Computer arithmetic and logic form the foundation of all computing processes, enabling computers to perform calculations, make decisions, and process data efficiently. Mastery of number systems, their operations, and conversions is essential for computer scientists, especially in Kenya's growing software development and data science industries. This chapter focuses on the key number systems used in computer architecture, how arithmetic operations are performed within these systems, and how to convert numbers between them for effective computation and programming.

5.1 Number systems

Number systems are the mathematical frameworks computers use to represent, store, and manipulate data. Understanding these systems is crucial for tasks such as programming, circuit design, and algorithm development in Kenya’s IT sector. This section covers the types of number systems, operations performed on them, and conversion techniques between different systems.

5.1.1 Types of number systems

Number systems define the base or radix used to represent numbers. The most common types in computer science are binary, octal, decimal, and hexadecimal. Each system uses a unique set of digits and positional values to express numbers.

Binary number system

The binary system uses base 2 and digits 0 and 1. It is the fundamental language of computers because digital circuits have two states: ON and OFF.

• Each digit is called a bit.
• The rightmost bit represents \(2^0\), the next \(2^1\), and so on.
• Binary is used in machine-level programming and hardware design.
• It represents logical values in computer logic gates.
• Binary arithmetic underlies all higher-level computations.

Octal number system

The octal system uses base 8 and digits 0 to 7.

• Each octal digit represents three binary bits.
• It provides a shorthand for binary numbers, simplifying reading and writing.
• Frequently used in older computing systems and permissions in Unix/Linux.
• It reduces the length of binary sequences.
• Useful in error detection and data compression techniques.

Decimal number system

The decimal system uses base 10 and digits 0 to 9.

• It is the standard human counting system.
• Computers convert decimal inputs into binary for processing.
• Most programming languages accept decimal literals.
• Decimal system is used in financial and business applications.
• Programming often requires conversion between decimal and other bases.

Hexadecimal number system

The hexadecimal system uses base 16 and digits 0 to 9 plus A to F.

• Each hex digit represents four binary bits.
• It is widely used in memory addressing and debugging.
• Hexadecimal simplifies large binary number representation.
• Common in color codes in web design and graphics programming.
• Used extensively in low-level programming and assembly language.

Worked Examples

Example 1: Convert the decimal number 13 to binary.

Given: Decimal number = 13

Divide by 2 repeatedly, recording remainders:

13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1

Reading remainders from bottom to top: 1101

Answer: Binary 1101

Example 2: Convert binary 101101 to octal.

Group binary digits in sets of 3 from right: 101 101

Convert each group:

101 = 5
101 = 5

Answer: Octal 55

Example 3: Convert hexadecimal 2F to decimal.

2F hex = (2 × 16) + (15 × 1) = 32 + 15 = 47

Answer: Decimal 47

5.1.2 Operations in number systems

Arithmetic operations such as addition, subtraction, multiplication, and division have distinct methods in different number systems. Mastering these operations is essential for algorithms and low-level programming in Kenya’s software development industry.

Binary addition

Binary addition follows rules similar to decimal addition but with base 2.

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (carry 1 to next higher bit)

Binary subtraction

Binary subtraction uses borrow similar to decimal subtraction.

• 0 - 0 = 0
• 1 - 0 = 1
• 1 - 1 = 0
• 0 - 1 requires borrowing 1 from next higher bit

Worked Example

Example: Subtract binary 1010 from 1101.

Given: \(1101_2 - 1010_2\)

Step 1: Align the numbers:

\( \begin{array}{cccc} & 1 & 1 & 0 & 1 \\ - & 1 & 0 & 1 & 0 \\ \end{array} \)

Step 2: Subtract bit by bit from right, handling borrows:

• Bit 0: \(1 - 0 = 1\)
• Bit 1: \(0 - 1\) (need to borrow)
  • Borrow 1 from bit 2: bit 2 becomes 0, bit 1 becomes \(2 - 1 = 1\)
• Bit 2: \(0 - 0 = 0\) (after borrowing)
• Bit 3: \(1 - 1 = 0\)

Final result: \(0011_2\)

Answer: \(11_2\) (decimal 3)

Binary multiplication

Binary multiplication is like decimal multiplication but simpler.

• Multiply each bit of the multiplier by the multiplicand.
• Shift results according to bit position.
• Add all shifted results.

Worked Example

Example: Multiply binary 101 by 11.

Given: \(101_2 \times 11_2\)

Step 1: Write 11 as two bits: \(11_2 = 3_{10}\)

Multiply each bit:

• \(101 \times 1 = 101\) (no shift)
• \(101 \times 1 = 101\) (shift left by 1: 1010)

Step 2: Add partial products:

\( \begin{array}{cccc} & & 1 & 0 & 1 \\ + & 1 & 0 & 1 & 0 \\ \hline & 1 & 1 & 1 & 1 \\ \end{array} \)

Answer: \(1111_2\) (decimal 15)

Binary division

Binary division involves repeated subtraction of divisor shifted appropriately.

