Convert 1 010 111 100 101 029 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 010 111 100 101 029(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 010 111 100 101 029 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;
  • 1 010 111 100 101 029 ÷ 2 = 505 055 550 050 514 + 1;
  • 505 055 550 050 514 ÷ 2 = 252 527 775 025 257 + 0;
  • 252 527 775 025 257 ÷ 2 = 126 263 887 512 628 + 1;
  • 126 263 887 512 628 ÷ 2 = 63 131 943 756 314 + 0;
  • 63 131 943 756 314 ÷ 2 = 31 565 971 878 157 + 0;
  • 31 565 971 878 157 ÷ 2 = 15 782 985 939 078 + 1;
  • 15 782 985 939 078 ÷ 2 = 7 891 492 969 539 + 0;
  • 7 891 492 969 539 ÷ 2 = 3 945 746 484 769 + 1;
  • 3 945 746 484 769 ÷ 2 = 1 972 873 242 384 + 1;
  • 1 972 873 242 384 ÷ 2 = 986 436 621 192 + 0;
  • 986 436 621 192 ÷ 2 = 493 218 310 596 + 0;
  • 493 218 310 596 ÷ 2 = 246 609 155 298 + 0;
  • 246 609 155 298 ÷ 2 = 123 304 577 649 + 0;
  • 123 304 577 649 ÷ 2 = 61 652 288 824 + 1;
  • 61 652 288 824 ÷ 2 = 30 826 144 412 + 0;
  • 30 826 144 412 ÷ 2 = 15 413 072 206 + 0;
  • 15 413 072 206 ÷ 2 = 7 706 536 103 + 0;
  • 7 706 536 103 ÷ 2 = 3 853 268 051 + 1;
  • 3 853 268 051 ÷ 2 = 1 926 634 025 + 1;
  • 1 926 634 025 ÷ 2 = 963 317 012 + 1;
  • 963 317 012 ÷ 2 = 481 658 506 + 0;
  • 481 658 506 ÷ 2 = 240 829 253 + 0;
  • 240 829 253 ÷ 2 = 120 414 626 + 1;
  • 120 414 626 ÷ 2 = 60 207 313 + 0;
  • 60 207 313 ÷ 2 = 30 103 656 + 1;
  • 30 103 656 ÷ 2 = 15 051 828 + 0;
  • 15 051 828 ÷ 2 = 7 525 914 + 0;
  • 7 525 914 ÷ 2 = 3 762 957 + 0;
  • 3 762 957 ÷ 2 = 1 881 478 + 1;
  • 1 881 478 ÷ 2 = 940 739 + 0;
  • 940 739 ÷ 2 = 470 369 + 1;
  • 470 369 ÷ 2 = 235 184 + 1;
  • 235 184 ÷ 2 = 117 592 + 0;
  • 117 592 ÷ 2 = 58 796 + 0;
  • 58 796 ÷ 2 = 29 398 + 0;
  • 29 398 ÷ 2 = 14 699 + 0;
  • 14 699 ÷ 2 = 7 349 + 1;
  • 7 349 ÷ 2 = 3 674 + 1;
  • 3 674 ÷ 2 = 1 837 + 0;
  • 1 837 ÷ 2 = 918 + 1;
  • 918 ÷ 2 = 459 + 0;
  • 459 ÷ 2 = 229 + 1;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 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.

1 010 111 100 101 029(10) = 11 1001 0110 1011 0000 1101 0001 0100 1110 0010 0001 1010 0101(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 1 010 111 100 101 029(10) converted to signed binary in two's complement representation:

1 010 111 100 101 029(10) = 0000 0000 0000 0011 1001 0110 1011 0000 1101 0001 0100 1110 0010 0001 1010 0101

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