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Contemporary Mathematics

4.1 Hindu-Arabic Positional System

Contemporary Mathematics4.1 Hindu-Arabic Positional System

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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
An old manuscript contains early written numerals.
Figure 4.2 This manuscript is an early example of Hindu numerals. (credit: modification of work “Bakshali manuscript”, Bodleian Libraries/ University of Oxford, public domain)

Learning Objectives

After completing this section, you should be able to:

  1. Evaluate an exponential expression.
  2. Convert a Hindu-Arabic numeral to expanded form.
  3. Convert a number in expanded form to a Hindu-Arabic numeral.

The modern system of counting and computing isn’t necessarily natural. That different symbols are used to indicate different quantities or amounts is a relatively new invention. Simple marking by scratches or dots, one for each item being counted, was the norm long into human history. The modern system doesn’t use repeated symbols to indicate more than one of a thing. It uses the place of a digit in a numeral to determine what that digit represents. A numeral is a symbol used to represent a number. A number is an abstract idea that represents quantity or amount.

Being clear about the difference between numeral and number is important. Just like a person can be called by various names, such as brother, father, husband, uncle, they are all representing the same person, John Smith. The person John Smith is the number, and the names brother, father, husband, and uncle are the numerals.

Who Knew?

Hindu-Arabic Numerals

The numerals we currently use are referred to as Hindu-Arabic numerals, although they have changed as time has passed. Early forms of the numerals for 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 began in India, and passed through Persia to the Middle East. Place value was also employed in the early systems of India. Once this system was in north Africa and the Middle East, it spread to Europe, eventually replacing Roman numerals. Over time, the original symbols transformed into our modern ones. Read this article for another perspective on how the symbols began (based on the moon!).

The system we use for counting and computing uses place values based on powers of 10. In this section, we review exponents and our positional system.

Evaluating Exponential Expressions

Most modern numerical systems depend on place values, where the quantity represented depends not only on the digit, but also on where the digit is in the number. The place value is a power of some specific number, which means most numbering systems are actually exponential expressions. An exponential expression is any mathematical expression that includes exponents. So, evaluating such an expression means performing the calculation. In this chapter, we will be using exponents that are positive integer values. Before we do so, let’s remind ourselves about exponents and what they represent. Suppose you want to multiply a number. Let’s label that number aa, by itself some number of times. Let’s label the number of times bb. We denote that as abab. We say aa, or the base, raised to the bbth power, or the exponent. For example, if we are multiplying 13 by itself eight times, we write 138138 and say 13 to the eighth power.

When computing exponential expressions, we should be careful to remember the order of operations. Using the order of operation rules, calculations inside the parentheses are done first, then exponents are calculated, then multiplication and division calculations are performed, and then addition and subtraction.

Example 4.1

Evaluating an Exponential Expression

Evaluate the following exponential expressions.

  1. 4×52+2×634×52+2×63
  2. 6×82+3×81+4×806×82+3×81+4×80
  3. 3×102+0×101+6×1003×102+0×101+6×100

Your Turn 4.1

Evaluate the following exponential expressions.
1.
3 \times {2^5} + 5 \times {8^2}
2.
5 \times {7^3} + 2 \times {7^2} + 5 \times {7^1} + 3 \times {7^0}
3.
1 \times {10^4} + 7 \times {10^3} + 4 \times {10^2} + 8 \times {10^1} + 8 \times {10^0}

Converting Hindu-Arabic Numerals to Expanded Form

When you see the number 738, and you speak the number out loud, what do you say? You probably said “seven hundred thirty-eight” while wondering what point could possibly be made by asking this. What you didn’t say was “seven, and three, and eight.” A pre-K student might say that. Which should make you wonder, why?

The reason is that you’ve been taught place values, or the positions of digits in a number that determine the values of those digits. You know that in a three-digit number, the first digit is hundreds, the second digit is tens, and the last digit is ones. These place values rely on powers of 10, which makes this system a base 10 system.

This sense of place value is what makes our system of numbers so useful. You’ve also been taught the Hindu-Arabic numeration system. This system, which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, and also employs place value based on powers of 10, is in use today.

Writing a number using these place values is writing them in expanded form. For a number with nn digits, the expanded form is the first digit times 10 raised to one less than nn, plus each following digit times 10 raised to one less than the previous power of 10. For example, the number 738 would be written as 7×102+3×101+8×1007×102+3×101+8×100.

What about a four-digit number, like 5,825? Out loud, we’d say five thousand, seven hundred twenty-five. In expanded form, it would be 5×103+8×102+2×101+5×1005×103+8×102+2×101+5×100. Notice that the largest exponent is one less than the number of digits, and that the exponents go down by one as we move through the number.

