Denary System

• Traditional number system of base 10, Digits from 0 to 9

Binary System

• Base 2 number system with values of 0 and 1
• 0 represents off/false/dark spot/low voltage
• 1 represents on/true/light spot/high voltage
• Any form of data needs to be converted to binary to be processed by a computer
• Data is processed using logic gates and stored in registers
• Each digit is called Bit (Bit = Binary Digit)
• Most left bit is called the MSB (Most Significant Bit)
• The right most bit is called the LSB (Least Significant Bit)
• 4 bits = a nibble
• 8 bits = a byte

Hexadecimal Number System

• Base 16 number system
• Values from 0 to 9 followed by A to F
• Advantages:
  • Easier to code, faster, and less error-prone than binary
• Applications: IP Address, Error Codes, URL, Assembly Language, Memory Dumps, Locations in Memory, Color Codes of HTML, MAC Address

Number Values

Prefixes

Binary Coded Decimals (BCD)

• Each positive denary digit is represented using a nibble (4 bits)
• Example: Convert 928 to binary
  • 9 = 1001 | 2 = 0010 | 8 = 1000
  • Binary Value: 1001 0010 1000₂
• Applications
  • Displaying numbers in electronic device (eg. Calculators)
  • Accurately measuring decimal fractions
  • Electronically coding denary numbers

Converting Number Systems (Examples)

Denary to Binary

Method 1

Break the decimal number down into a sum of powers of 2 and then write the binary value
Example: Representing 553 in binary
553 = 512 + 32 + 8 + 1
Binary: 0010 0010 1001

Method 2

• Successively divide the denary value by 2, noting every remainder
• Repeat until quotient is 0
• Write the remainders in reverse order
• Example: Convert 35 to Binary
  Refer to the table to the right
  Answer: 100011₂

Binary To Denary

• Identify the position (or power of 2) of each digit, starting from the rightmost digit (which is 2⁰) and moving to the left.
• Multiply each binary digit by its corresponding power of 2.
• Sum all the products to get the denary equivalent.
• Example: 1011 1100₂ to Denary

1011 1100₂ = 2⁷ + 2⁵ + 2⁴ + 2³ + 2² = 128 + 32 + 16 + 8 + 4 = 188₁₀

Binary To Hexadecimal

• Make groups of four bits and convert each to hexadecimal
• Example: 1100 1110 = C E ₁₆

Hexadecimal To Binary

• Convert each Hex digit to binary and write serially
• Example: F B = 1111 1011₂

Denary to Hexadecimal

• Method 1: Convert the value to binary, and then convert it to hex
• Method 2: Break the decimal number down into a sum of powers of 16
  Example: 554 to Hexadecimal
  554₁₀ = 256 * 2 + 16 * 2 + 10 = 16² * 2 + 16¹ * 2 + 10 = 2 2 A₁₆

Hexadecimal to Denary

• Find the denary value for each digit
• Multiply with appropriate 16’s power and add up
• Example: E4₁₆
  • E = 14, 4 = 4
  • Answer: 14 * 16¹ + 4 * 16⁰ = 14 * 16 + 4 = 224 + 4 = 228₁₀

Binary Operations

Adding two positive 8-bit binary

Normal Binary Addition

Overflow

• A computer or a device has a predefined limit that it can represent or store, for example 16-bit
• When adding two values, an overflow error occurs when a value outside this limit should be returned
• E.g. The solution has 9 bits (Value greater than 255), but the question has 8 bits per value (8-bit register), the 9th bit (most left bit) is called overflow.
• Shows that memory doesn’t have enough space to store the answer

Example of Addition

Add: 1011 0111 and 0111 1111

Note: When adding values, move from RHS to LHS using above rules. For overflow bit denote using bracket
Answer: (1) 0011 0110

Logical Shifts

• Moving a binary value to the left or the right
• Most significant bits (MSB) or the least significant bits (LSB) are lost
• Empty positions are filled with 0.
• Shifting 1100 1010 two places to the left = 0010 1000
• Shifting 1100 1010 two places to the right = 0011 0010
• Each left shift multiplies the original number by 2 (Data may be lost)
• Each right shift divides the original number by 2 (Data may be lost)

Arithmetic Binary Shift

Moves all bits 1-place to the right and copies the sign bit into the MSB

Cyclic Shift

Bits which are removed from one end is added to the other

One’s Complement

• The negative version of a binary number is formed by flipping all the bits

• Example: 47₁₀ = 0101111₂

  One’s Complement Representation: 101000₂

Two’s Complement

• Used to represent signed integer
• The most significant bit (MBS) becomes a sign bit denoting positive or negative number
• Maximum positive number in 8 bits: 127
• Maximum negative number in 8 bits: -128

Converting negative denary to two’s complement (Example: –61):

• Find the binary equivalent of the denary number (ignoring the negative sign) | 61 = 111101
• Add extra 0 bits before the MSB | 00111101
• Convert to one’s complement (flip the bits) | 11000010
• Convert to two’s complement (add 1) | 11000011

Converting binary two’s complement into denary (example 11010110):

• Flip all the bits | 00101001
• Add 1 | 00101010
• Convert to denary and put a negative sign | -42

Binary Subtraction

• 1) For number with different length, add 0 to the smaller to give same length
• 2) Add a 0 as MSB (will represent the sign bit) to both numbers
• 3) Convert the subtrahend to its 2’s complement
• 4) Add with the minuend
• 5) Omit the carry if working with signed two’s complement numbers.
• Example: 1100 - 10001
  • 1) The question becomes 01100 -10001
  • 2) The question becomes 001100 - 010001
  • 3) 2’s complement of subtrahend = 101111
  • 4) 001100 + 101111 = 111011
  • 5) No carry so answer is 111011 which is a 2’s complement representation

Text

• A character set generally includes upper & lower case letters, number digits, punctuation marks and other characters
• Character sets use different binary representations for each character via character encoding
• Value of A = 65, a = 97, 0 = 48
• Character Encoding Standards: