Numbers Representation and Files

Data is represented as 1 and 0 in computers. This is done with transistor devices that can maintain a constant state of 1 (positive voltage) and 0 (no voltage). We can represent numbers with 1s and 0s by using the binary number system.

Binary numbers (base 2) are very similar to decimals (base 10)

• Decimals (base 10): 3089 = 3 * 1000 + 0 * 100 + 8 * 10 + 9 * 1
• Binary (base 2): 1011 = 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0
• 12 (base 10) = 8 + 4 = 1010 (base 2)

Note that n^0 = 1 for any n.

Binary addition

1001 + 0011 = 1100

• xxx1 + xxx1 = xx10 (carry the one)
• xx0x + xx1x + 1x = x10x
• x0xx + x0xx + 1xx = x1xx
• 1xxx + 0xxx = 1xxx
• Together: 1110

Hexadecimals

Hexadecimals is a number in base 16. The digits in hexadecimal are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

• 14 (base 10) = E (base 16)
• 17 (base 10) = F1 (base 16)

Converting between binary and hexadecimal is actually really convenient, because 16 = 2^4, each hexadecimal digit represents 4 bits in binary

• 1111 (15 in base 10) = F (base 16)
• 0001 1111 (31 in base 10) = 1F (base 16)
• 1010 0001 1111 (2591 in base 10) = A1F (base 16)

Twos Complement

How do we represent negative numbers? One method is to dedicate a bit at the start of the number to indicate the sign of the number (0 - positive, 1 - negative).

But using a sign bit has two problems:

• There are two representations of zero e.g. 1000 and 0000 (for a 4 bit number)
• It is more complicated to add numbers with different signs e.g. 1001 + 0010 = ?

Twos complement is commonly used to address these two problems. For a number with width w, the first bit has a value of -2^(w-1)

For a 4 bit number:

• 1000 = -2^3 = -8
• 0001 = 1 (all other bits are treated normally)
• 1001 = -2^3 + 1 = -7
• 1111 = -2^3 + 4 + 2 + 1 = -1
• 0000 = 0

There is only one representation of zero: 0000, and we can add negative and positive numbers using normal binary addition.

1001 (-7) + 0010 (2) = 1011 (-5)

Converting between positive to negative

There is a trick to calculate the twos complement of -n if you already know the binary of n

1. Flip all the bits
2. Add 1

Example for a 4 bit binary:

• 5 = 0101
• -5 = 1010 + 0001 = 1011 (-8 + 2 + 1 = -5)

The same holds for converting a negative number positive

• -3 = 1101
• 3 = 0010 + 0001 = 0011

Proof:

Flipping all the bits of XXXX is equivalent to 1111 - XXXX

Flip(XXXX) + 0001 = 1111 - XXXX + 0001 = -1 - n + 1 = -n

Files

File handling in C revolves around the FILE struct defined by stdio.h

You can open a file using FILE *file_ptr = fopen("./filename.txt", mode);

The mode parameter determines what can be done to the opened file

• "r" - file is read-only
• "w" - file is write-only, starts writing at the start of file. If the file does not exist, a new file is created

• "a" - same as "w", except if the file exists, the program will append content to the end of the file

• "r+" - opens a file for reading and writing
• "w+", "a+" - refer to resources online

To close a file, use fclose(file_ptr);

Always close files at the end of your program.

Reading and writing text to a file is very similar to functions for stdin/stdout

    char c = fgetc(file_ptr);
    fputc(c, file_ptr);
    fprintf(file_ptr, "%d...", num, ...);
    fscanf(file_ptr, "%d...", &num, ...);

Sometimes we want to read and write binary directly. This is useful if the file is not a text file (e.g. executables, object files, data files, video files…)

In that case use

    size_t fread(void *ptr, size_t size_of_elements, size_t number_of_elements, FILE *a_file);
    size_t fwrite(const void *ptr, size_t size_of_elements, size_t number_of_elements, FILE *a_file);

Note that fread and fwrite can be used to write arrays.