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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 .

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number representation and calculation 4 1 our hindu arabic system and early positional systems
Number Representation and Calculation

4.1 Our Hindu-Arabic System and Early Positional Systems

Thinking Mathematically
exponential notation
“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
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
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 system1
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
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 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)…
counting in a positional system
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, ...

converting to from base 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
computation in other bases
Computation in Other Bases

Remember how its done in base 10

  • Carry (addition and multiplication)
  • Borrow (subtraction)
  • Long Division
examples computation in other bases
Examples: Computation in Other Bases

Exercise Set 4.3 #5, #17

  • 342five + 413five =
  • 475eight – 267eight =

Hexadecimal Arithmetic

  • 4C6sixteen + 198sixteen =
  • 694sixteen – 53Bsixteen =