we convert from Babylonian numbers to Hindu-Arabic numbers? To do so, we need to use the symbols from Table 4.1, and then place values based on powers of 60. If you have nn digits in the Babylonian number, you multiply the first “digit” by 60 raised to one less than the number of “digits.” You then continue through the “digits,” multiplying each by 60 raised to a power that is one smaller. However, be careful of spaces, since they represent a zero in that place.

Example 4.4

Converting Two-Digit Babylonian Numbers to Hindu-Arabic Numbers

Convert the Babylonian number

Babylonian numeral 4 is displayed.Babylonian numeral 27 is displayed.

into a Hindu-Arabic number.

Your Turn 4.4

1.
Convert the Babylonian number Babylonian numeral 21 is displayed.Babylonian numeral 9 is displayed. into a Hindu-Arabic number.

Example 4.5

Converting Three-Digit Babylonian Numbers to Hindu-Arabic Numbers

Convert the Babylonian number

Babylonian numeral 13 is displayed.Babylonian numeral 8 is displayed.Babylonian numeral 54 is displayed.

into a Hindu-Arabic number.

Your Turn 4.5

1.
Convert the Babylonian number Babylonian numeral 11 is displayed.Babylonian numeral 42 is displayed. Babylonian numeral 16 is displayed. into a Hindu-Arabic number.

Example 4.6

Converting Four-Digit Babylonian Numbers to Hindu-Arabic Numbers

Convert the Babylonian number

Babylonian numeral 12 is displayed.Babylonian numeral 42 is displayed.Babylonian numeral 39 is displayed.

into a Hindu-Arabic number.

Your Turn 4.6

1.
Convert the Babylonian number Babylonian numeral 28 is displayed. Babylonian numeral 16 is displayed. Babylonian numeral 43 is displayed. into a Hindu-Arabic number.

Who Knew?

The Legacy of Babylonian System

The Babylonian system can still be seen today. An hour is 60 minutes, and a minute is 60 seconds. Additionally, when measuring angles in degrees, each degree can be split into 60 minutes (1/60th of a degree) and 60 seconds (1/60th of a minute).

Understand and Convert Mayan Numerals to Hindu-Arabic Numerals

The Mayans employed a positional system just as we do and the Babylonians did, but they based their position values on powers of 20 and they had a dedicated symbol for zero. Similar to the Babylonians, the Mayans would repeat symbols to indicate certain values. A single dot was a 1, two dots were a 2, up to four dots. Then a five was a horizontal bar. The horizontal bars could be used three times, since the fourth horizontal bar would make a 20, which was a new position in the number. The 0 was a special picture, which appears like a turtle lying on its back. The shell would then be "empty," so maybe that’s why the symbol was 0. The complete list is provided in Table 4.2. Another feature of Mayan numbers was that they were written vertically. The powers of 20 increased from bottom to top.

Mayan numerals 0 to 19 are displayed.
Table 4.2 Mayan Numerals

To summarize, the Mayan system of numbers used repeating symbol to indicate more than one, used place values, and employed a 0. So how do we convert from Mayan numbers to Hindu-Arabic numbers? To do so, we need to use the symbols from Table 4.2 and then place values based on powers of 20. If you have nn digits in the Mayan number, you multiply the first “digit” by 20 raised to one less than the number of “digits.” You then continue through the “digits,” multiplying each by 20 raised to a power that is one smaller than the previous power. Fortunately, there is an explicit 0, so there is no ambiguity about numbers like 110, 101, and 11.

Example 4.7

Converting Two-Digit Mayan Numbers to Hindu-Arabic Numbers

Convert the Mayan number

Mayan numeral 15 is displayed.


Mayan numeral 9 is displayed.

into a Hindu-Arabic number.

Your Turn 4.7

1.
Convert the Mayan number into a Hindu-Arabic number.
Mayan numeral 12 is displayed.
Mayan numeral 17 is displayed.

Example 4.8

Converting Three-Digit Mayan Numbers to Hindu-Arabic Numbers

Convert the Mayan number

Mayan numeral 6 is displayed.


Mayan numeral 8 is displayed.


Mayan numeral 4 is displayed.

into a Hindu-Arabic number.

Your Turn 4.8

1.
Convert the Mayan number into a Hindu-Arabic number.
Mayan numeral 15 is displayed.
Mayan numeral 2 is displayed.
Mayan numeral 14 is displayed.

Example 4.9

Converting Four-Digit Mayan Numbers to Hindu-Arabic Numbers

Convert the Mayan number

Mayan numeral 8 is displayed.


Mayan numeral 0 is displayed.


Mayan numeral 16 is displayed.


Mayan numeral 5 is displayed.

into a Hindu-Arabic number.

Your Turn 4.9

1.
Convert the Mayan number into a Hindu-Arabic number.
Mayan numeral 7 is displayed.
Mayan numeral 16 is displayed.
Mayan numeral 0 is displayed.
Mayan numeral 3 is displayed.
Mayan numeral 13 is displayed.

Who Knew?

The Mayan Calendar

The Mayans used this base 20 system for everyday situations. But their culturally important, and extremely accurate, calendar system used a slightly different system. For their calendars, they used a system where the place values were 1, 20, then 20*18, then 20*18*18. The reason for this is 20*18 is 360, which is closer to the number of days in a year. Had they used a purely base 20 system for their calendar, they’d be very far off with 400 days in a year.

Three hundred sixty days still left the Mayans a bit short, as there are 365 days in a year (ignoring leap years). The Mayan calendar also included 5 days, called Wayeb days, which brings their calendar to 365 days. As it happens, Wayeb is the Mayan god of misfortune, so these 5 days were considered the bad luck days.

