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Number Systems Decimal Binary Denary Octal Hexadecimal Click the mouse or Press the space bar to Continue

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Exponents 2 0 = 1 2 1 = 2 2 2 = 2 x 2 =4 2 3 = 2 x 2 x 2 = 8 x 5 + x 10 = x 15 1 / x 2 = x -2 Click to Continue

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Decimal Numbering systems Base: 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Representation 5234 ThousandsHundredsTensUnits 5 2 3 4 Click to Continue

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Decimal Numbering systems Example: 5234 10 10 3 = 1000 10 2 = 100 10 1 = 10 10 0 = 1 5 2 3 4 5,234 = 5 x 1000 + 2 x 100 + 3 x 10 + 4 x 1 Click to Continue

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Binary Numbering systems Base: 2 Digits: 0, 1 binary number:110101 2 positional powers of 2: 2 5 2 4 2 3 2 2 2 1 2020 decimal positional value: 32 16 8 4 2 1 binary number: 1 1 0 1 0 1 Click to Continue

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Binary to Decimal Conversion To convert to base 10, add all the values where a one digit occurs. Ex: 110101 2 positional powers of 2: 2 5 2 4 2 3 2 2 2 1 2020 decimal positional value: 32 16 8 4 2 1 binary number: 1 1 0 1 0 1 32 +16 + 4 + 1 = 53 10 Click to Continue

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Binary to Decimal Conversion Ex: 101011 2 positional powers of 2: 2 5 2 4 2 3 2 2 2 1 2 0 decimal positional value: binary number: Click to Continue

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Decimal to Binary Conversion The Division Method. Divide by 2 until you reach zero, and then collect the remainders in reverse. Ex 1: 56 10 = 111000 2 2 ) 56 Rem: 2 ) 28 0 2 ) 14 0 2 ) 7 0 2 ) 3 1 2 ) 1 1 0 1 Click to Continue

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Decimal to Binary Conversion Ex 2:35 10 = 2 ) Rem: 2 ) Answer: 35 10 = 2 Click to Continue

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Decimal to Binary Conversion The Subtraction Method: Subtract out largest power of 2 possible (without going below zero) each time until you reach 0. Place a one in each position where you were able to subtract the value, and a 0 in each position that you could not subtract out the value without going below zero. Click to Continue

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Decimal to Binary Conversion Ex 1: 56 10 562 6 | 2 5 2 4 2 3 2 2 2 1 2 0 - 32 64 | 32 16 8 4 2 1 24 | 1 1 1 0 0 0 - 16 8 - 8 0 Answer: 56 10 = 111000 2 Click to Continue

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Decimal to Binary Conversion Ex 2: 38 10 382 6 | 2 5 2 4 2 3 2 2 2 1 2 0 | | Answer: 38 10 = 2 Click to Continue

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Character Representation ASCII Table RightmostLeftmost Three Bits Four Bits000001010011100101110111 0000NULDLESpace0@P`p 0001SOHDC1!1AQaq 0010STXDC2"2BRbr 0011ETXDC3#3CScs 0100EOTDC4$4DTdt 0101ENQNAK%5EUeu 0110ACKSYN&6FVfv 0111BELETB'7GWgw 1000BSCAN(8HXhx 1001HTEM)9IYIy 1010LFSUB*:JZjz 1011VTESC+;K[k{ 1100FFFS,

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Character Representation Ex: Find the binary ASCII and decimal ASCII values for the ‘&’ character. RightmostLeftmost Three Bits Four Bits000001010011100101110111 0000NULDLESpace0@P`p 0001SOHDC1!1AQaq 0010STXDC2"2BRbr 0011ETXDC3#3CScs 0100EOTDC4$4DTdt 0101ENQNAK%5EUeu & 0110ACKSYN & 6FVfv 0111BELETB'7GWgw 1000BSCAN(8HXhx 1001HTEM)9IYIy 1010LFSUB*:JZjz 1011VTESC+;K[k{ 1100FFFS,

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Character Representation ASCII Table From the chart: ‘&’ = 0100110(binary ASCII value) Convert the binary value to decimal: 0100110 2 = 32 + 4 + 2 = 38 10 Therefore: ‘&’ = 38(decimal ASCII value) Click to Continue

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Octal Numbering systems Base: 8 Digits: 0, 1, 2, 3, 4, 5, 6, 7 Octal number:1246 8 powers of : 8 4 8 3 8 2 8 1 8080 decimal value: 4096 512 64 8 1 Octal number: 1 2 4 6 Click to Continue

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Octal to Decimal Conversion To convert to base 10, beginning with the rightmost digit multiply each nth digit by 8 (n-1), and add all of the results together. Ex: 1246 8 positional powers of 8: 8 3 8 2 8 1 8 0 decimal positional value: 512 64 8 1 Octal number: 1 2 4 6 512 + 128 + 32 + 6 = 678 10 Click to Continue

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Octal to Decimal Conversion Ex: 10352 8 positional powers of 8: 8 4 8 3 8 2 8 1 8 0 decimal positional value: Octal number: Click to Continue

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Decimal to Octal Conversion The Division Method. Divide by 8 until you reach zero, and then collect the remainders in reverse. Ex 1: 4330 10 = 10352 8 8 ) 4330 Rem: 8 ) 541 2 8 ) 67 5 8 ) 8 3 8 ) 1 0 0 1 Click to Continue

