1.  2012 Pearson Education, Inc. Slide 4-4-1 Chapter 4 NumerationSystems

2.  2012 Pearson Education, Inc. Slide 4-4-2 Chapter 4:Numeration Systems 4.1 Historical Numeration Systems 4.2 More Historical Numeration Systems 4.3Arithmetic in the Hindu-Arabic System 4.4 Conversion Between Number Bases

3.  2012 Pearson Education, Inc. Slide 4-4-3 Section 4-4 Conversion Between Number Bases

4.  2012 Pearson Education, Inc. Slide 4-4-4 General Base Conversions Computer Mathematics Conversion Between Number Bases

5.  2012 Pearson Education, Inc. Slide 4-4-5 We consider bases other than ten. Bases other than ten will have a spelled-out subscript as in the numeral 54 eight. When a number appears without a subscript assume it is base ten. Note that 54 eight is read “five four base eight.” Do not read it as “fifty-four.” General Base Conversions

6.  2012 Pearson Education, Inc. Slide 4-4-6 Fourth Power Third Power Second Power First Power Zero Power Base two168421 Base five6251252551 Base seven24013434971 Base eight40965126481 Base sixteen65,5364096256161 Powers of Alternative Bases

7.  2012 Pearson Education, Inc. Slide 4-4-7 Convert 2134 five to decimal form. Solution 2134 five Example: Converting Bases

8.  2012 Pearson Education, Inc. Slide 4-4-8 To convert from another base to decimal form: Start with the first digit on the left and multiply by the base. Then add the next digit, multiply again by the base, and so on. The last step is to add the last digit on the right. Do not multiply it by the base. Calculator Shortcut for Base Conversion

9.  2012 Pearson Education, Inc. Slide 4-4-9 Use the calculator shortcut to convert 432134 five to decimal form. Solution 432134 five Example: Calendar Shortcut

10.  2012 Pearson Education, Inc. Slide 4-4-10 Convert 7508 to base seven. 7508 10724 1531 216 30 03 Solution Divide by 7, then divide the resulting quotient by 7, until a quotient of 0 results. From the remainders (bottom to top) we get the answer: 7508 = 30614 seven Remainder Example: Converting Bases

11.  2012 Pearson Education, Inc. Slide 4-4-11 Many people feel the most comfortable handling conversions between arbitrary bases (where neither is ten) by going from the given base to base ten and then to the desired base. Converting Between Two Bases Other Than Ten

12.  2012 Pearson Education, Inc. Slide 4-4-12 There are three alternative base systems that are most useful in computer applications. These are binary (base two), octal (base eight), and hexadecimal (base sixteen) systems. Computers and handheld calculators use the binary system. Computer Mathematics

13.  2012 Pearson Education, Inc. Slide 4-4-13 Convert 111001 two to decimal form. Solution 111001 two Example: Convert Binary to Decimal

14.  2012 Pearson Education, Inc. Slide 4-4-14 Convert 8B4F sixteen to binary form. Solution Each hexadecimal digit yields a 4-digit binary equivalent. 8B4F sixteen = 1000101101001111 two. 8 B 4 F sixteen 1000 1011 0100 1111 two Combine to get Example: Convert Hexadecimal to Binary