Why Does Multiplying 15 and -7 in Signed Binary Result in -105?

[snoggerT]

15X-7

The Attempt at a Solution

- I can't seem to figure this one out for some reason. My first attempt, I used 1111(15)X1001(2's complement of 7), but I can't get the right answer. I believe that the answer should come out to be the 2's complement of 105 since the actual answer is -105, but I can't get that. Please help.

 

[Uraptor]

Your problem is due to the way in which you have defined your numbers. When using signed numbers the MSB is actually -8.

So, 1111 = -8+(7) = -1 and 1001 = -8+(1) = -7.

To get the correct answer you need to add another MSB such that the MSB now equals -16.

Then, 01111 = 15 and 11001 = -16 +(9) = -7.

To do the multiplication;

01111 X 11001 = 15X(-16) + 15X9 = -105.

 

[piercas]

Signed binary multiplication involves multiplying two binary numbers, taking into account their signed values. In this case, we have 15 and -7, which in binary form are 1111 and 1001 (using 2's complement for negative numbers).

To multiply these two numbers, we can use the following steps:

1. Start by multiplying the digits from right to left, just like in regular binary multiplication. In this case, we have 1x1, which gives us 1 as the rightmost digit of our answer.

2. For the next digit, we have 1x0, which gives us 0. However, since we are dealing with signed numbers, we need to also add the carry from the previous step, which is 1. This gives us a final value of 1 for the second digit.

3. For the third digit, we have 1x0, which gives us 0. Again, we need to add the carry from the previous step, which is 1. This gives us a final value of 1 for the third digit.

4. For the fourth and final digit, we have 1x1, which gives us 1. We also need to add the carry from the previous step, which is 1. This gives us a final value of 0 for the fourth digit.

5. Our final answer in binary form is 0111. However, this is not the correct answer since we are dealing with signed numbers. To get the correct answer, we need to convert this back to decimal form.

6. To convert from binary to decimal, we can use the following formula: (-1)^n x (2^n-1) x (binary number), where n is the number of digits in the binary number. In this case, n=4, so our formula becomes (-1)^4 x (2^4-1) x (0111), which gives us the final answer of -105.

Therefore, the signed binary multiplication of 15X-7 is -105.