People in Mathematics

Aryabhata of Kusumapura and Brahmagupta

The Hindu-Arabic numeral system developed in India, and Aryabhata of Kusumapura is credited with the place value notation in the 5th century. However, the system wasn’t as complete as it could be, until. roughly a century later, when Brahmagupta introduced the symbol for 0. The 0 is necessary to indicate that a given place value has been skipped, as in 4,098. In 4,098, the 102 power is skipped. Without such a symbol, 4,098 and 498 look similar. The value of both the place value notation and the introduction of the symbol 0 cannot be overstated, for math and the sciences.

Example 4.2

Writing a Number in Expanded Form

Write the following in expanded form.

  1. 563
  2. 4,821
  3. 903,786

Your Turn 4.2

Write the following in expanded form.
1.
924
2.
1,279
3.
4,130,045

Converting Numbers in Expanded Form to Hindu-Arabic Numerals

Converting from expanded form back into a Hindu-Arabic numeral is the reverse process of expanding a number, and is equivalent to evaluating the exponential expression.

Example 4.3

Converting a Number from Expanded Form to a Hindu-Arabic Numeral

Convert the following into Hindu-Arabic numerals.

  1. 3×102+4×101+8×1003×102+4×101+8×100
  2. 5×103+0×102+9×101+9×1005×103+0×102+9×101+9×100
  3. 6×106+2×105+0×104+9×103+1×102+1×101+7×1006×106+2×105+0×104+9×103+1×102+1×101+7×100

Your Turn 4.3

Convert the following to Hindu-Arabic Numerals.
1.
6 \times {10^2} + 2 \times {10^1} + 1 \times {10^0}
2.
3 \times {10^3} + 2 \times {10^2} + 0 \times {10^1} + 3 \times {10^0}
3.
4 \times {10^7} + 0 \times {10^6} + 6 \times {10^5} + 3 \times {10^4} + 0 \times {10^3} + 8 \times {10^2} + 9 \times {10^1} + 1 \times {10^0}

Check Your Understanding

1.
What is meant by a place value system?
2.
Evaluate the following exponential expression: 4 \times {8^2} + 2 \times {8^1} + 7 \times {8^0}.
3.
Express the following number in expanded form: 45,209.
4.
What number provides the value of a digit in our system of numeration?
5.
How are numerals and numbers different?
6.
Express as a Hindu-Arabic number: 6 \times {10^5} + 0 \times {10^4} + 1 \times {10^3} + 9 \times {10^2} + 4 \times {10^1} + 7 \times {10^0}.

Section 4.1 Exercises

1.
What does it mean for a system to be a place value system?
2.
In the system we use today, what number are the place values based on?
3.
How are numerals and numbers different?
4.
What relates numerals to numbers?
For the following exercises, evaluate the exponential expression.
5.
3 \times {4^2} + 5 \times {2^3}
6.
5 \times {6^3} + 7 \times {3^2}
7.
7 \times {5^2} + 2 \times {4^5}
8.
10 \times {11^2} + 7 \times {3^4}
9.
5 \times {6^2} + 3 \times {6^1} + 4 \times {6^0}
10.
4 \times {12^2} + 11 \times {12^1} + 2 \times {12^0}
11.
14 \times {8^3} + 19 \times {5^4} + 2 \times {3^1}
12.
8 \times {10^4} + 3 \times {5^3} + 9 \times {4^2}
13.
1 \times {2^3} + 0 \times {2^2} + 1 \times {2^1} + 1 \times {2^0}
14.
5 \times {8^4} + 1 \times {8^3} + 0 \times {8^2} + 7 \times {8^1} + 3 \times {8^0}
For the following exercises, express the Hindu-Arabic number in expanded form.
15.
13
16.
25
17.
82
18.
99
19.
131
20.
408
21.
651
22.
3,901
23.
5,098
24.
12,430
For the following exercises, express the expanded number as a Hindu-Arabic number.
25.
3 \times {10^1} + 2 \times {10^0}
26.
5 \times {10^1} + 7 \times {10^0}
27.
2 \times {10^2} + 4 \times {10^1} + 9 \times {10^0}
28.
6 \times {10^2} + 0 \times {10^1} + 1 \times {10^0}
29.
1 \times {10^3} + 4 \times {10^2} + 4 \times {10^1} + 0 \times {10^0}
30.
7 \times {10^3} + 0 \times {10^2} + 1 \times {10^1} + 8 \times {10^0}
31.
6 \times {10^4} + 7 \times {10^3} + 0 \times {10^2} + 0 \times {10^1} + 0 \times {10^0}
32.
9 \times {10^4} + 8 \times {10^3} + 7 \times {10^2} + 3 \times {10^1} + 4 \times {10^0}
33.
7 \times {10^7} + 3 \times {10^6} + 4 \times {10^5} + 0 \times {10^4} + 4 \times {10^3} + 1 \times {10^2} + 5 \times {10^1} + 1 \times {10^0}
34.
8 \times {10^8} + 0 \times {10^7} + 4 \times {10^6} + 9 \times {10^5} + 9 \times {10^4} + 2 \times {10^3} + 2 \times {10^2} + 6 \times {10^1} + 0 \times {10^0}
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