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Thinking Mathematically Number Representation and Calculation 4.1 Our Hindu-Arabic System and Early Positional Systems

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“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. 10 2 = 10*10 = 100 10 3 = 10*10*10 = 1000 10 4 = 10*10*10*10 = 10,000

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Exponents, cont. Exercise Set 4.1, #3 2 3 = ? We will re-visit exponents in a more general sense in section 5.6 –0 exponent –Negative exponents –Fractional exponents

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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?

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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 = (3x10 3 )+(4x10 2 )+(0x10 1 )+(7x1) or (3x1000)+(4x100)+(0x10)+(7x1) 53,525=(5x10 4 )+(3x10 3 )+(5x10 2 )+(2x10 1 )+(5x1) or (5x10,000)+(3x1000)+(5x100)+(2x10)+(5x1)

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Examples: Expanded Form Exercise Set 4.1 #17, #29 Write in expanded form –3,070 Express as a Hindu-Arabic numeral –(7 x 10 3 ) + (0 x 10 2 ) + (0 x 10 1 ) + (2 x 1)

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Thinking Mathematically Number Representation and Calculation 4.2 Number Bases in Positional Systems

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Base of a Positional System Base n n numerals (0 through n-1) Powers of n define the place values Example – base 10 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) Positional values (right to left) 10 0 (=1), 10 1 (=10), 10 2 (=100), 10 3 (=1,000)… Example – base 16 (hexadecimal) 16 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f) Positional values (right to left) 16 0 (=1), 16 1 (=16), 16 2 (=256), 16 3 (=4,096)…

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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,... Base 16 (hexadecimal) 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,...

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Converting to/from Base 10 Exercise Set 4.2 #3, #21, #37 –Convert 52 eight to base 10 –Convert 11 to base seven –Convert 19 to base two

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Thinking Mathematically Number Representation and Calculation 4.3 Computation in Positional Systems

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Computation in Other Bases Remember how its done in base 10 –Carry (addition and multiplication) –Borrow (subtraction) –Long Division

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Examples: Computation in Other Bases Exercise Set 4.3 #5, #17 342 five + 413 five = 475 eight – 267 eight = Hexadecimal Arithmetic 4C6 sixteen + 198 sixteen = 694 sixteen – 53B sixteen =

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Thinking Mathematically Chapter 4: Number Representation and Calculation

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