• Compare divisor with leftmost bits of dividend.
• Subtract if divisor less or equal, else bring down next bit.
• Continue until all bits processed.

Worked Example

Example: Divide binary 1101 by 10.

Given: \(1101_2 \div 10_2\)

Step 1: Convert to decimal for verification: \(1101_2 = 13_{10}\), \(10_2 = 2_{10}\)

Step 2: Binary long division:

• 2 fits into 13 six times, remainder 1.
• In binary:
  • 10 fits into 11 (first two bits of 1101), quotient bit 1, subtract 10 from 11: 11 - 10 = 1
  • Bring down next bit: 1 becomes 10 (after shifting), 10 fits into 10 once, quotient bit 1, subtract 10 from 10: 10 - 10 = 0
  • Bring down last bit: 0 becomes 1, 10 does not fit, quotient bit 0, remainder 1

Quotient: \(110_2\) (decimal 6), Remainder: \(1_2\)

Answer: Quotient = \(110_2\), Remainder = \(1_2\)

Worked Examples

Example 1: Add binary numbers 1101 and 1011.

Given: 1101 + 1011

Add bit by bit from right:

1 + 1 = 10 (write 0 carry 1)
0 + 1 + carry 1 = 10 (write 0 carry 1)
1 + 0 + carry 1 = 10 (write 0 carry 1)
1 + 1 + carry 1 = 11 (write 1 carry 1)

Write carry 1 to leftmost position.

Result: 11000

Answer: Binary 11000

Example 2: Subtract binary 1010 from 1101.

Given: 1101 - 1010

Subtract bit by bit:

1 - 0 = 1
0 - 1 (borrow 1) = 1
0 - 0 = 0
1 - 1 = 0

Result: 0011

Answer: Binary 11

Example 3: Multiply binary 101 by 11.

Given: 101 × 11

101 × 1 = 101
101 × 1 (shifted left 1) = 1010

Add:
0101
+1010
=1111

Answer: Binary 1111

5.1.3 Conversion between number systems

Converting numbers accurately between decimal, binary, octal, and hexadecimal is vital for programming, debugging, and data representation in Kenyan IT environments.

Decimal to binary conversion

Use successive division by 2, recording remainders.

Binary to decimal conversion

Multiply each bit by \(2^{position}\) and sum all.

Binary to octal and hexadecimal conversion

Group binary digits into sets of 3 (octal) or 4 (hexadecimal) starting from right, then convert each group to respective digit.

Octal and hexadecimal to binary conversion

Convert each octal or hex digit to its binary equivalent.

Worked Examples

Example 1: Convert decimal 45 to binary.

Divide by 2 repeatedly:

45 ÷ 2 = 22 remainder 1
22 ÷ 2 = 11 remainder 0
11 ÷ 2 = 5 remainder 1
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1

Read remainders bottom-up: 101101

Answer: Binary 101101

Example 2: Convert binary 110101 to decimal.

Calculate:

\(1 \times 2^5 = 32\)
\(1 \times 2^4 = 16\)
\(0 \times 2^3 = 0\)
\(1 \times 2^2 = 4\)
\(0 \times 2^1 = 0\)
\(1 \times 2^0 = 1\)

Sum: 32 + 16 + 0 + 4 + 0 + 1 = 53

Answer: Decimal 53

Example 3: Convert binary 101110111 to hexadecimal.

Group into 4 bits from right: 0001 0111 0111

Convert each group:

0001 = 1
0111 = 7
0111 = 7

Answer: Hexadecimal 177

Chapter Summary

This chapter explored the fundamentals of computer arithmetic and logic starting with number systems, including their types, basic operations, and methods for converting between different systems. It then examined IEEE-based standards for representing integers and floating-point numbers, which are essential for consistent data processing in computers. The chapter detailed arithmetic operations such as addition, subtraction, and multiplication for both integer and floating-point formats, highlighting their differences and computational approaches. Logic operators including OR, AND, NAND, NOR, and NOT were introduced, explaining their roles in binary logic manipulation. Further, the chapter covered logic operations corresponding to arithmetic functions like addition, multiplication, subtraction, and division within logical contexts. Finally, it discussed various ways to represent logic operations, focusing on truth tables, Karnaugh maps for simplification, and the implementation of logic gates in digital circuits. Together, these topics provide a comprehensive understanding of how computers perform arithmetic and logical functions at the hardware level.