CSE111 Spring 2009 H Kershner Binary Arithmetic Decimal Numbers Base 10 2 98310 2x1000 9x100 8x10 3x1 3 2 1 or 0 2x10 9x10 8x10 3x10 Remember that 103 means 10 x 10 x 10 or 10 multiplied by itself 3 times Effectively 1 followed by 3 zeros because multiplying anything by 10 is the same as adding a 0 to the end 58 75210 5x10 000 8x1000 7x100 5x10 2x1 or 5x104 8x103 7x102 5x101 2x100 So 104 means 10 x 10 x 10 x 10 or effectively 10 times itself 4 times 1 followed by 4 zeros 10 000 2 And 10 means 10 x 10 or effectively 10 times itself 2 times 1 followed by 2 zeros 100 To determine the correct power count the number of digits to the right of that number REMEMBER 100 1 the mathematical rule states that any number raised to the zero0 power is one Hence 210 1 160 1 AND 20 1 Let s expand these 1 123 45610 1x105 2x104 3x103 4x102 5x101 6x100 2 7 26910 7x103 2x102 6x101 9x100 3 3 720 452 3x106 7x105 2x104 0x103 4x102 5x101 2x100 Binary Numbers Base 2 1 11012 1 x 23 1 x 22 0 x 21 1 x 20 2 110112 1x24 1x23 0x22 1x21 1x20 3 1010112 1x25 0x24 1x23 0x22 1x21 1x20 Expand these binary numbers for answers see end of document a 111102 b 10101012 c 11100112 Copyright 2008 by Helene G Kershner CSE111 Spring 2009 H Kershner Binary Arithmetic Binary to Decimal Conversion To convert Binary numbers to their Decimal equivalent you need to be able to translate the powers of 2 20 1 21 2 22 2 x 2 4 23 2 x 2 x 2 8 24 2 x 2 x 2 x 2 16 25 2 x 2 x 2 x 2 x 2 32 there are 5 twos 6 2 2 x 2 x 2 x 2 x 2 x 2 64 there are 6 twos In general it is just easiest to remember at least the first five powers Working with our first expansion above 1 11012 1 x 23 1 x 22 0 x 21 1 x 20 8 4 0 1 1310 So 11012 1310 2 110112 1x24 1x23 0x22 1x21 1x20 16 8 0 2 1 2710 So 110112 2710 3 1010112 1x25 0x24 1x23 0x22 1x21 1x20 32 0 8 0 2 1 So 1010112 4310 Now try these examples Expand and convert these examples to decimal Answers found at end of document a 11011012 b 111002 c 10101012 d 1000012 Copyright 2008 by Helene G Kershner CSE111 Spring 2009 H Kershner Binary Arithmetic Decimal to Binary Conversion Decimal to Binary conversion is performed by a series of short division short division is where you have a remainder Quotient Divisor Remainder Dividend Examples 1 2710 Binary 2 27 13 R 1 Repeat until the quotient is zero 0 2 27 2 13 2 6 2 3 1 R R R R 1 1 0 1 Translated number is read bottom up Recording the bits from left to right 2710 Binary 2710 1 1 0 1 12 Let s check our results Expand the binary number into powers of 2 and convert to decimal 1 1 0 1 12 1 x 24 1 x 23 0 x 22 1 x 21 1 x 20 16 2 8 0 2 1 2710 5210 Binary 2 52 2 26 R 0 2 13 R 0 Recording from the bottom up 2 2 Recording the bits from left to right 6 R 1 3 R 0 1 R 1 5210 1101002 Copyright 2008 by Helene G Kershner CSE111 Spring 2009 H Kershner Binary Arithmetic Let s check this example Expand the binary number and convert 1101002 1x25 1x24 0x23 1x22 0x21 0x20 32 16 0 4 0 0 1101002 5210 3 4410 Binary 2 44 2 22 R 0 2 11 R 0 2 5 R 1 2 2 R 1 1 R 0 Recording from the bottom up Recording the bits from left to right 4410 1011002 Check this example by expanding the binary and converting 1011002 1x25 0x24 1x23 1x22 0x21 0x20 32 0 8 4 0 0 4410 4 10310 Binary 2 103 2 51 R 1 2 25 R 1 2 12 R 1 2 6 R 0 2 3 R 0 1 R 1 Recording from the bottom up Recording the bits from left to right 10310 11001112 Check this example by expanding the binary and converting 11001112 1x26 1x25 0x24 0x23 1x22 1x21 1x20 64 32 0 0 4 2 1 10310 Copyright 2008 by Helene G Kershner CSE111 Spring 2009 H Kershner Binary Arithmetic Try these examples a 3310 binary c 9410 binary b 7610 binary d 6710 binary Copyright 2008 by Helene G Kershner CSE111 Spring 2009 H Kershner Binary Arithmetic Solutions Expanded binary numbers a 111102 1x24 1x23 1x22 1x21 0x20 b 10101012 1x26 0x25 1x24 0x23 1x22 0x21 1x20 c 11100112 1x26 1x25 1x24 0x23 0x22 1x21 1x20 Binary to Decimal Conversion a 11011012 10910 Work 11011012 1x26 1x25 0x24 1x23 1x22 0x21 1x20 64 32 0 b 8 4 0 1 10910 111002 2810 Work 111002 1x24 1x23 1x22 0x21 0x20 16 8 c 4 0 0 2810 10101012 8510 Work 10101012 1x26 0x25 1x24 0x23 1x22 0x21 1x20 64 0 d 16 0 4 0 1 8510 1000012 3310 Work 1000012 1x25 0x24 0x23 0x22 0x21 1x20 32 0 0 0 0 1 3310 Decimal to Binary Conversion a 3310 binary 1000012 2 33 2 16 R 1 2 8 R 0 2 4 R 0 2 2 R 0 Record bottom up 1000012 1 R 0 Copyright 2008 by Helene G Kershner CSE111 Spring 2009 H Kershner Binary Arithmetic b 7610 binary 10011002 2 76 2 38 R 0 2 19 R 0 2 9 R 1 2 4 R 1 2 2 R 0 Record bottom up 10011002 1 R 0 Let s check this answer 10011002 1x26 0x25 0x24 1x23 1x22 0x21 0x20 64 0 c 0 8 4 0 0 7610 9410 binary 10111102 2 94 2 47 R 0 2 23 R 1 Record bottom up 2 11 R 1 2 5 R 1 2 2 R 1 10111102 1 R 0 Check 10111102 1x26 0x25 1x24 1x23 1x22 1x21 0x20 d 64 0 16 8 4 2 0 9410 6710 binary 10000112 2 67 2 33 R 1 2 16 R 1 2 8 R 0 2 4 R 0 2 2 R 0 Record bottom up 10000112 1 R 0 Check 10000112 1x26 0x25 0x24 0x23 0x22 1x21 1x20 64 0 0 0 0 2 1 6710 Copyright 2008 by Helene G Kershner
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