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;
- 987 125 070 ÷ 2 = 493 562 535 + 0;
- 493 562 535 ÷ 2 = 246 781 267 + 1;
- 246 781 267 ÷ 2 = 123 390 633 + 1;
- 123 390 633 ÷ 2 = 61 695 316 + 1;
- 61 695 316 ÷ 2 = 30 847 658 + 0;
- 30 847 658 ÷ 2 = 15 423 829 + 0;
- 15 423 829 ÷ 2 = 7 711 914 + 1;
- 7 711 914 ÷ 2 = 3 855 957 + 0;
- 3 855 957 ÷ 2 = 1 927 978 + 1;
- 1 927 978 ÷ 2 = 963 989 + 0;
- 963 989 ÷ 2 = 481 994 + 1;
- 481 994 ÷ 2 = 240 997 + 0;
- 240 997 ÷ 2 = 120 498 + 1;
- 120 498 ÷ 2 = 60 249 + 0;
- 60 249 ÷ 2 = 30 124 + 1;
- 30 124 ÷ 2 = 15 062 + 0;
- 15 062 ÷ 2 = 7 531 + 0;
- 7 531 ÷ 2 = 3 765 + 1;
- 3 765 ÷ 2 = 1 882 + 1;
- 1 882 ÷ 2 = 941 + 0;
- 941 ÷ 2 = 470 + 1;
- 470 ÷ 2 = 235 + 0;
- 235 ÷ 2 = 117 + 1;
- 117 ÷ 2 = 58 + 1;
- 58 ÷ 2 = 29 + 0;
- 29 ÷ 2 = 14 + 1;
- 14 ÷ 2 = 7 + 0;
- 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.
987 125 070(10) = 11 1010 1101 0110 0101 0101 0100 1110(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 30.
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, 30,
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.
987 125 070(10) = 0011 1010 1101 0110 0101 0101 0100 1110
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.
!(0011 1010 1101 0110 0101 0101 0100 1110)
= 1100 0101 0010 1001 1010 1010 1011 0001
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
1100 0101 0010 1001 1010 1010 1011 0001
(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):
-987 125 070 =
1100 0101 0010 1001 1010 1010 1011 0001 + 1
Number -987 125 070(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:
-987 125 070(10) = 1100 0101 0010 1001 1010 1010 1011 0010
Spaces were used to group digits: for binary, by 4, for decimal, by 3.