3. Binary Arithmetics

Digital electronics processes information using separate or discrete values.

The word digital comes from digit, which means a number symbol.

Digital circuits normally use two logic levels:

• 0 – LOW
• 1 – HIGH

Positive and Negative Logic

Logic defines how voltage levels represent binary values.

Table 1: Logic Conventions

Positive logic is used more widely in digital circuits.

Importance of Number Systems

Number systems are used to:

• Represent digital data
• Store values in registers
• Perform arithmetic operations
• Write memory addresses
• simplify long binary values

A number system uses a fixed set of digits and a base.

The value of each digit depends on its position:

Value = Digit × Baseᵖᵒˢⁱᵗⁱᵒⁿ

Table 2: Common Number Systems

In hexadecimal:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

The same value will be used in all examples:

45₁₀ = 101101₂ = 55₈ = 2D₁₆

Decimal to Other Number Systems

Decimal to Binary

Repeatedly divide the decimal number by 2.

Write the remainders from bottom to top.

Example: Convert 45₁₀ to Binary

Reading upward:

45₁₀ = 101101₂

Decimal to Octal

Repeatedly divide the decimal number by 8.

45 ÷ 8 = 5, remainder 5

5 ÷ 8 = 0, remainder 5

Therefore:

45₁₀ = 55₈

Decimal to Hexadecimal

Repeatedly divide the decimal number by 16.

45 ÷ 16 = 2, remainder 13

13 = D

Therefore:

45₁₀ = 2D₁₆

Table 3: Decimal Conversion Summary

Binary to Other Number Systems

Binary to Decimal

Multiply every binary bit by its positional weight.

101101₂
= 1×2⁵ + 0×2⁴ + 1×2³ + 1×2² + 0×2¹ + 1×2⁰
= 32 + 8 + 4 + 1
= 45₁₀

Binary to Octal

Group binary bits into sets of 3, starting from the right.

101 101

• 101₂ = 5₈
• 101₂ = 5₈

Therefore:

101101₂ = 55₈

Binary to Hexadecimal

Group binary bits into sets of 4, starting from the right.

Add leading zeros when required:

0010 1101

• 0010₂ = 2₁₆
• 1101₂ = D₁₆

Therefore:

101101₂ = 2D₁₆

Table 4: Binary Conversion Summary

Octal to Other Number Systems

Octal to Binary

Replace each octal digit with its 3-bit binary value.

5₈ = 101₂

Therefore:

55₈ = 101 101₂

55₈ = 101101₂

Octal to Decimal

Multiply every digit by its positional weight.

55₈
= 5×8¹ + 5×8⁰
= 40 + 5
= 45₁₀

Octal to Hexadecimal

First convert octal to binary:

55₈ = 101101₂

Group the binary value into 4 bits:

0010 1101

Therefore:

55₈ = 2D₁₆

Table 5: Octal Conversion Summary

Hexadecimal to Other Number Systems

Hexadecimal to Binary

Replace each hexadecimal digit with its 4-bit binary value.

• 2₁₆ = 0010₂
• D₁₆ = 1101₂

Therefore:

2D₁₆ = 0010 1101₂

Removing leading zeros:

2D₁₆ = 101101₂

Hexadecimal to Decimal

Multiply every digit by its positional weight.

2D₁₆
= 2×16¹ + 13×16⁰
= 32 + 13
= 45₁₀

Hexadecimal to Octal

First convert hexadecimal to binary:

2D₁₆ = 00101101₂

Group the bits into sets of 3:

00 101 101

101₂ = 5₈

Therefore:

2D₁₆ = 55₈

Table 6: Hexadecimal Conversion Summary

Conversion Shortcuts

Table 7: Number-System Conversion Methods

Signed Binary Representation

Signed binary numbers represent both positive and negative values.

The most significant bit is used as the sign bit:

• Sign bit 0: Positive
• Sign bit 1: Negative

The remaining bits represent the value.

Sign-Magnitude Range

For an n-bit sign-magnitude number:

Range = −(2ⁿ⁻¹ − 1) to +(2ⁿ⁻¹ − 1)

For 4 bits:

Range = −7 to +7

Sign-magnitude has two zero values:

• 0000 = +0
• 1000 = −0

Binary Addition

Binary addition follows four basic rules.

