Convert -41 111 482 to a Signed Binary in Two's (2's) Complement Representation
How to convert decimal number -41 111 482(10) to a signed binary in two's (2's) complement representation
What are the steps to convert decimal number
-41 111 482 to a signed binary in two's (2's) complement representation?
- A signed integer, written in base ten, or a decimal system number, is a number written using the digits 0 through 9 and the sign, which can be positive (+) or negative (-). If positive, the sign is usually not written. A number written in base two, or binary, is a number written using only the digits 0 and 1.
1. Start with the positive version of the number:
|-41 111 482| = 41 111 482
2. Divide the number repeatedly by 2:
Keep track of each remainder.
We stop when we get a quotient that is equal to zero.
- division = quotient + remainder;
- 41 111 482 ÷ 2 = 20 555 741 + 0;
- 20 555 741 ÷ 2 = 10 277 870 + 1;
- 10 277 870 ÷ 2 = 5 138 935 + 0;
- 5 138 935 ÷ 2 = 2 569 467 + 1;
- 2 569 467 ÷ 2 = 1 284 733 + 1;
- 1 284 733 ÷ 2 = 642 366 + 1;
- 642 366 ÷ 2 = 321 183 + 0;
- 321 183 ÷ 2 = 160 591 + 1;
- 160 591 ÷ 2 = 80 295 + 1;
- 80 295 ÷ 2 = 40 147 + 1;
- 40 147 ÷ 2 = 20 073 + 1;
- 20 073 ÷ 2 = 10 036 + 1;
- 10 036 ÷ 2 = 5 018 + 0;
- 5 018 ÷ 2 = 2 509 + 0;
- 2 509 ÷ 2 = 1 254 + 1;
- 1 254 ÷ 2 = 627 + 0;
- 627 ÷ 2 = 313 + 1;
- 313 ÷ 2 = 156 + 1;
- 156 ÷ 2 = 78 + 0;
- 78 ÷ 2 = 39 + 0;
- 39 ÷ 2 = 19 + 1;
- 19 ÷ 2 = 9 + 1;
- 9 ÷ 2 = 4 + 1;
- 4 ÷ 2 = 2 + 0;
- 2 ÷ 2 = 1 + 0;
- 1 ÷ 2 = 0 + 1;
3. Construct the base 2 representation of the positive number:
Take all the remainders starting from the bottom of the list constructed above.
41 111 482(10) = 10 0111 0011 0100 1111 1011 1010(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 26.
- A signed binary's bit length must be equal to a power of 2, as of:
- 21 = 2; 22 = 4; 23 = 8; 24 = 16; 25 = 32; 26 = 64; ...
- The first bit (the leftmost) indicates the sign:
- 0 = positive integer number, 1 = negative integer number
The least number that is:
1) a power of 2
2) and is larger than the actual length, 26,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 32.
5. Get the positive binary computer representation on 32 bits (4 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32.
41 111 482(10) = 0000 0010 0111 0011 0100 1111 1011 1010
6. Get the negative integer number representation. Part 1:
- To write the negative integer number on 32 bits (4 Bytes), as a signed binary in one's complement representation, replace all the bits on 0 with 1s and all the bits set on 1 with 0s.
Reverse the digits, flip the digits:
Replace the bits set on 0 with 1s and the bits set on 1 with 0s.
!(0000 0010 0111 0011 0100 1111 1011 1010)
= 1111 1101 1000 1100 1011 0000 0100 0101
7. Get the negative integer number representation. Part 2:
- To write the negative integer number on 32 bits (4 Bytes), as a signed binary in two's complement representation, add 1 to the number calculated above 1111 1101 1000 1100 1011 0000 0100 0101 (to the signed binary in one's complement representation).
Binary addition carries on a value of 2:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 1 = 10
- 1 + 10 = 11
- 1 + 11 = 100
Add 1 to the number calculated above
(to the signed binary number in one's complement representation):
-41 111 482 =
1111 1101 1000 1100 1011 0000 0100 0101 + 1
Decimal Number -41 111 482(10) converted to signed binary in two's complement representation:
-41 111 482(10) = 1111 1101 1000 1100 1011 0000 0100 0110
Spaces were used to group digits: for binary, by 4, for decimal, by 3.