Understand and Convert Between Roman Numerals and Hindu-Arabic Numerals

The Mayan and Babylonian systems shared two features, one of which we are familiar with (place value) and one that we don’t use (repeated symbols). The Roman system of numbers used repeated symbols, but does not employ a place value. It also lacks a 0. The Roman system is built on the following symbols in Table 4.3.

Roman Numeral Hindu-Arabic Value
I 1
V 5
X 10
L 50
C 100
D 500
M 1,000
Table 4.3 Roman Numerals

As in the Mayan and Babylonian systems, a symbol may be repeated to indicate a larger value. However, at 4, they did not use IIII. They instead used IV. Since the I came before the V, the number stands for “one before five.” A similar process was used for 9, which was written IX, or “one before ten.” The value 40 was written XL, or “ten before fifty,” while 49 was written XLIX, or “forty plus nine.”

The following are the rules for writing and reading Roman numerals.

  • The representations for bigger values precede those for smaller values.
  • Up to three symbols may be grouped together; for example, III for 3, or XXX for 30, or CC for 200.
  • A larger value followed by a smaller value indicated addition; for example, VII for 7, XIII for 13, LV for 55, and MCC for 1200.
  • I can be placed before V to indicate 4, or before X, to indicate 9. These are the only ways I is used as a subtraction.
  • X can be placed before L to indicate 40, and before C to indicate 90. These are the only ways X is used as a subtraction.
  • C can be placed before D to indicate 400, and before M to indicate 900. These are the only ways C is used as a subtraction.
  • If multiple symbols are used, and a subtraction involving that symbol, the subtraction part comes after the multiple symbols. For example, XXIX for 29 and CCXC for 290.

Who Knew?

Legacy of Roman Numerals

The Roman numbering system is still used today in some situations. Many cornerstones of buildings have the year written in Roman numerals. Movie titles often represent the year the movie was produced as Roman numerals. The most recognizable might be that the Super Bowl is numbered using Roman numerals.

Example 4.10

Converting Roman Numerals to Hindu-Arabic Numbers

Convert the following Roman numerals into Hindu-Arabic numerals.

  1. XXVII
  2. XXXIV
  3. MMCMXLVIII

Your Turn 4.10

Convert the following Roman numerals into Hindu-Arabic numerals.
1.
LXXVII
2.
CCXL
3.
MMMCDXLVII

Of course, we can convert from Hindu-Arabic numerals, to Roman numerals, too.

Example 4.11

Converting Hindu-Arabic Numbers to Roman Numerals

Convert the following Hindu-Arabic numerals into Roman numerals.

  1. 38
  2. 94
  3. 846
  4. 2,987

Your Turn 4.11

Convert the following Hindu-Arabic numerals into Roman numerals.
1.
27
2.
49
3.
739
4.
3,647

Check Your Understanding

7.
What is the place value for Babylonian numerals?
8.
What place value is used in the Mayan numeration system?
9.
What place value is used for Roman numerals?
10.
Convert the Babylonian numeral Babylonian numeral 5 is displayed. Babylonian numeral 41 is displayed. into a Hindu-Arabic numeral.
11.
Convert the Mayan numeral into a Hindu-Arabic numeral.
Mayan numeral 10 is displayed.
Mayan numeral 9 is displayed.
12.
Convert the Roman numeral CCXLVII into a Hindu-Arabic numeral.
13.
Convert 479 into a Roman numeral.
For the following exercises, convert the Babylonian numeral into a Hindu-Arabic numeral.
1 .
Babylonian numeral 12 is displayed.
2 .
Babylonian numeral 20 is displayed.
3 .
Babylonian numeral 31 is displayed.
4 .
Babylonian numeral 48 is displayed.
5 .
Babylonian numeral 53 is displayed. Babylonian numeral 4 is displayed.
6 .
Babylonian numeral 3 is displayed. Babylonian numeral 27 is displayed.
7 .
Babylonian numeral 5 is displayed. Babylonian numeral 40 is displayed.
8 .
Babylonian numeral 24 is displayed. Babylonian numeral 10 is displayed. Babylonian numeral 41 is displayed.
For the following exercises, express the Mayan numeral as a Hindu-Arabic numeral. Use the common system, which is based on powers of 20 only.
9 .
Mayan numeral 10 is displayed.
10 .
Mayan numeral 19 is displayed.
11 .
Mayan numeral 3 is displayed.
Mayan numeral 11 is displayed.
12 .
Mayan numeral 9 is displayed.
Mayan numeral 3 is displayed.
13 .
Mayan numeral 2 is displayed.
Mayan numeral 18 is displayed.
14 .
Mayan numeral 6 is displayed.
Mayan numeral 0 is displayed.
15 .
Mayan numeral 5 is displayed.
Mayan numeral 5 is displayed.
Mayan numeral 0 is displayed.
Mayan numeral 2 is displayed.
16 .
Mayan numeral 18 is displayed.
Mayan numeral 0 is displayed.
Mayan numeral 9 is displayed.
Mayan numeral 11 is displayed.
For the following exercises, express the Roman numeral as a Hindu-Arabic numeral.
17 .
VII
18 .
XI
19 .
IX
20 .
XXIV
21 .
MCXLII
22 .
CXXII
23 .
DCCXLIV
24 .
MCMLIX
For the following exercises, express the Hindu-Arabic numeral as a Roman numeral.
25 .
8
26 .
14
27 .
27
28 .
94
29 .
274
30 .
487
31 .
936
32 .
2,481
33 .
What uses a place value system for numbers: Roman, Babylonian, Egyptian, Greek?
34 .
What uses an additive system: Roman, Mayan, Egyptian, Greek?
35 .
What uses a 0: Roman, Mayan, Egyptian, Greek?
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