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;
- 1 657 000 259 ÷ 2 = 828 500 129 + 1;
- 828 500 129 ÷ 2 = 414 250 064 + 1;
- 414 250 064 ÷ 2 = 207 125 032 + 0;
- 207 125 032 ÷ 2 = 103 562 516 + 0;
- 103 562 516 ÷ 2 = 51 781 258 + 0;
- 51 781 258 ÷ 2 = 25 890 629 + 0;
- 25 890 629 ÷ 2 = 12 945 314 + 1;
- 12 945 314 ÷ 2 = 6 472 657 + 0;
- 6 472 657 ÷ 2 = 3 236 328 + 1;
- 3 236 328 ÷ 2 = 1 618 164 + 0;
- 1 618 164 ÷ 2 = 809 082 + 0;
- 809 082 ÷ 2 = 404 541 + 0;
- 404 541 ÷ 2 = 202 270 + 1;
- 202 270 ÷ 2 = 101 135 + 0;
- 101 135 ÷ 2 = 50 567 + 1;
- 50 567 ÷ 2 = 25 283 + 1;
- 25 283 ÷ 2 = 12 641 + 1;
- 12 641 ÷ 2 = 6 320 + 1;
- 6 320 ÷ 2 = 3 160 + 0;
- 3 160 ÷ 2 = 1 580 + 0;
- 1 580 ÷ 2 = 790 + 0;
- 790 ÷ 2 = 395 + 0;
- 395 ÷ 2 = 197 + 1;
- 197 ÷ 2 = 98 + 1;
- 98 ÷ 2 = 49 + 0;
- 49 ÷ 2 = 24 + 1;
- 24 ÷ 2 = 12 + 0;
- 12 ÷ 2 = 6 + 0;
- 6 ÷ 2 = 3 + 0;
- 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.
1 657 000 259(10) = 110 0010 1100 0011 1101 0001 0100 0011(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 31.
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, 31,
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.
1 657 000 259(10) = 0110 0010 1100 0011 1101 0001 0100 0011
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.
!(0110 0010 1100 0011 1101 0001 0100 0011)
= 1001 1101 0011 1100 0010 1110 1011 1100
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
1001 1101 0011 1100 0010 1110 1011 1100
(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):
-1 657 000 259 =
1001 1101 0011 1100 0010 1110 1011 1100 + 1
Number -1 657 000 259(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:
-1 657 000 259(10) = 1001 1101 0011 1100 0010 1110 1011 1101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.