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 (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 – base 10 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, ... 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 Exercises 1-34, save some for table use

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