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;
- 15 919 000 ÷ 2 = 7 959 500 + 0;
- 7 959 500 ÷ 2 = 3 979 750 + 0;
- 3 979 750 ÷ 2 = 1 989 875 + 0;
- 1 989 875 ÷ 2 = 994 937 + 1;
- 994 937 ÷ 2 = 497 468 + 1;
- 497 468 ÷ 2 = 248 734 + 0;
- 248 734 ÷ 2 = 124 367 + 0;
- 124 367 ÷ 2 = 62 183 + 1;
- 62 183 ÷ 2 = 31 091 + 1;
- 31 091 ÷ 2 = 15 545 + 1;
- 15 545 ÷ 2 = 7 772 + 1;
- 7 772 ÷ 2 = 3 886 + 0;
- 3 886 ÷ 2 = 1 943 + 0;
- 1 943 ÷ 2 = 971 + 1;
- 971 ÷ 2 = 485 + 1;
- 485 ÷ 2 = 242 + 1;
- 242 ÷ 2 = 121 + 0;
- 121 ÷ 2 = 60 + 1;
- 60 ÷ 2 = 30 + 0;
- 30 ÷ 2 = 15 + 0;
- 15 ÷ 2 = 7 + 1;
- 7 ÷ 2 = 3 + 1;
- 3 ÷ 2 = 1 + 1;
- 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.
15 919 000(10) = 1111 0010 1110 0111 1001 1000(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 24.
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, 24,
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.
15 919 000(10) = 0000 0000 1111 0010 1110 0111 1001 1000
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 0000 1111 0010 1110 0111 1001 1000)
= 1111 1111 0000 1101 0001 1000 0110 0111
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 1111 0000 1101 0001 1000 0110 0111
(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):
-15 919 000 =
1111 1111 0000 1101 0001 1000 0110 0111 + 1
Number -15 919 000(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:
-15 919 000(10) = 1111 1111 0000 1101 0001 1000 0110 1000
Spaces were used to group digits: for binary, by 4, for decimal, by 3.