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;
- 92 233 720 368 449 ÷ 2 = 46 116 860 184 224 + 1;
- 46 116 860 184 224 ÷ 2 = 23 058 430 092 112 + 0;
- 23 058 430 092 112 ÷ 2 = 11 529 215 046 056 + 0;
- 11 529 215 046 056 ÷ 2 = 5 764 607 523 028 + 0;
- 5 764 607 523 028 ÷ 2 = 2 882 303 761 514 + 0;
- 2 882 303 761 514 ÷ 2 = 1 441 151 880 757 + 0;
- 1 441 151 880 757 ÷ 2 = 720 575 940 378 + 1;
- 720 575 940 378 ÷ 2 = 360 287 970 189 + 0;
- 360 287 970 189 ÷ 2 = 180 143 985 094 + 1;
- 180 143 985 094 ÷ 2 = 90 071 992 547 + 0;
- 90 071 992 547 ÷ 2 = 45 035 996 273 + 1;
- 45 035 996 273 ÷ 2 = 22 517 998 136 + 1;
- 22 517 998 136 ÷ 2 = 11 258 999 068 + 0;
- 11 258 999 068 ÷ 2 = 5 629 499 534 + 0;
- 5 629 499 534 ÷ 2 = 2 814 749 767 + 0;
- 2 814 749 767 ÷ 2 = 1 407 374 883 + 1;
- 1 407 374 883 ÷ 2 = 703 687 441 + 1;
- 703 687 441 ÷ 2 = 351 843 720 + 1;
- 351 843 720 ÷ 2 = 175 921 860 + 0;
- 175 921 860 ÷ 2 = 87 960 930 + 0;
- 87 960 930 ÷ 2 = 43 980 465 + 0;
- 43 980 465 ÷ 2 = 21 990 232 + 1;
- 21 990 232 ÷ 2 = 10 995 116 + 0;
- 10 995 116 ÷ 2 = 5 497 558 + 0;
- 5 497 558 ÷ 2 = 2 748 779 + 0;
- 2 748 779 ÷ 2 = 1 374 389 + 1;
- 1 374 389 ÷ 2 = 687 194 + 1;
- 687 194 ÷ 2 = 343 597 + 0;
- 343 597 ÷ 2 = 171 798 + 1;
- 171 798 ÷ 2 = 85 899 + 0;
- 85 899 ÷ 2 = 42 949 + 1;
- 42 949 ÷ 2 = 21 474 + 1;
- 21 474 ÷ 2 = 10 737 + 0;
- 10 737 ÷ 2 = 5 368 + 1;
- 5 368 ÷ 2 = 2 684 + 0;
- 2 684 ÷ 2 = 1 342 + 0;
- 1 342 ÷ 2 = 671 + 0;
- 671 ÷ 2 = 335 + 1;
- 335 ÷ 2 = 167 + 1;
- 167 ÷ 2 = 83 + 1;
- 83 ÷ 2 = 41 + 1;
- 41 ÷ 2 = 20 + 1;
- 20 ÷ 2 = 10 + 0;
- 10 ÷ 2 = 5 + 0;
- 5 ÷ 2 = 2 + 1;
- 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.
92 233 720 368 449(10) = 101 0011 1110 0010 1101 0110 0010 0011 1000 1101 0100 0001(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 47.
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, 47,
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.
92 233 720 368 449(10) = 0000 0000 0000 0000 0101 0011 1110 0010 1101 0110 0010 0011 1000 1101 0100 0001
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 0101 0011 1110 0010 1101 0110 0010 0011 1000 1101 0100 0001)
= 1111 1111 1111 1111 1010 1100 0001 1101 0010 1001 1101 1100 0111 0010 1011 1110
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 1010 1100 0001 1101 0010 1001 1101 1100 0111 0010 1011 1110
(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):
-92 233 720 368 449 =
1111 1111 1111 1111 1010 1100 0001 1101 0010 1001 1101 1100 0111 0010 1011 1110 + 1
Number -92 233 720 368 449(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:
-92 233 720 368 449(10) = 1111 1111 1111 1111 1010 1100 0001 1101 0010 1001 1101 1100 0111 0010 1011 1111
Spaces were used to group digits: for binary, by 4, for decimal, by 3.