03_Integers

# 03_Integers - Integer Numbers The Number Bases of Integers...

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Integer Numbers The Number Bases of Integers Textbook Chapter 2 +

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CMPE12 – Fall 2009 03-2 Number Systems Unary, or marks: /////// = 7 /////// + ////// = ///////////// Grouping lead to Roman Numerals: VII + V = VVII = XII Better: Arabic Numerals: 7 + 5 = 12 = 1·10 + 2
CMPE12 – Fall 2009 03-3 The value represented by a digit depends on its position in the number. Ex: 1832 Positional Number System

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CMPE12 – Fall 2009 03-4 Number = ( d i · b p ) i =0 num digits Positional Number Systems Select a number as the base b Define an alphabet of b –1 symbols plus a symbol for zero to represent all numbers Use an ordered sequence of 2 or more digits d to represent numbers The represented number is the sum of all digits, each multiplied by b to the power of the digit‟s position p
CMPE12 – Fall 2009 03-5 First used over 4000 years ago in Mesopotamia Base 60 (Sexagesimal), alphabet: 0. .59, written as 60 different symbols But the Babylonians used only two symbols, 1 and 10, and didn‟t have the zero Needed context to tell 1 from 60! Example 5,45 60 = Sexagesimal: A Positional Number System

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CMPE12 – Fall 2009 03-6 Babylonian Numbers
CMPE12 – Fall 2009 03-7 Arabic/Indic Numerals Base (or radix): 10 (decimal) The alphabet (digits or symbols) is 0. .9 We use the Arabic symbols for the 10 digits Has the ZERO Numerals introduced to Europe by Leonardo Fibonacci in his Liber Abaci In 1202 So useful!

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CMPE12 – Fall 2009 03-8 Arabic/Indic Numerals The Italian mathematician Leonardo Fibonacci Also known for the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21
CMPE12 – Fall 2009 03-9 Base Conversion Three cases: I. From any base b to base 10 II. From base 10 to any base b III. From any base b to any other base c

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CMPE12 – Fall 2009 03-10 •Base (radix): b •Digits (symbols): 0 … (b – 1) •S n-1 S n-2 ….S 2 S 1 S 0 Use summation to transform any base to decimal Value = Σ (S i b i ) n-1 i=0 From Base b to Base 10
CMPE12 – Fall 2009 03-11 From Base b to Base 10 Example: 1234 5 = ? 10

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CMPE12 – Fall 2009 03-12 From Base 10 to Base b Use successive divisions Remember the remainders Divide again with the quotients
CMPE12 – Fall 2009 03-13 From Base 10 to Base b Example: 2010 10 = ? 5

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CMPE12 – Fall 2009 03-14 From Base b to Base c Use a known intermediate base The easiest way is to convert from base b to base 10 first, and then from 10 to c Or, in some cases, it is easier to use base 2 as the intermediate base (we‟ll see them soon)
CMPE12 – Fall 2009 03-15 Roman Multiplication XXXIII (33 in decimal) XII (12 in decimal) --------------

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CMPE12 – Fall 2009 03-16 Positional Multiplication 113 5 42 5 --------------------------- ( a lot easier!)
CMPE12 – Fall 2009 03-17 Numbers for Computers There are many ways to represent a number Representation does not affect computation result Representation affects

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03_Integers - Integer Numbers The Number Bases of Integers...

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