Why is -1L * -9223372036854775808L == -9223372036854775808L

Question

I understand this has something to do with the way processors treat overflows, but I fail to see it. Multiplication with different negative numbers gives either zero or -2^63:

In C# Interactive:

> return unchecked (-1L * -9223372036854775808L);
-9223372036854775808
> return unchecked (-2L * -9223372036854775808L);
0
> return unchecked (-3L * -9223372036854775808L);
-9223372036854775808
> return unchecked (-4L * -9223372036854775808L);
0
> return unchecked (-5L * -9223372036854775808L);
-9223372036854775808

In F# Interactive:

> -1L * -9223372036854775808L;;
val it : int64 = -9223372036854775808L
> -2L * -9223372036854775808L;;
val it : int64 = 0L
> -3L * -9223372036854775808L;;
val it : int64 = -9223372036854775808L
> -4L * -9223372036854775808L;;
val it : int64 = 0L

I came to this because it surprised me in F# until I realized that F# by default operates in unchecked contexts. Still, I couldn't readily explain the behavior.

I do understand why 9223372036854775807L + 1L == -9223372036854775808L, I just don't get it for multiplication with a negative number and why it alternates between 0 (binary all zeroes) and -2^63 (binary most significant bit 1, rest zeroes).

Interestingly, this holds with the rule of multiplicative identity, i.e., since -1L * -9223372036854775808L == -9223372036854775808L, it follows that -1L * -1L * -9223372036854775808L == -9223372036854775808L and since -1L * -1L = 1L, it shows that the identity law still holds.

Answer 1

The answers you are getting are all correct modulo 2^64: that is, they differ from the mathematically correct answers by a multiple of 2^64, and that's a reasonable definition of a correctly truncated answer.

I'll use ≅ to relate two numbers that are congruent modulo 2^64. Thus

• -1 * -2^63 = 2^63 ≅ -2^63
• -2 * -2^63 = 2^64 ≅ 0
• -3 * -2^63 = 3 * 2^63 = 2^64 + 2^63 ≅ -2^63

and so on. Note that 2^63 and -2^63 are congruent, but it is -2^63 that is representable according to the conventions of twos-complement arithmetic.