Thinking
Mathematically
Number Representation and
Calculation
4.1 Our Hindu-Arabic System and Early
Positional Systems
“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)
Thinking
Mathematically
Number Representation and
Calculation
4.2 Number Bases in Positional
Systems
Base of a Positional System
Base n
• n numerals (0 through n-1)
• Powers of n define the place values
Example – base 16
10 (hexadecimal)
•16
•10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
9) a, b, c, d, e, f)
•Positional values (right to left) 16
1000 (=1), 16
1011 (=16),
(=10),
16
1022 (=256),
(=100), 16
1033 (=4,096)…
(=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, ...
•
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
Exercise Set 4.2 #3, #21, #37
– Convert 52eight to base 10
– Convert 11 to base seven
– Convert 19 to base two
Thinking
Mathematically
Number Representation and
Calculation
4.3 Computation in Positional Systems
Computation in Other Bases
Remember how its done in base 10
– Carry (addition and multiplication)
– Borrow (subtraction)
– Long Division
Examples: Computation in Other
Bases
Exercise Set 4.3 #5, #17
• 342five + 413five =
• 475eight – 267eight =
Hexadecimal Arithmetic
• 4C6sixteen + 198sixteen =
• 694sixteen – 53Bsixteen =
Thinking
Mathematically
Chapter 4: Number Representation
and Calculation