Overview of this chapter

• Binary System
  • Basic unit
  • Conversion between Binary and Denary
• Hexadecimal System
  • Advantage & Usage
  • Conversion between Hex and Denary
• Conversion between Binary and Hex

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1.1 Binary System

The binary number system (base 2) which comprises of only 0s and 1s is used by computers to process all data.

UNIT

• Bit: a binary digit
  E.g. 0 or 1 is represented by 1 bit
• Nibble = 4 bits
• Byte = 8 bits
  1 character = 1 byte
• 1 KB = 2^10 B = 1024 bytes
• 1 MB = 2^20 B = 1024 KB
• 1 GB = 2^30 B = 1024 MB
• 1 TB = 2^40 B = 1024 GB

Note: In the exam, because of lack of calculator, examiners usually permits using 1000 in the conversion between units.

CONVERSION

From Denary to Binary

Process: Divide the denary number by two repeatedly, meanwhile keep track of remainders after each division, until the quotient is 0. Assemble the set of remainders reversely

Example: Convert 25(base 10) from denary to binary:

1. Divide by 2
   the first remainder is 0
2. Divide by 2
   the second remainder is 1
   …
3. Divide by 2
   the fifth remainder is 1
   the quotient is 0
4. Assemble all remainders in opposite direction, we get 11001(base2)
   

From Binary to Denary

• Multiply each bit by 2n, where n is the position of the bit, start from 0 on the right
• Add up the results
• Example: Convert 11001 to denary
  1 * 2^4 + 1* 2^3 + 0* 2^2 + 0 * 2^1 + 1 * 2^0 = 16 + 8 + 1 = 25

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1.2 Hexadecimal System

ADVANTAGE

• Easier for humans to communicate with computersCOMPARE TO THE BINARY SYSTEM, 4 BITS IS REPRESENTED BY A SINGLE SYMBOL
• Fewer digits to use
• Smaller Screen is required
• Easier to find errors / less likely to make errors
• Faster than entering binary digits
• Conversion to binary is easier than denary to binary

USAGE

• Display error code
  E.g. #404 – page not found
• HTML colour code
  E.g. FF0000 is red
• Media Access Control (MAC) addressesPHYSICAL UNIQUE ADDRESS OF EACH NETWORK INTERFACE CARD THAT MADE UP OF 2 PARTS, MANUFACTURE ID AND SERIAL NUMBERE.g. A4-50-B4-C2-60-3E
• Web address
  E.g. www: %77%77%77 (%means hexadecimal is used)
• Address of memory location
  E.g. STO FFA4

CONVERSION

From Denary to Hexadecimal

• Divide the number by 16
• Keep track of the remainder
• E.g. Convert 504 to Hexadecimal
  16 | 504
  16 | 31 | 8
  16 | 01 | 15
  —- | 00 | 1
  Result: 1F8

From Hexadecimal to Denary

• Multiply each bit by 16^n, where n is the position of the bit, start from 0 on the right
• Add up the results
• E.g. Convert A to denary
  10 * 16^0 = 10

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1.3 Conversion between Hexadecimal and Binary

FROM HEX TO BINARY

• Convert each digit to denary
• Convert each denary digit to 4-bits binary digits
• Merge the digit
• Example: 18F -> 0001 1000 1111 -> 110001111

FROM BINARY TO HEX

• Group bits by 4 from right to left
• Add additional zeros to the left if necessary
• Convert each group to denary
• Represents them in hexadecimal
• Example: 110001111 -> 0001 1000 1111 -> 1 8 15 -> 18F