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;
- 8 740 470 485 ÷ 2 = 4 370 235 242 + 1;
- 4 370 235 242 ÷ 2 = 2 185 117 621 + 0;
- 2 185 117 621 ÷ 2 = 1 092 558 810 + 1;
- 1 092 558 810 ÷ 2 = 546 279 405 + 0;
- 546 279 405 ÷ 2 = 273 139 702 + 1;
- 273 139 702 ÷ 2 = 136 569 851 + 0;
- 136 569 851 ÷ 2 = 68 284 925 + 1;
- 68 284 925 ÷ 2 = 34 142 462 + 1;
- 34 142 462 ÷ 2 = 17 071 231 + 0;
- 17 071 231 ÷ 2 = 8 535 615 + 1;
- 8 535 615 ÷ 2 = 4 267 807 + 1;
- 4 267 807 ÷ 2 = 2 133 903 + 1;
- 2 133 903 ÷ 2 = 1 066 951 + 1;
- 1 066 951 ÷ 2 = 533 475 + 1;
- 533 475 ÷ 2 = 266 737 + 1;
- 266 737 ÷ 2 = 133 368 + 1;
- 133 368 ÷ 2 = 66 684 + 0;
- 66 684 ÷ 2 = 33 342 + 0;
- 33 342 ÷ 2 = 16 671 + 0;
- 16 671 ÷ 2 = 8 335 + 1;
- 8 335 ÷ 2 = 4 167 + 1;
- 4 167 ÷ 2 = 2 083 + 1;
- 2 083 ÷ 2 = 1 041 + 1;
- 1 041 ÷ 2 = 520 + 1;
- 520 ÷ 2 = 260 + 0;
- 260 ÷ 2 = 130 + 0;
- 130 ÷ 2 = 65 + 0;
- 65 ÷ 2 = 32 + 1;
- 32 ÷ 2 = 16 + 0;
- 16 ÷ 2 = 8 + 0;
- 8 ÷ 2 = 4 + 0;
- 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.
8 740 470 485(10) = 10 0000 1000 1111 1000 1111 1110 1101 0101(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 34.
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, 34,
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.
8 740 470 485(10) = 0000 0000 0000 0000 0000 0000 0000 0010 0000 1000 1111 1000 1111 1110 1101 0101
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 0010 0000 1000 1111 1000 1111 1110 1101 0101)
= 1111 1111 1111 1111 1111 1111 1111 1101 1111 0111 0000 0111 0000 0001 0010 1010
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 1101 1111 0111 0000 0111 0000 0001 0010 1010
(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):
-8 740 470 485 =
1111 1111 1111 1111 1111 1111 1111 1101 1111 0111 0000 0111 0000 0001 0010 1010 + 1
Number -8 740 470 485(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:
-8 740 470 485(10) = 1111 1111 1111 1111 1111 1111 1111 1101 1111 0111 0000 0111 0000 0001 0010 1011
Spaces were used to group digits: for binary, by 4, for decimal, by 3.