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;
- 167 772 147 ÷ 2 = 83 886 073 + 1;
- 83 886 073 ÷ 2 = 41 943 036 + 1;
- 41 943 036 ÷ 2 = 20 971 518 + 0;
- 20 971 518 ÷ 2 = 10 485 759 + 0;
- 10 485 759 ÷ 2 = 5 242 879 + 1;
- 5 242 879 ÷ 2 = 2 621 439 + 1;
- 2 621 439 ÷ 2 = 1 310 719 + 1;
- 1 310 719 ÷ 2 = 655 359 + 1;
- 655 359 ÷ 2 = 327 679 + 1;
- 327 679 ÷ 2 = 163 839 + 1;
- 163 839 ÷ 2 = 81 919 + 1;
- 81 919 ÷ 2 = 40 959 + 1;
- 40 959 ÷ 2 = 20 479 + 1;
- 20 479 ÷ 2 = 10 239 + 1;
- 10 239 ÷ 2 = 5 119 + 1;
- 5 119 ÷ 2 = 2 559 + 1;
- 2 559 ÷ 2 = 1 279 + 1;
- 1 279 ÷ 2 = 639 + 1;
- 639 ÷ 2 = 319 + 1;
- 319 ÷ 2 = 159 + 1;
- 159 ÷ 2 = 79 + 1;
- 79 ÷ 2 = 39 + 1;
- 39 ÷ 2 = 19 + 1;
- 19 ÷ 2 = 9 + 1;
- 9 ÷ 2 = 4 + 1;
- 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.
167 772 147(10) = 1001 1111 1111 1111 1111 1111 0011(2)
4. Determine the signed binary number bit length:
The base 2 number's actual length, in bits: 28.
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, 28,
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.
167 772 147(10) = 0000 1001 1111 1111 1111 1111 1111 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.
!(0000 1001 1111 1111 1111 1111 1111 0011)
= 1111 0110 0000 0000 0000 0000 0000 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
1111 0110 0000 0000 0000 0000 0000 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):
-167 772 147 =
1111 0110 0000 0000 0000 0000 0000 1100 + 1
Number -167 772 147(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:
-167 772 147(10) = 1111 0110 0000 0000 0000 0000 0000 1101
Spaces were used to group digits: for binary, by 4, for decimal, by 3.