© 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 4 Number Representation and Calculation

© 2010 Pearson Prentice Hall. All rights reserved. 2 §4.1, Our Hindu-Arabic System and Early Positional Systems

© 2010 Pearson Prentice Hall. All rights reserved. I will evaluate an exponential expression. I will write a Hindu-Arabic numeral in expanded form. I will express a number’s expanded form as a Hindu- Arabic numeral. I will understand and use the Babylonian numeration system. I will understand and use the Mayan numeration system. 3 Learning Targets:

© 2010 Pearson Prentice Hall. All rights reserved. VOCABULARY exponential notation exponential expression Hindu-Arabic number system digits expanded form place value

© 2010 Pearson Prentice Hall. All rights reserved. Crititcal Thinking: A number system has only two digits, 0 and 1. What place values (using our current number system) would you use for each digit for the number ?

© 2010 Pearson Prentice Hall. All rights reserved. Exponential Notation 6

© 2010 Pearson Prentice Hall. All rights reserved. Example 1: Understanding Exponential Notation Evaluate the following: Solution: Since the exponent is 8, we multiply eight 10’s together: 10 8 = 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 100,000,000 Notice the relationship: 7

© 2010 Pearson Prentice Hall. All rights reserved. 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 write 663 in an expanded form such that 663 = (6 × 100) + (6 × 10) + (3 × 1) = (6 × 10 2 ) + (6 × 10 1 ) + (3 × 1) 8

© 2010 Pearson Prentice Hall. All rights reserved. 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: …,10 5, 10 4, 10 3, 10 2, 10 1,1 Our Hindu-Arabic Numeration System 9

© 2010 Pearson Prentice Hall. All rights reserved. Write 3407 in expanded form. Solution: 3407 = (3 × 10 3 ) + (4 × 10 2 ) + (0 × 10 1 ) + (7 × 1) or = (3 × 1000) + (4 × 100) + (0 × 10) + (7 × 1) Example 2: Writing Hindu-Arabic Numerals in Expanded Form 10 GP: Pg 199, #17, 19

© 2010 Pearson Prentice Hall. All rights reserved. Express the expanded form as a Hindu-Arabic numeral: (7 × 10 3 ) + (5 × 10 1 ) + (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 × 10 3 ) + (5 × 10 1 ) + (4 × 1) = (7 × 10 3 ) + (0 × 10 2 ) + (5 × 10 1 ) + (4 × 1) = 7,054 Example 3: Expressing a Number’s Expanded Form as a Hindu-Arabic Numeral 11 GP: Pg 199, #27, 31

© 2010 Pearson Prentice Hall. All rights reserved. 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. 12

© 2010 Pearson Prentice Hall. All rights reserved. Example 4: Converting from a Babylonian Numeral to a Hindu-Arabic Numeral Write as a Hindu-Arabic numeral. Solution: From right to left the place values are 1, 60 1, and

© 2010 Pearson Prentice Hall. All rights reserved. Remember Tallying Data? Sometimes it is necessary to tally when you are counting, like when your homeroom teacher tallies the Homecoming court ballots. 1 = | 5 = |||| 34 = |||| |||| |||| |||| |||| |||| ||||

© 2010 Pearson Prentice Hall. All rights reserved. The Mayans had a very similar way of counting and writing their totals. It is much like tallying but they use vertical lines and dots…

© 2010 Pearson Prentice Hall. All rights reserved. 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. 16

© 2010 Pearson Prentice Hall. All rights reserved. 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. Example 5: Using the Mayan Numeration System 17

© 2010 Pearson Prentice Hall. All rights reserved. Homework GP: Pg 200, #37, 41, 51, 55 Pg 199 – 200, #16, 18, 24, 26, 34 – 44(e), 48 – 60 (e)