Grokking Signed Overflows

Transcript

1. Grokking Signed Overflows Detecting an overflow when adding two signed numbers
2. Grokking Signed Numbers
3. Grokking Signed Numbers • When a number is signed, the high digit represents the additive inverse of what it typically would • For instance, with a 4 bit signed nibble: 0 0 0 0
 -8 4 2 1 • Instead of the high bit representing 8, with a signed number it would represent negative 8
4. Grokking Signed Numbers • With our four bit signed nibble, 7 is the highest integer our signed nibble can represent 0 1 1 1 = 7
 -8 4 2 1 .
 ( 0 + 4 + 2 + 1) .
5. Grokking Signed Numbers • If we set the -8 place only, we get the lowest integer our signed nibble can represent, -8 1 0 0 0 =-8
 -8 4 2 1 .
 (-8 + 0 + 0 + 0) .
6. Grokking Signed Numbers • If we set the -8 place and the 1 place, we add one to negative eight, for a total of -7 1 0 0 1 =-7
 -8 4 2 1 .
 (-8 + 0 + 0 + 1) .
7. Grokking Signed Numbers • With our four bit signed nibble, we can represent all integers from -8 to +7 1000 -8 1001 -7 1010 -6 1011 -5 1100 -4 1101 -3 1110 -2 1111 -1 0000 0 0001 +1 0010 +2 0011 +3 0100 +4 0101 +5 0110 +6 0111 +7
8. Grokking Signed Numbers • Because of how the numbers are arranged, we can do binary addition without worrying about the sign 1101 -3 13
 +0010 +2 2
 ────00 00000
 1111 -1 15
9. Grokking Signed Overflows
10. Grokking Signed Overflows • When adding two numbers with a different sign bit, the sum will NEVER overflow 1000 -8
 +0001 +1
 ──── .
 1001 -7 No overflow
11. • Sums will only overflow when both addends have the same sign bit. • Adding two positive numbers should never result in a negative number • Adding two negative numbers should never result in a positive number Grokking Signed Overflows
12. • When adding two signed positive numbers, if the sign bit of the sum is set, an overflow has occurred • When adding two negative numbers, if the sign bit of the sum is not set, an overflow has occurred Grokking Signed Overflows
13. Grokking Signed Overflows Detecting an overflow when adding two signed numbers Grokking Signed Overflows by Pat Hawks is licensed under a
 Creative Commons Attribution 4.0 International License