1. Multidimensional Arrays

1.1 Definition

• Example: int m[3][4];

  • This defines an array containing 3 arrays, each having 4 integers.

1.2 Pointer Representation

• m is the name of the entire array and is treated as a pointer to the first row.

• m[0] names the first row and acts as a pointer to its first element.

• m[0][0] refers to the first element.

1.3 Pointer Arithmetic Examples

• m + 1, m + 2: Point to the second and third rows respectively.

• *m, *m + 1: Dereference the first row.

  • *(m + 1), *(m + 1) + 1: Access elements in the second row.

• Addressing:

  • &m[0][0] - Address of first element.

  • &m[1][0], &m[2][0]: Addresses of the first element in the second and third rows.

2. Indexing Multi-dimensional Arrays

2.1 Array Indexing Mechanics

• Address calculation:

  • General formula:

    • address = base + index × element_size

  • Pointer arithmetic automatically accounts for element size.

2.2 For Two-dimensional Arrays

• Offset formula for accessing elements:

  • offset = (row_number × row_length + column_number) × element_size

3. Passing Multi-dimensional Arrays to Functions

3.1 Passing Mechanism

• Pass a pointer when sending a 1D array to a function.

• For multi-dimensional arrays, the length of inner arrays must be known.

• Example Function:

  • void transpose(double matrix[][3]);

  • Length of outer array is unused (optional).

4. Multi-level Arrays

4.1 Characteristics

• Rectangular Design: Multidimensional arrays must have all rows of the same length.

• Difficulty with heap allocation:

  • Non-trivial to specify dimensions.

• Data structure:

  • Arrays of pointers to arrays (e.g., row pointers).

4.2 Indexing Representation

• Accessing data remains similar to multidimensional indexing:

  • int **p; p serves as a pointer to the outer array.

4.3 Dynamic Allocation Example

• Memory allocation example:

int **M = malloc(ROWS * sizeof(int *));
for (i = 0; i < ROWS; i++) {
    M[i] = malloc(COLS * sizeof(int));
}

• Alternatively, allocate all rows:

M[0] = malloc(ROWS * COLS * sizeof(int));
for (i = 1; i < ROWS; i++) {
    M[i] = &M[0][i * COLS];
}

5. Compilation Process

5.1 Overview of Compilation Stages

• Primary components:

  • Source File: hello.c

  • Pre-processor, Compiler, Assembler, Linker components.

5.2 Compilation Steps

1. Pre-processing: gcc -E hello.c

2. Compilation: gcc -S hello.c -> Produces hello.s

3. Assembly: gcc -c hello.c -> Produces hello.o

4. Linking: Combines object files to create an executable (e.g. a.out).

6. ISA (Instruction Set Architecture)

6.1 Definition and Function

• Bridge between hardware and software.

• Defines data representation (e.g., integers, pointers).

• Instructions are also structured as data.

7. Number Representation in Computing

7.1 Abstract Number Representation

• Manipulation of data, primarily represented as numbers.

7.2 Base-10 Decimal System

• Structure:

  • Each digit represents powers of 10 (weights) based on position.

  • Example: 86042 = 8*10^4 + 6*10^3 + ...

7.3 General Base-n Notation

• Numbers expressed by digits representing powers of base n.

• Formula: x = dkn^k + ... + d0n^0 where digit values range from 0 to n-1.

8. Binary and Other Number Systems

8.1 Base-2 (Binary)

• Utilizes only two digits (0 and 1).

• Ideal for computer logic and representation.

8.2 Base-16 (Hexadecimal)

• Use of digits 0-9 and letters A-F.

• More compact than binary, each hex digit represents 4 bits.

8.3 Conversion Tables

• Hex to Binary Conversion: | Hex | Binary | |-----|---------| | 0 | 0000 | | 1 | 0001 | | 2 | 0010 | | 3 | 0011 | | ... | ... | | F | 1111 |

9. Special Numeric Values

9.1 Floating Point Representation

• Compact notation for large and small numbers via significant and exponent.

9.2 IEEE Floating-Point Standard

• Structure includes sign bit, exponent, and significand.

• Handles non-finite values such as ±infinity and NaN (not a number).

9.3 Subnormal Numbers

• Represent values very close to zero to fill the gap between zero and the smallest positive normalized number.