Convert 1 000 000 110 101 367 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 000 000 110 101 367(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 000 000 110 101 367 to a signed binary in two's (2's) complement representation?

  • A signed integer, written in base ten, or a decimal system number, is a number written using the digits 0 through 9 and the sign, which can be positive (+) or negative (-). If positive, the sign is usually not written. A number written in base two, or binary, is a number written using only the digits 0 and 1.

1. 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;
  • 1 000 000 110 101 367 ÷ 2 = 500 000 055 050 683 + 1;
  • 500 000 055 050 683 ÷ 2 = 250 000 027 525 341 + 1;
  • 250 000 027 525 341 ÷ 2 = 125 000 013 762 670 + 1;
  • 125 000 013 762 670 ÷ 2 = 62 500 006 881 335 + 0;
  • 62 500 006 881 335 ÷ 2 = 31 250 003 440 667 + 1;
  • 31 250 003 440 667 ÷ 2 = 15 625 001 720 333 + 1;
  • 15 625 001 720 333 ÷ 2 = 7 812 500 860 166 + 1;
  • 7 812 500 860 166 ÷ 2 = 3 906 250 430 083 + 0;
  • 3 906 250 430 083 ÷ 2 = 1 953 125 215 041 + 1;
  • 1 953 125 215 041 ÷ 2 = 976 562 607 520 + 1;
  • 976 562 607 520 ÷ 2 = 488 281 303 760 + 0;
  • 488 281 303 760 ÷ 2 = 244 140 651 880 + 0;
  • 244 140 651 880 ÷ 2 = 122 070 325 940 + 0;
  • 122 070 325 940 ÷ 2 = 61 035 162 970 + 0;
  • 61 035 162 970 ÷ 2 = 30 517 581 485 + 0;
  • 30 517 581 485 ÷ 2 = 15 258 790 742 + 1;
  • 15 258 790 742 ÷ 2 = 7 629 395 371 + 0;
  • 7 629 395 371 ÷ 2 = 3 814 697 685 + 1;
  • 3 814 697 685 ÷ 2 = 1 907 348 842 + 1;
  • 1 907 348 842 ÷ 2 = 953 674 421 + 0;
  • 953 674 421 ÷ 2 = 476 837 210 + 1;
  • 476 837 210 ÷ 2 = 238 418 605 + 0;
  • 238 418 605 ÷ 2 = 119 209 302 + 1;
  • 119 209 302 ÷ 2 = 59 604 651 + 0;
  • 59 604 651 ÷ 2 = 29 802 325 + 1;
  • 29 802 325 ÷ 2 = 14 901 162 + 1;
  • 14 901 162 ÷ 2 = 7 450 581 + 0;
  • 7 450 581 ÷ 2 = 3 725 290 + 1;
  • 3 725 290 ÷ 2 = 1 862 645 + 0;
  • 1 862 645 ÷ 2 = 931 322 + 1;
  • 931 322 ÷ 2 = 465 661 + 0;
  • 465 661 ÷ 2 = 232 830 + 1;
  • 232 830 ÷ 2 = 116 415 + 0;
  • 116 415 ÷ 2 = 58 207 + 1;
  • 58 207 ÷ 2 = 29 103 + 1;
  • 29 103 ÷ 2 = 14 551 + 1;
  • 14 551 ÷ 2 = 7 275 + 1;
  • 7 275 ÷ 2 = 3 637 + 1;
  • 3 637 ÷ 2 = 1 818 + 1;
  • 1 818 ÷ 2 = 909 + 0;
  • 909 ÷ 2 = 454 + 1;
  • 454 ÷ 2 = 227 + 0;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

1 000 000 110 101 367(10) = 11 1000 1101 0111 1110 1010 1011 0101 0110 1000 0011 0111 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


4. 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.


Decimal Number 1 000 000 110 101 367(10) converted to signed binary in two's complement representation:

1 000 000 110 101 367(10) = 0000 0000 0000 0011 1000 1101 0111 1110 1010 1011 0101 0110 1000 0011 0111 0111

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed integers from decimal system to signed binary in two's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in two's complement representation:

  • 1. If the number to be converted is negative, start with the positive version of the number.
  • 2. Divide repeatedly by 2 the positive representation of the integer number, keeping track of each remainder, until we get a quotient that is zero.
  • 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Binary numbers represented in computer language must have 4, 8, 16, 32, 64, ... bit length (a power of 2) - if needed, add extra bits on 0 in front (to the left) of the base 2 number above, up to the required length, so that the first bit (the leftmost) will be 0, correctly representing a positive number.
  • 5. To get the negative integer number representation in signed binary one's complement, replace all 0 bits with 1s and all 1 bits with 0s (reversing the digits).
  • 6. To get the negative integer number, in signed binary two's complement representation, add 1 to the number above.

Example: convert the negative number -60 from the decimal system (base ten) to signed binary in two's complement:

  • 1. Start with the positive version of the number: |-60| = 60
  • 2. Divide repeatedly 60 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 60 ÷ 2 = 30 + 0
    • 30 ÷ 2 = 15 + 0
    • 15 ÷ 2 = 7 + 1
    • 7 ÷ 2 = 3 + 1
    • 3 ÷ 2 = 1 + 1
    • 1 ÷ 2 = 0 + 1
  • 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:
    60(10) = 11 1100(2)
  • 4. Bit length of base 2 representation number is 6, so the positive binary computer representation of a signed binary will take in this particular case 8 bits (the least power of 2 larger than 6) - add extra 0 digits in front of the base 2 number, up to the required length:
    60(10) = 0011 1100(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all the 0 bits with 1s and all 1 bits with 0s (reversing the digits):
    !(0011 1100) = 1100 0011
  • 6. To get the negative integer number, signed binary in two's complement representation, add 1 to the number above:
    -60(10) = 1100 0011 + 1 = 1100 0100
  • Number -60(10), signed integer, converted from decimal system (base 10) to signed binary two's complement representation = 1100 0100