Comprehensive Guide to Binary Arithmetic, Two's Complement, and Hexadecimal Systems

Fundamental Computer Logic and Base Number Systems

• Binary Processing in Computers:

  • Computers process all data exclusively in binary.

  • This occurs because data processing relies on logic gates, which operate with only two distinct states: $0$ or $1$

• Base System Definitions:

  • Denary System: Base-$10$ number system (using digits $0$ through $9$

  • Binary System: Base-$2$ number system (using digits $0$ and $1$

  • Hexadecimal System: Base-$16$ number system (using digits $0$ through $9$ and letters $A$ through $F$

  • Exactly $4$ binary digits (bits) correspond to $1$ hexadecimal digit.

• Padding Binary Numbers:

  • When adding extra digits to a binary number to extend it to a full byte (such as an $8$-bit representation), zeros must always be added to the left side of the value, never to the right side.

Binary and Denary Conversions

• Binary to Denary Conversion:

  • Position each binary digit under its corresponding positional base-$2$ heading ($128, 64, 32, 16, 8, 4, 2, 1$

  • Sum the positional values where the bit is set to $1$

  • Conversion Example: Convert binary $11101110_2$ to denary:

  • $128 + 64 + 32 + 8 + 4 + 2 = 238$

  • Denary value = $238$

• Denary to Binary Conversion (Successive Division by 2):

  • Divide the denary integer continuously by $2$

  • Record the remainder ($0$ or $1$) at each division step.

  • Continue division until the quotient is $0$

  • Write out the remainders from bottom to top (last remainder recorded to first remainder recorded).

  • Conversion Example: Convert denary $142$ to binary:

  • $142$

  • $71$

  • $35$

  • $17$

  • $8$

  • $4$

  • $2$

  • $1$

  • Binary result (reading remainders bottom to top): $10001110_2$

Binary Arithmetic, Overflow, and Shifts

• Binary Addition Rules:

  • $0 + 0 = 0$

  • $0 + 1 = 1$

  • $1 + 0 = 1$

  • $1 + 1 = 10$ (result $0$, carry $1$

  • $1 + 1 + 1 = 11$ (result $1$, carry $1$

• Overflow Error:

  • Definition: Occurs when the result of a calculation exceeds the maximum capacity of the allocated number of bits.

  • Limits: In an $8$-bit system, the maximum representable denary value is $255$ ($11111111_2$

  • Cause: An overflow error happens when a calculation produces a result greater than $255$, which cannot be stored within the available $8$ bits.

• Binary Shift Operations:

  • Left Binary Shift:

  • Moves all bits to the left by a specified number of positions.

  • Effect: Multiplies the denary value by $2$ for each place shifted left.

  • Example:

    • Starting value: $00010100_2$ ($16 + 4 = 20$

    • Shifted 1 place left: $00101000_2$ ($32 + 8 = 40$

  • Right Binary Shift:

  • Moves all bits to the right by a specified number of positions.

  • Effect: Divides the denary value by $2$ for each place shifted right.

Two's Complement Representation

• Purpose and Usage:

  • Two's complement is used to represent negative denary numbers in binary.

  • Simplifies the architectural implementation of binary subtraction in computers.

• Bit Identification:

  • Most Significant Bit (MSB): The leftmost bit in the sequence.

  • Least Significant Bit (LSB): The rightmost bit in the sequence.

• Positive Denary in Two's Complement:

  • For positive numbers, the Most Significant Bit (MSB) always remains $0$

  • Example: Convert positive denary $38$ to an $8$-bit Two's complement binary representation:

  • Place values: $128, 64, 32, 16, 8, 4, 2, 1$

  • Representation: $00100110_2$ ($32 + 4 + 2 = 38$, where MSB is $0$

• Converting Positive Denary to Negative Denary in Two's Complement:

  1. Convert the positive denary number to standard binary.

  2. Invert all the bits (flip all $0$s to $1$s and all $1$s to $0$s).

  3. Add $1$ to the inverted binary result.

Hexadecimal Conversion Methods

• Hexadecimal Mapping Table:

  • Values $0$ through $9$ are represented as digits $0$ to $9$

  • Denary $10 = A$

  • Denary $11 = B$

  • Denary $12 = C$

  • Denary $13 = D$

  • Denary $14 = E$

  • Denary $15 = F$

• Binary to Hexadecimal Conversion:

  1. Divide the binary string into groups of $4$ bits (nibbles) starting from the right.

  2. Convert each $4$-bit group into its equivalent hexadecimal character.

  • Conversion Example: Convert $101111100001_2$ to hexadecimal:

  • Grouping: $1011 \quad 1110 \quad 0001$

  • $1011_2 = 11_{10} = B$

  • $1110_2 = 14_{10} = E$

  • $0001_2 = 1_{10} = 1$

  • Hexadecimal value = $BE1_{16}$

• Hexadecimal to Denary Conversion:

  • Multiply each hexadecimal digit by its corresponding base-$16$ positional heading ($4096, 256, 16, 1$) and sum the products.

  • Conversion Example: Convert hexadecimal $45A_{16}$ to denary:

  • Headings: $256, 16, 1$

  • Hex Digits: $4, 5, A$ (where $A = 10$

  • Calculation: $(4 \times 256) + (5 \times 16) + (10 \times 1) = 1024 + 80 + 10 = 1114$

  • Denary value = $1114$

• Denary to Hexadecimal Conversion (Successive Division by 16):

  • Divide the denary integer continuously by $16$

  • Record the remainder at each step, mapping remainders between $10$ and $15$ to letters $A$ through $F$

  • Read the remainders from bottom to top (last remainder recorded to first remainder recorded).

  • Conversion Example: Convert denary $2004$ to hexadecimal:

  • $2004 \div 16 = 125 \text{ remainder } 4$

  • $125 \div 16 = 7 \text{ remainder } 13 \quad (13 = D)$

  • $7 \div 16 = 0 \text{ remainder } 7$

  • Hexadecimal value (read bottom to top) = $7D4_{16}$

Practical Applications of Hexadecimal

• Error Codes:

  • Automatically generated by the computer system.

  • Refer directly to the memory location where an error has occurred.

• Media Access Control (MAC) Addresses:

  • A unique identifier assigned to a network interface controller (NIC) or device on a network.

  • Made up of $48$ bits.

  • Displayed as $6$ groups of $2$ hexadecimal digits.

  • Rarely changed, enabling persistent device identification regardless of network connection location.

• IP Addresses (IPv6):

  • The network address assigned to each device connected to a network.

  • IPv6 consists of a $128$-bit number broken down into $16$-bit chunks, represented using hexadecimal format.

• HTML Color Codes:

  • Used to specify colors of text and elements on computer displays.

  • Intensity levels of different color components are defined by hexadecimal values.