Table 8: Binary Addition Rules

Example

   1011
 + 0110
 -------
  10001

1011₂ = 11₁₀
0110₂ = 6₁₀
10001₂ = 17₁₀

1's Complement

The 1's complement of a binary number is found by inverting every bit.

• Change 0 to 1.
• Change 1 to 0.

Example

Original number:

0101

Invert all bits:

1010

Therefore, in 4-bit 1's complement:

• +5 = 0101
• −5 = 1010

Limitation

1's complement has two representations of zero:

• Positive zero: 0000
• Negative zero: 1111

Addition may also require an end-around carry.

2's Complement

The 2's complement is found in two steps:

1. Find the 1's complement.
2. Add 1.

Example: Find −5

Positive 5:

0101

1's complement:

1010

Add 1:

  1010
+ 0001
------
  1011

Therefore:

−5 = 1011

Range

For an n-bit 2's complement number:

Range = −2ⁿ⁻¹ to +(2ⁿ⁻¹ − 1)

For 4 bits:

Range = −8 to +7

Benefits

• Only one representation of zero
• Simple addition and subtraction
• Widely used in processors and digital circuits

Table 9: Signed Binary Representations

Binary Subtraction

Binary subtraction uses borrowing, similar to decimal subtraction.

Basic rules:

• 0 − 0 = 0
• 1 − 0 = 1
• 1 − 1 = 0
• 0 − 1 requires a borrow

After borrowing:

10₂ − 1₂ = 1₂

Example

   10110
 - 00101
 --------
   10001

10110₂ = 22₁₀
00101₂ = 5₁₀
10001₂ = 17₁₀

Subtraction Using 2's Complement

Binary subtraction can be changed into addition:

A − B = A + 2's complement of B

Example: 10110 − 00101

Step 1: Find 2's Complement of 00101

1's complement:

11010

Add 1:

11011

Step 2: Add It to 10110

   10110
 + 11011
 --------
 1 10001

Discard the final carry:

10001

Therefore:

10110₂ − 00101₂ = 10001₂

Result Rule

• Final carry present: Discard the carry. The result is positive.
• No final carry: Take the 2's complement of the result and add a negative sign.

Binary codes represent numbers, characters or changing states using binary bits.

Gray Code

Gray code is a binary code in which two consecutive values differ by only one bit.

It is also called a unit-distance code.

Example

Properties

• Only one bit changes between adjacent values.
• It is a non-weighted code.
• It reduces transition errors.
• It is not normally used for arithmetic.

Applications

• Rotary encoders
• Position sensors
• Shaft-angle measurement
• Error reduction during state changes

BCD Code

BCD means Binary-Coded Decimal.

Each decimal digit is represented separately using four binary bits.

BCD normally uses the 8421 weighted code.

Example

Decimal value:

59

• 5 = 0101
• 9 = 1001

Therefore:

59₁₀ = 0101 1001 in BCD

BCD 1010 to 1111 are invalid decimal-digit codes.

Properties

• Each decimal digit uses four bits.
• It is a weighted code.
• Decimal digits remain separate.
• It requires more bits than pure binary.

Applications

• Calculators
• Digital clocks
• 7-segment displays
• Decimal measurement systems

Excess-3 Code

Excess-3 represents each decimal digit after adding 3 to it.

Example

Convert decimal 59:

For digit 5:

5 + 3 = 8 = 1000₂

For digit 9:

9 + 3 = 12 = 1100₂

Therefore:

59₁₀ = 1000 1100 in Excess-3

Properties

• It is a non-weighted code.
• It is a self-complementing code.
• Each decimal digit uses four bits.
• It is derived from BCD by adding 0011.

Applications

• Decimal arithmetic circuits
• Code converters
• Complement-based decimal operations

ASCII Code

ASCII means American Standard Code for Information Interchange.

It represents:

• Letters
• Numbers
• Symbols
• Control characters

Standard ASCII uses 7 bits.

Examples

• A = 65₁₀ = 1000001₂
• a = 97₁₀ = 1100001₂
• 0 = 48₁₀ = 0110000₂

Properties

• Represents 128 standard characters.
• It is used for character data, not arithmetic values.
• Extended ASCII systems commonly use 8 bits.
• Uppercase and lowercase letters have different codes.

Applications

• Computer keyboards
• Text files
• Serial communication
• Computer terminals
• Embedded-system displays

Binary-Code Comparison

Table 10: Binary Code Comparison