03_Integers

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

Info iconThis preview shows pages 1–18. Sign up to view the full content.

View Full Document Right Arrow Icon
Integer Numbers The Number Bases of Integers Textbook Chapter 2 +
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
CMPE12 – Fall 2009 03-3 The value represented by a digit depends on its position in the number. Ex: 1832 Positional Number System
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
CMPE12 – Fall 2009 03-6 Babylonian Numbers
Background image of page 6
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!
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 8
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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 10
CMPE12 – Fall 2009 03-11 From Base b to Base 10 Example: 1234 5 = ? 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
CMPE12 – Fall 2009 03-12 From Base 10 to Base b Use successive divisions Remember the remainders Divide again with the quotients
Background image of page 12
CMPE12 – Fall 2009 03-13 From Base 10 to Base b Example: 2010 10 = ? 5
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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)
Background image of page 14
CMPE12 – Fall 2009 03-15 Roman Multiplication XXXIII (33 in decimal) XII (12 in decimal) --------------
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

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

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 18
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 67

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

This preview shows document pages 1 - 18. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online