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Thinking Mathematically

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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Thinking Mathematically

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

  2. “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

  3. 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

  4. 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?

  5. 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)

  6. 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)

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

  8. Base of a Positional System Base n • n numerals (0 through n-1) • Powers of n define the place values 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) 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)…

  9. 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, ...

  10. 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

  11. Number Representation and Calculation 4.3 Computation in Positional Systems Thinking Mathematically

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

  13. Examples: Computation in Other Bases Exercise Set 4.3 #5, #17 • 342five + 413five = • 475eight – 267eight = Hexadecimal Arithmetic • 4C6sixteen + 198sixteen = • 694sixteen – 53Bsixteen =

  14. Chapter 4: Number Representation and Calculation Thinking Mathematically

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