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Decimal to Octal Conversion Ex 2:810 10 = 8 ) 810 Rem: 8 ) Answer: 810 10 = 8 Click to Continue

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Decimal to Octal Conversion The Subtraction Method: Subtract out multiples of the largest power of 8 possible (without going below zero) each time until you reach 0. Place the multiple value in each position where you were able to subtract the value, and a 0 in each position that you could not subtract out the value without going below zero. Click to Continue

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Decimal to Octal Conversion Ex 1: 2018 10 2018 8 4 | 8 3 8 2 8 1 8 0 - 1536 4096 | 512 64 8 1 482 | 3 7 4 2 - 448 34 - 32 2 - 2 0Answer: 2018 10 = 3742 8

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Decimal to Octal Conversion Ex 2: 765 10 765 8 4 | 8 3 8 2 8 1 8 0 | | Answer: 765 10 = 8

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Hexadecimal Numbering systems Base: 16 Digits: 0, 1, 2, 3, 4, 5, 6, 7,8,9,A,B,C,D,E,F Hexadecimal number:1F4 16 powers of : 16 4 16 3 16 2 16 1 16 0 decimal value: 65536 4096 256 16 1 Hexadecimal number: 1 F 4

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Hexadecimal Numbering systems Four-bit GroupDecimal DigitHexadecimal Digit 0000 0 0 0001 1 1 0010 2 2 0011 3 3 0100 4 4 0101 5 5 0110 6 6 0111 7 7 1000 8 8 1001 9 9 1010 10 A 1011 11 B 1100 12 C 1101 13 D 1110 14 E 1111 15 F

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Hexa to Decimal Conversion To convert to base 10, beginning with the rightmost digit multiply each nth digit by 16 (n-1), and add all of the results together. Ex: 1F4 16 positional powers of 16: 16 3 16 2 16 1 16 0 decimal positional value: 4096 256 16 1 Hexadecimal number: 1 F 4 256 + 240 + 4 = 500 10

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Hexa to Decimal Conversion Ex: 7E 16 positional powers of 16: 16 3 16 2 16 1 16 0 decimal positional value: Hexa number:

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Decimal to Hexa Conversion The Division Method. Divide by 16 until you reach zero, and then collect the remainders in reverse. Ex 1: 126 10 = 7E 16 16) 126 Rem: 16) 7 14=E 0 7

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Decimal to Hexa Conversion Ex 2:810 10 = 16 ) 810 Rem: 16 ) Answer: 810 10 = 16

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Decimal to Hexa Conversion The Subtraction Method: Subtract out multiples of the largest power of 16 possible (without going below zero) each time until you reach 0. Place the multiple value in each position where you were able to subtract the value, and a 0 in each position that you could not subtract out the value without going below zero.

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Decimal to Hexa Conversion Ex 1: 810 10 810 16 3 | 16 2 16 1 16 0 - 768 4096 | 256 16 1 42 | 3 2 A - 32 10 - 10 0Answer: 810 10 = 32A 16

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Decimal to Hexa Conversion Ex 2: 156 10 156 16 2 | 16 1 16 0 | | Answer: 156 10 = 16

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Binary to Octal Conversion Since the maximum value represented in 3 bit is equal to: 2 3 – 1 = 7 i.e. using 3 bits we can represent values from 0 –7 which are the digits of the Octal numbering system. Thus, three binary digits can be converted to one octal digit.

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Binary to Octal Conversion Three-bit GroupDecimal DigitOctal Digit 000 0 0 001 1 1 010 2 2 011 3 3 100 4 4 101 5 5 110 6 6 111 7 7

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Octal to Binary Conversion Ex : Convert 742 8 = 2 7 = 111 4 = 100 2 = 010 742 8 = 111 100 010 2

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Binary to Octal Conversion Ex : Convert 10100110 2 = 8 110 = 6 100 = 4 010 = 2 ( pad empty digits with 0) 10100110 2 = 246 8

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Binary to Hexa Conversion Since the maximum value represented in 4 bit is equal to: 2 4 – 1 = 15 i.e. using 4 bits we can represent values from 0 –15 which are the digits of the Hexadecimal numbering system. Thus, Four binary digits can be converted to one Hexadecimal digit.

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Binary to Hexa conversion Four-bit GroupDecimal DigitHexadecimal Digit 0000 0 0 0001 1 1 0010 2 2 0011 3 3 0100 4 4 0101 5 5 0110 6 6 0111 7 7 1000 8 8 1001 9 9 1010 10 A 1011 11 B 1100 12 C 1101 13 D 1110 14 E 1111 15 F

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Hexa to Binary Conversion Ex : Convert 3D9 8 = 2 3 = 0011 D = 1101 9 = 1001 3D9 8 = 0011 1101 1001 2

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Binary to Hexa Conversion Ex : Convert 10100110 2 = 8 0110 = 6 1010 = A 10100110 2 = A6 16

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Octal to Hexa Conversion To convert between Octal to Hexadecimal numbering systems and visa versa convert from one system to binary first then convert from binary to the new numbering system

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Hexa to Octal Conversion Ex : Convert E8A 16 = 8 1110 1000 1010 2 111010001010(group by 3 bits) 72127212 E8A 16 = 7217 8

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Octal to Hexa Conversion Ex : Convert 752 8 = 16 111 101 010 2 (group by 4 bits) 000111101010 1EA1EA 752 8 = 1EA 16

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