Thinking Mathematically

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# Thinking Mathematically - PowerPoint PPT Presentation

Number Representation and Calculation 4.1 Our Hindu-Arabic System and Early Positional Systems. Thinking Mathematically. “Exponential” Notation. An “exponent” is a small number written slightly above and just to the right of a number or an expression .

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Number Representation and Calculation

4.1 Our Hindu-Arabic System and Early Positional Systems

Thinking Mathematically
“Exponential” Notation

An “exponent” is a small number written slightly above and just to the right of a number or an expression.

When an exponent is a positive integer it stands for repeated multiplication.

102 = 10*10 = 100

103 = 10*10*10 = 1000

104 = 10*10*10*10 = 10,000

Exponents, cont.
• Exercise Set 4.1, #3

23 = ?

• We will re-visit exponents in a more general sense in section 5.6
• 0 exponent
• Negative exponents
• Fractional exponents
Our Hindu-Arabic Numeration System

Introduced to Europe ~1200A.D. by Fionacci

A base 10 system:

• 10 numerals

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

• The value of each position is a power of 10

Why 10? How about 12 or 60?

Our Hindu-Arabic Numeration System

With the use of exponents, Hindu-Arabic numerals can be written in expanded form in which the value of the digit in each position is made clear.

3407 = (3x103)+(4x102)+(0x101)+(7x1)

or (3x1000)+(4x100)+(0x10)+(7x1)

53,525=(5x104)+(3x103)+(5x102)+(2x101)+(5x1)

or (5x10,000)+(3x1000)+(5x100)+(2x10)+(5x1)

Examples: Expanded Form

Exercise Set 4.1 #17, #29

Write in expanded form

• 3,070

Express as a Hindu-Arabic numeral

• (7 x 103) + (0 x 102) + (0 x 101) + (2 x 1)
Base of a Positional System

Base n

• n numerals (0 through n-1)
• Powers of n define the place values

• 16 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f)
• Positional values (right to left) 160 (=1), 161 (=16), 162 (=256), 163 (=4,096)…

Example – base10

• 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• Positional values (right to left) 100 (=1), 101 (=10), 102 (=100), 103 (=1,000)…
Counting in a Positional System
• Base 10

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

• Base 4

0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, ...

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f, 10, 11, ...

• Base 2 (binary)

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, ...

Converting to/from Base 10

Exercise Set 4.2 #3, #21, #37

• Convert 52eight to base 10
• Convert 11 to base seven
• Convert 19 to base two
Computation in Other Bases

Remember how its done in base 10