© 2003 Dr. Kevin Chouinard Edited by Jean Pacelli Section 4.3 Converting Between Number Bases

2 Place Values for the Decimal System Also Called Base Ten __ __ __ thousandshundreds 10º tensones 10¹10²10³10 4 ten thousands 10 5 hundred thousands Digits for Base Ten 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

3 Value of a multi-digit number in Base Ten The base ten number 406,391 is 6 digits long, and is written in standard form. It is read four hundred and six thousand, three hundred ninety one. In expanded form, 406,391 = (4x10 5 ) + (0x10 4 ) + (6x10 3 ) + (3x10 2 ) + (9x10 1 ) + (1x10 0 ) = 4x100, x10, x1, x x10 + 1x1 This number system is multiplicative, and positional. The digit is in a specific place value, and is the multiplier for that place value. The 4 in the number 4000 has a different value from the 4 in the number 400.

4 Using Other Bases When we work in a base other than base 10, we must calculate the place values for that base. The place values will be powers of the base, starting with the 0 th power, 1 st power, 2 nd power, etc… working from right to left.

5 Place Values for Base Five Let’s try counting in base five. First, we need the place values. _ _ _ _ _ _ twenty- fives fivesones 5¹5¹5²5²5º5º

6 Counting in Base Five _ _ _ _ _ _ twenty- fives fivesones 5¹5¹5²5²5º5º Base Base Base Base 10

7 Conversion from base 5 to base five = _____ ten Answer: 1  5³ 4  5² 3  5¹ +2  5º 1   25 3  5 +2  ten

8 Conversion from base 10 to base ten = _____ five Answer: Start by finding the powers of five which are  698 5º = 1 5¹ = 5 5² = 25 5³ = 125 How many 625’s are there in 698? 698  625 = 1 (R 73) How many 125’s are there in 73? 73  125 = 0 (R 73) How many 25’s are there in 73? 73  25 = 2 (R 23) How many 5’s are there in 23? 23  5 = 4 (R 3) How many 1’s are there in 3? 3  1 = 3 (R 0) 698 ten = five

9 Place Values for Base Eight Let’s examine the first four place values for base eight _ _ _ __ five- hundred- twelves sixty- fours 8º8º eightsones 8¹8¹8²8²8³8³

10 Conversion from base 8 to base eight = _____ ten Answer: 2  8² 7  8¹ +0  8º 2  64 7  8 +0  ten

11 Conversion from base 10 to base ten = _____ eight Answer: Start by finding the powers of eight which are  497 8º = 1 8¹ = 8 8² = 64 8³ = 512 How many 64’s are there in 497? 497  64 = 7 (R 49) How many 8’s are there in 49? 49  8 = 6 (R 1) How many 1’s are there in 1? 1  1 = 1 (R 0) 497 ten = 761 eight

12 Place Values for Base Two Let’s examine the first four place values for base two _ _ _ __ eightsfours 2º2º twosones 2¹2¹2²2²2³2³

13 Conversion from base 2 to base two = _____ ten Answer: 1  2³ 0  2² 1  2¹ +1  2º 1  8 0  4 1  2 +1  ten

14 Octal (base eight) Binary (base two) Binary to Octal Conversion

15 Examples two = 561 eight two = two = 33 eight

16 Place Values for Base Sixteen Let’s examine the first four place values for base sixteen _ _ _ _ º sixteenones 16¹16²16³

17 Digits in various bases Base 100, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base 80, 1, 2, 3, 4, 5, 6, 7 Base 50, 1, 2, 3, 4 Base 20, 1 Base 160, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

18 Digits in various bases (cont.) Note that each base has digits from 0 up to a number one less than the base Also, note that in base 16, A = 10 B = 11 C = 12 D = 13 E = 14 F = 15

19 Binary to Hexadecimal Conversion Hexadecimal (Base 16) A B C D E F Binary (Base 2)

20 Convert to base five 317 eight two = 113 ten = 207 ten = 87 ten

21 Convert to the indicated base 21 ten to base two 396 ten to base eight 392 ten to base five = two = 614 eight = 3032 five

22 Cereal Box Magic Trick