Get overflow from arithmetic operations

Assuming unsigned type operands, you can write:

bool cf = a+b<a;

or

bool cf = a>-1-b;

These work regardless of the existence of a larger type to work with.

Multiplication is harder; without a larger type, there is no way to access the upper half of the result. If you do have one you can use it. For example, if your operands are uint32_t,

uint32_t upper = ((uint64_t)a * b) >> 32;
uint32_t lower = a*b;

Otherwise, you're stuck dropping to a half-sized type and using long multiplication. For example, with uint64_t a,b;

uint32_t al = a, ah = a>>32;
uint32_t bl = b, bh = b>>32;

And then the upper part of the result is ah*bh plus the carry out of adding al*bh, ah*bl, and the upper bits of al*bl.

Bigint libraries can avoid the pain of this by just choosing a limb type that's at most half the width of the largest integer type.

How do I get the 1 that overflowed?

To do it afterwards in a portable way (not forgetting that unsigned int might only be 16 bits):

    uint32_t a = 0Xffffffff;
    uint32_t b = 0xffffffff;
    uint32_t c_low = a + b;
    uint32_t c_high;
    if(c_low >= a) {
        c_high = 0;
    } else {
        c_high = 1;
    }

To do it beforehand in a portable way (without branches):

    uint32_t a = 0Xffffffff;
    uint32_t b = 0xffffffff;
    uint32_t c_low;
    uint32_t c_high;

    c_high = (a&b) >> 31;
    c_low = (a ^ (c_high<<31)) + b;

How can I do the same for multiplication?

Multiplication doesn't have a carry, it has an "upper half". Specifically; if you multiply an unsigned integer that has N bits with an unsigned integer that has M bits then the result will have N+M bits; and if both numbers had the same size then the result will be twice as big.

Sadly C doesn't support "result type is larger than source/s types", so you need to "pre-promote" the source types, like:

    uint32_t a = 0Xffffffff;
    uint32_t b = 0xffffffff;
    uint64_t temp = (uint64_t)a * (uint64_t)b;
    uint32_t c_low = temp;
    uint32_t c_high = temp >> 32;

Of course if the compiler doesn't support a larger type then you have to split it into smaller pieces, like:

    uint32_t a = 0Xffffffff;
    uint32_t b = 0xffffffff;

    uint32_t a_low = a & 0xFFFF;
    uint32_t a_high = a >> 16;
    uint32_t b_low = a & 0xFFFF;
    uint32_t b_high = b >> 16;

    uint32_t temp_0 = a_low * b_low;
    uint32_t temp_16a = a_high * b_low;
    uint32_t temp_16b = a_low * b_high;
    uint32_t temp_32 = a_high * b_high;

    uint32_t c_low = temp_0 + (temp16a << 16) + (temp16b << 16);
    uint32_t c_high = (temp16a >> 16) + (temp16b >> 16) + temp_32;

How do bigint libraries manage this?

Mainly; they use inline assembly language because most CPUs support instructions to work on bigger integers efficiently and/or because you can access the carry flag directly. For example; for 80x86; the CPU has adc/sbb, shld/shrd, mul (with double width result)/div (with double width numerator); plus maybe extensions (adcx and adox).

In 32-bit 80x86 assembly language, the addition might look like:

    xor edx,0
    add eax,ebx      ;c_low = a + b
    adc edx,0        ;c_high = carry

..and the multiplication might look like:

    mul ebx          ;edx:eax = a * b