Convert 110 011 100 110 010 276 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 110 011 100 110 010 276(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
110 011 100 110 010 276 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;
  • 110 011 100 110 010 276 ÷ 2 = 55 005 550 055 005 138 + 0;
  • 55 005 550 055 005 138 ÷ 2 = 27 502 775 027 502 569 + 0;
  • 27 502 775 027 502 569 ÷ 2 = 13 751 387 513 751 284 + 1;
  • 13 751 387 513 751 284 ÷ 2 = 6 875 693 756 875 642 + 0;
  • 6 875 693 756 875 642 ÷ 2 = 3 437 846 878 437 821 + 0;
  • 3 437 846 878 437 821 ÷ 2 = 1 718 923 439 218 910 + 1;
  • 1 718 923 439 218 910 ÷ 2 = 859 461 719 609 455 + 0;
  • 859 461 719 609 455 ÷ 2 = 429 730 859 804 727 + 1;
  • 429 730 859 804 727 ÷ 2 = 214 865 429 902 363 + 1;
  • 214 865 429 902 363 ÷ 2 = 107 432 714 951 181 + 1;
  • 107 432 714 951 181 ÷ 2 = 53 716 357 475 590 + 1;
  • 53 716 357 475 590 ÷ 2 = 26 858 178 737 795 + 0;
  • 26 858 178 737 795 ÷ 2 = 13 429 089 368 897 + 1;
  • 13 429 089 368 897 ÷ 2 = 6 714 544 684 448 + 1;
  • 6 714 544 684 448 ÷ 2 = 3 357 272 342 224 + 0;
  • 3 357 272 342 224 ÷ 2 = 1 678 636 171 112 + 0;
  • 1 678 636 171 112 ÷ 2 = 839 318 085 556 + 0;
  • 839 318 085 556 ÷ 2 = 419 659 042 778 + 0;
  • 419 659 042 778 ÷ 2 = 209 829 521 389 + 0;
  • 209 829 521 389 ÷ 2 = 104 914 760 694 + 1;
  • 104 914 760 694 ÷ 2 = 52 457 380 347 + 0;
  • 52 457 380 347 ÷ 2 = 26 228 690 173 + 1;
  • 26 228 690 173 ÷ 2 = 13 114 345 086 + 1;
  • 13 114 345 086 ÷ 2 = 6 557 172 543 + 0;
  • 6 557 172 543 ÷ 2 = 3 278 586 271 + 1;
  • 3 278 586 271 ÷ 2 = 1 639 293 135 + 1;
  • 1 639 293 135 ÷ 2 = 819 646 567 + 1;
  • 819 646 567 ÷ 2 = 409 823 283 + 1;
  • 409 823 283 ÷ 2 = 204 911 641 + 1;
  • 204 911 641 ÷ 2 = 102 455 820 + 1;
  • 102 455 820 ÷ 2 = 51 227 910 + 0;
  • 51 227 910 ÷ 2 = 25 613 955 + 0;
  • 25 613 955 ÷ 2 = 12 806 977 + 1;
  • 12 806 977 ÷ 2 = 6 403 488 + 1;
  • 6 403 488 ÷ 2 = 3 201 744 + 0;
  • 3 201 744 ÷ 2 = 1 600 872 + 0;
  • 1 600 872 ÷ 2 = 800 436 + 0;
  • 800 436 ÷ 2 = 400 218 + 0;
  • 400 218 ÷ 2 = 200 109 + 0;
  • 200 109 ÷ 2 = 100 054 + 1;
  • 100 054 ÷ 2 = 50 027 + 0;
  • 50 027 ÷ 2 = 25 013 + 1;
  • 25 013 ÷ 2 = 12 506 + 1;
  • 12 506 ÷ 2 = 6 253 + 0;
  • 6 253 ÷ 2 = 3 126 + 1;
  • 3 126 ÷ 2 = 1 563 + 0;
  • 1 563 ÷ 2 = 781 + 1;
  • 781 ÷ 2 = 390 + 1;
  • 390 ÷ 2 = 195 + 0;
  • 195 ÷ 2 = 97 + 1;
  • 97 ÷ 2 = 48 + 1;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

110 011 100 110 010 276(10) = 1 1000 0110 1101 0110 1000 0011 0011 1111 0110 1000 0011 0111 1010 0100(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 57.

  • 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, 57,

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 110 011 100 110 010 276(10) converted to signed binary in two's complement representation:

110 011 100 110 010 276(10) = 0000 0001 1000 0110 1101 0110 1000 0011 0011 1111 0110 1000 0011 0111 1010 0100

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