The document discusses signed numbers and overflow detection in signed addition. It explains that signed numbers use the most significant bit to represent positive or negative values. It shows how to represent numbers from -8 to 7 in 4 bits and that addition works the same regardless of sign. Overflows only occur when adding two numbers with the same sign, which would result in a negative sum for two positives or a positive sum for two negatives. The document concludes by stating overflows can be detected by checking if the sign bit is set for two positives added or not set for two negatives added.

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