Hammond Prospectus text2

Numbers, Numerals, and Computations
Mathematics 15: Lecture 3
Dan Sloughter
Furman University
September 15, 2006
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 1 / 9I Egyptian symbols (page 445): ∩∩∩|| = 32
I Greek symbols (page 446)
I Roman numerals (page 448)
Denoting numbers
I Babylonian symbols (page 444)
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 2 / 9I Greek symbols (page 446)
I Roman numerals (page 448)
Denoting numbers
I Babylonian symbols (page 444)
I Egyptian symbols (page 445): ∩∩∩|| = 32
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 2 / 9I Roman numerals (page 448)
Denoting numbers
I Babylonian symbols (page 444)
I Egyptian symbols (page 445): ∩∩∩|| = 32
I Greek symbols (page 446)
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 2 / 9Denoting numbers
I Babylonian symbols (page 444)
I Egyptian symbols (page 445): ∩∩∩|| = 32
I Greek symbols (page 446)
I Roman numerals (page 448)
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 2 / 9I The Babylonians developed a sexagesimal, that is, base 60, positional
system (unlike our decimal, that is, base 10, system).
I Example: In our decimal system, writing 278 is shorthand for
2 1 02×10 +7×10 +8×10 = 200+70+8,
0remembering that 10 = 1.
I Example: In a base 2 positional system, 10011 represents the base 10
number
4 3 2 1 01×2 +0×2 +0×2 +1×2 +1×2 = 16+0+0+2+1 = 19.
I Example: To represent numbers in base 12 we would have to
introduce new digits, say A for 10 and B for 11. Then the base 12
number AB8 would represent the base 10 number
2 1 010×12 +11×12 +8×12 = 1440+132+8 = 1580.
Positional systems
I The Egyptian, Greek, and Roman systems are not positional systems.
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 3 / 9I Example: In our decimal system, writing 278 is shorthand for
2 1 02×10 +7×10 +8×10 = 200+70+8,
0remembering that 10 = 1.
I Example: In a base 2 positional system, 10011 represents the base 10
number
4 3 2 1 01×2 +0×2 +0×2 +1×2 +1×2 = 16+0+0+2+1 = 19.
I Example: To represent numbers in base 12 we would have to
introduce new digits, say A for 10 and B for 11. Then the base 12
number AB8 would represent the base 10 number
2 1 010×12 +11×12 +8×12 = 1440+132+8 = 1580.
Positional systems
I The Egyptian, Greek, and Roman systems are not positional systems.
I The Babylonians developed a sexagesimal, that is, base 60, positional
system (unlike our decimal, that is, base 10, system).
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 3 / 9I Example: In a base 2 positional system, 10011 represents the base 10
number
4 3 2 1 01×2 +0×2 +0×2 +1×2 +1×2 = 16+0+0+2+1 = 19.
I Example: To represent numbers in base 12 we would have to
introduce new digits, say A for 10 and B for 11. Then the base 12
number AB8 would represent the base 10 number
2 1 010×12 +11×12 +8×12 = 1440+132+8 = 1580.
Positional systems
I The Egyptian, Greek, and Roman systems are not positional systems.
I The Babylonians developed a sexagesimal, that is, base 60, positional
system (unlike our decimal, that is, base 10, system).
I Example: In our decimal system, writing 278 is shorthand for
2 1 02×10 +7×10 +8×10 = 200+70+8,
0remembering that 10 = 1.
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 3 / 9I Example: To represent numbers in base 12 we would have to
introduce new digits, say A for 10 and B for 11. Then the base 12
number AB8 would represent the base 10 number
2 1 010×12 +11×12 +8×12 = 1440+132+8 = 1580.
Positional systems
I The Egyptian, Greek, and Roman systems are not positional systems.
I The Babylonians developed a sexagesimal, that is, base 60, positional
system (unlike our decimal, that is, base 10, system).
I Example: In our decimal system, writing 278 is shorthand for
2 1 02×10 +7×10 +8×10 = 200+70+8,
0remembering that 10 = 1.
I Example: In a base 2 positional system, 10011 represents the base 10
number
4 3 2 1 01×2 +0×2 +0×2 +1×2 +1×2 = 16+0+0+2+1 = 19.
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 3 / 9Positional systems
I The Egyptian, Greek, and Roman systems are not positional systems.
I The Babylonians developed a sexagesimal, that is, base 60, positional
system (unlike our decimal, that is, base 10, system).
I Example: In our decimal system, writing 278 is shorthand for
2 1 02×10 +7×10 +8×10 = 200+70+8,
0remembering that 10 = 1.
I Example: In a base 2 positional system, 10011 represents the base 10
number
4 3 2 1 01×2 +0×2 +0×2 +1×2 +1×2 = 16+0+0+2+1 = 19.
I Example: To represent numbers in base 12 we would have to
introduce new digits, say A for 10 and B for 11. Then the base 12
number AB8 would represent the base 10 number
2 1 010×12 +11×12 +8×12 = 1440+132+8 = 1580.
Dan Sloughter (Furman University) Numbers, Numerals, and Computations September 15, 2006 3 / 9