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;
- 4 093 640 697 ÷ 2 = 2 046 820 348 + 1;
- 2 046 820 348 ÷ 2 = 1 023 410 174 + 0;
- 1 023 410 174 ÷ 2 = 511 705 087 + 0;
- 511 705 087 ÷ 2 = 255 852 543 + 1;
- 255 852 543 ÷ 2 = 127 926 271 + 1;
- 127 926 271 ÷ 2 = 63 963 135 + 1;
- 63 963 135 ÷ 2 = 31 981 567 + 1;
- 31 981 567 ÷ 2 = 15 990 783 + 1;
- 15 990 783 ÷ 2 = 7 995 391 + 1;
- 7 995 391 ÷ 2 = 3 997 695 + 1;
- 3 997 695 ÷ 2 = 1 998 847 + 1;
- 1 998 847 ÷ 2 = 999 423 + 1;
- 999 423 ÷ 2 = 499 711 + 1;
- 499 711 ÷ 2 = 249 855 + 1;
- 249 855 ÷ 2 = 124 927 + 1;
- 124 927 ÷ 2 = 62 463 + 1;
- 62 463 ÷ 2 = 31 231 + 1;
- 31 231 ÷ 2 = 15 615 + 1;
- 15 615 ÷ 2 = 7 807 + 1;
- 7 807 ÷ 2 = 3 903 + 1;
- 3 903 ÷ 2 = 1 951 + 1;
- 1 951 ÷ 2 = 975 + 1;
- 975 ÷ 2 = 487 + 1;
- 487 ÷ 2 = 243 + 1;
- 243 ÷ 2 = 121 + 1;
- 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.
4 093 640 697(10) = 1111 0011 1111 1111 1111 1111 1111 1001(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 32.
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, 32,
3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)
=== is: 64.
5. Get the positive binary computer representation on 64 bits (8 Bytes):
If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 64.
4 093 640 697(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1111 0011 1111 1111 1111 1111 1111 1001
6. Get the negative integer number representation. Part 1:
To write the negative integer number on 64 bits (8 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 0000 0000 0000 0000 0000 0000 1111 0011 1111 1111 1111 1111 1111 1001)
= 1111 1111 1111 1111 1111 1111 1111 1111 0000 1100 0000 0000 0000 0000 0000 0110
7. Get the negative integer number representation. Part 2:
To write the negative integer number on 64 bits (8 Bytes),
as a signed binary in two's complement representation,
add 1 to the number calculated above
1111 1111 1111 1111 1111 1111 1111 1111 0000 1100 0000 0000 0000 0000 0000 0110
(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):
-4 093 640 697 =
1111 1111 1111 1111 1111 1111 1111 1111 0000 1100 0000 0000 0000 0000 0000 0110 + 1
Number -4 093 640 697(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:
-4 093 640 697(10) = 1111 1111 1111 1111 1111 1111 1111 1111 0000 1100 0000 0000 0000 0000 0000 0111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.