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4.1

4.1. Our Hindu-Arabic System and Early Positional Systems. Objectives Evaluate an exponential expression. Write a Hindu-Arabic numeral in expanded form. Express a number’s expanded form as a Hindu-Arabic numeral. Understand and use the Babylonian numeration system.

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4.1

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  1. 4.1 Our Hindu-Arabic System and Early Positional Systems

  2. Objectives Evaluate an exponential expression. Write a Hindu-Arabic numeral in expanded form. Express a number’s expanded form as a Hindu-Arabic numeral. Understand and use the Babylonian numeration system. Understand and use the Mayan numeration system. 2

  3. Exponential Notation 3

  4. Example 1: Understanding Exponential Notation Evaluate the following: 108. Solution:Since the exponent is 8, we multiply eight 10’s together: 108 = 10 × 10 × 10 × 10 × 10 ×10 × 10 ×10 = 100,000,000 Notice the relationship: 4

  5. Our Hindu-Arabic Numeration System An important characteristic is that we can write the numeral for any number, large or small, using only ten symbols called digits: 0,1,2,3,4,5,6,7,8,9 Hindu-Arabic numerals can be written in expanded form, in which the value of the digit in each position is made clear. We can write663 in an expanded form such that 663 = (6 × 100) + (6 × 10) + (3 × 1) = (6 × 102) + (6 × 101) + (3 × 1) 5

  6. Our Hindu-Arabic Numeration System Hindu Arabic numeration system is called a positional-value,or place-value, system. The positional values in the system are based on the powers of ten: …,105, 104, 103, 102, 101,1 6

  7. Example 2: Writing Hindu-Arabic Numerals in Expanded Form Write 3407 in expanded form. Solution: 3407 = (3 × 103) + (4 × 102) + (0 × 101) + (7 × 1) or = (3 × 1000) + (4 × 100) + (0 × 10) + (7 × 1) 7

  8. Example 3: Expressing a Number’s Expanded Form as a Hindu-Arabic Numeral Express the expanded form as a Hindu-Arabic numeral: (7 × 103) + (5 × 101) + (4 × 1). Solution:We start by showing all powers of ten, starting with the highest exponent given. Any power left out is expressed as 0 times that power of ten. (7 × 103) + (5 × 101) + (4 × 1) = (7 × 103) + (0 × 102) + (5 × 101) + (4 × 1) = 7054 8

  9. The Babylonian Numeration System The place values in the Babylonian system use powers of 60. The place values are The Babylonians left a space to distinguish the various place values in a numeral from one another. 9

  10. Example 4: Converting from a Babylonian Numeral to a Hindu-Arabic Numeral Write as a Hindu-Arabic numeral. Solution: From left to right the place values are 602, 601, and 1. Represent the numeral in each place as a familiar Hindu-Arabic numeral. Multiply each Hindu-Arabic numeral by its respective place value. Find the sum of these products 10

  11. The Mayan Numeration System The place values in the Mayan system are Numerals in the Mayan system are expressed vertically. The place value at the bottom of the column is 1. 11

  12. Example 5: Using the Mayan Numeration System Write as a Hindu-Arabic numeral. Solution:The given Mayan numeral has four places. From top to bottom, the place values are 7200, 360, 20, and 1. Represent the numeral in each row as a familiar Hindu-Arabic numeral. Multiply each Hindu-Arabic numeral by its respective place value. Find the sum of these products. 12

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