Negative numbers and overflow

Computers store signed integers in two's complement. The highest bit counts as negative: in 8 bits it is worth -128 instead of +128. So 11111111 is -128 + 127 = -1, and 10000000 is -128.

Two's complement

#include <stdio.h>

void show(signed char x) {
    unsigned char bits = (unsigned char)x;
    printf("%4d = ", x);
    for (int i = 7; i >= 0; i--) printf("%d", (bits >> i) & 1);
    printf("\n");
}

int main(void) {
    show(5);
    show(-5);
    show(-1);
    show(-128);
    show(127);
    return 0;
}

To negate a number: flip every bit, then add 1. 5 is 00000101; flipping gives 11111010; adding 1 gives 11111011, which is -5. The trick makes addition simple: the same adder circuit works for positive and negative numbers.

The bits of x are read through an unsigned char, which has no sign, so you see the raw pattern.

Wrapping round and overflowing

#include <stdio.h>
#include <limits.h>

int main(void) {
    unsigned char small = 0;
    small = small - 1;
    printf("unsigned char 0 - 1 = %d\n", small);
    unsigned int big = 0;
    big = big - 1;
    printf("unsigned int 0 - 1 = %u\n", big);
    int x = INT_MAX;
    printf("INT_MAX = %d\n", x);
    x = x + 1;   // signed overflow: a bug
    printf("%d\n", x);
    return 0;
}

• Unsigned numbers wrap round, as C promises: 0 - 1 gives the largest value (255 for 8 bits, 4294967295 for 32), and 255 + 1 gives 0.
• Signed overflow is a bug: INT_MAX + 1 has no right answer. In compiled C it usually wraps to a large negative number, silently; StepVioz stops the program and says what happened.

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