1. © 2003 Dr. Kevin Chouinard Edited by Jean Pacelli 20041 Section 4.3 Converting Between Number Bases

2. 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. 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 ) = 4x100,000 + 0x10,000 + 6x1,000 + 3x100 + 9x10 + 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. 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. 5 Place Values for Base Five Let’s try counting in base five. First, we need the place values. _ _ _ _ _ _ twenty- fives fivesones 5¹5¹5²5²5º5º

6. 6 Counting in Base Five _ _ _ _ _ _ twenty- fives fivesones 5¹5¹5²5²5º5º 242322212014131211104321Base 5 1413121110987654321Base 10 10210110044434241403433323130Base 5 27262524232221201918171615Base 10

7. 7 Conversion from base 5 to base 10 1432 five = _____ ten Answer: 1  5³ 4  5² 3  5¹ +2  5º 1  125 4  25 3  5 +2  1 125 100 15 + 2 242 ten

8. 8 Conversion from base 10 to base 5 698 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 = 10243 five

9. 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. 10 Conversion from base 8 to base 10 270 eight = _____ ten Answer: 2  8² 7  8¹ +0  8º 2  64 7  8 +0  1 128 56 + 0 184 ten

11. 11 Conversion from base 10 to base 8 497 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. 12 Place Values for Base Two Let’s examine the first four place values for base two _ _ _ __ eightsfours 2º2º twosones 2¹2¹2²2²2³2³

13. 13 Conversion from base 2 to base 10 1011 two = _____ ten Answer: 1  2³ 0  2² 1  2¹ +1  2º 1  8 0  4 1  2 +1  1 8 0 2 + 1 11 ten

14. 14 Octal (base eight) 0 1 2 3 4 5 6 7 Binary (base two) 000 001 010 011 100 101 110 111 Binary to Octal Conversion

15. 15 Examples 101110001 two = 561 eight 11011 two = 011011 two = 33 eight

16. 16 Place Values for Base Sixteen Let’s examine the first four place values for base sixteen _ _ _ _ 4096256 16º sixteenones 16¹16²16³

17. 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. 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. 19 Binary to Hexadecimal Conversion Hexadecimal (Base 16) 0 1 2 3 4 5 6 7 8 9 A B C D E F Binary (Base 2) 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

20. 20 Convert to base 10 423 five 317 eight 1010111 two = 113 ten = 207 ten = 87 ten

21. 21 Convert to the indicated base 21 ten to base two 396 ten to base eight 392 ten to base five = 10101 two = 614 eight = 3032 five

22. 22 Cereal Box Magic Trick