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

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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

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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.

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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.

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

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

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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

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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

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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

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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

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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

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

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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

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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

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

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

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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

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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

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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

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

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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

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22 Cereal Box Magic Trick

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Arithmetic Chapter 4 Subject: Digital System Year: 2009.

Arithmetic Chapter 4 Subject: Digital System Year: 2009.

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