CHAPTER 4 Number Representation and Calculation 4.1 Our Hindu-Arabic System and Early Positional Systems Modified from Blitzer, “Thinking Mathematically,” Pearson, 2010

Numbers vs Numerals Number Numeral two books, two dogs, two cars, two houses, two committees, two marriages, two lies, two yells, etc… “Two” is a number that conveys a quantity. “Two” is an idea. Numeral Two, 2, II, ニ、dos, deux, · ·, zwei, elua, etc… Numeral is a symbol to represent a number. E.g., 0.5 and ½ represent the same number.

Numeral System How do you represent the number five? In decimal system: 5 In binary system: 101 How do you represent the number fourteen? In decimal system: 14 In binary system: 1110

Exponential Notation

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.

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:

Our Hindu-Arabic Numeration System Any number can be represented by a combination of on ten symbols, called digits: 0,1,2,3,4,5,6,7,8,9 The position of each digits implies 100, 101, 102, etc. Example: 673 673 = (6 × 100) + (7 × 10) + (3 × 1) = (6 × 102) + (7 × 101) + (3 × 1)

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)

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: (7 × 103) + (5 × 101) + (4 × 1) = (7 × 103) + (0 × 102) + (5 × 101) + (4 × 1) = 7054

Short History of Numeral Systems Babylonian Numeral System (2000 B.C.) Mayan Numeral System Arabic-Hindu Numeral System

Babylonian Empire Babylonian Numeration System by 2000 BC

Mayan Empire 2000 BCE ~ 900 CE By 30 A.D., had invented the concept of zero Calculated the number of days in a year to 365.242 (compared to the modern value of 365.242198)

Hindu-Arabic Numeral System Invented by Indian mathematicians around 1st – 4th cent. Adopted by Persians around 9th cent. Adopted by Arabs by 9th cent. Spread to Europe by 11th to 12th cent. Uses symbols: 0, 1, 2, 3, 4, 5, 6, 7, 9 Each position in a numeral implies power of 10.

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.

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.

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.

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.

Example: Mayan Number Given the Maya numeral at left, express it as a Hindu-Arabic numeral. Hints

Solution: Mayan Number