Convert 44 999 401 713 989 277 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 44 999 401 713 989 277(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
44 999 401 713 989 277 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;
  • 44 999 401 713 989 277 ÷ 2 = 22 499 700 856 994 638 + 1;
  • 22 499 700 856 994 638 ÷ 2 = 11 249 850 428 497 319 + 0;
  • 11 249 850 428 497 319 ÷ 2 = 5 624 925 214 248 659 + 1;
  • 5 624 925 214 248 659 ÷ 2 = 2 812 462 607 124 329 + 1;
  • 2 812 462 607 124 329 ÷ 2 = 1 406 231 303 562 164 + 1;
  • 1 406 231 303 562 164 ÷ 2 = 703 115 651 781 082 + 0;
  • 703 115 651 781 082 ÷ 2 = 351 557 825 890 541 + 0;
  • 351 557 825 890 541 ÷ 2 = 175 778 912 945 270 + 1;
  • 175 778 912 945 270 ÷ 2 = 87 889 456 472 635 + 0;
  • 87 889 456 472 635 ÷ 2 = 43 944 728 236 317 + 1;
  • 43 944 728 236 317 ÷ 2 = 21 972 364 118 158 + 1;
  • 21 972 364 118 158 ÷ 2 = 10 986 182 059 079 + 0;
  • 10 986 182 059 079 ÷ 2 = 5 493 091 029 539 + 1;
  • 5 493 091 029 539 ÷ 2 = 2 746 545 514 769 + 1;
  • 2 746 545 514 769 ÷ 2 = 1 373 272 757 384 + 1;
  • 1 373 272 757 384 ÷ 2 = 686 636 378 692 + 0;
  • 686 636 378 692 ÷ 2 = 343 318 189 346 + 0;
  • 343 318 189 346 ÷ 2 = 171 659 094 673 + 0;
  • 171 659 094 673 ÷ 2 = 85 829 547 336 + 1;
  • 85 829 547 336 ÷ 2 = 42 914 773 668 + 0;
  • 42 914 773 668 ÷ 2 = 21 457 386 834 + 0;
  • 21 457 386 834 ÷ 2 = 10 728 693 417 + 0;
  • 10 728 693 417 ÷ 2 = 5 364 346 708 + 1;
  • 5 364 346 708 ÷ 2 = 2 682 173 354 + 0;
  • 2 682 173 354 ÷ 2 = 1 341 086 677 + 0;
  • 1 341 086 677 ÷ 2 = 670 543 338 + 1;
  • 670 543 338 ÷ 2 = 335 271 669 + 0;
  • 335 271 669 ÷ 2 = 167 635 834 + 1;
  • 167 635 834 ÷ 2 = 83 817 917 + 0;
  • 83 817 917 ÷ 2 = 41 908 958 + 1;
  • 41 908 958 ÷ 2 = 20 954 479 + 0;
  • 20 954 479 ÷ 2 = 10 477 239 + 1;
  • 10 477 239 ÷ 2 = 5 238 619 + 1;
  • 5 238 619 ÷ 2 = 2 619 309 + 1;
  • 2 619 309 ÷ 2 = 1 309 654 + 1;
  • 1 309 654 ÷ 2 = 654 827 + 0;
  • 654 827 ÷ 2 = 327 413 + 1;
  • 327 413 ÷ 2 = 163 706 + 1;
  • 163 706 ÷ 2 = 81 853 + 0;
  • 81 853 ÷ 2 = 40 926 + 1;
  • 40 926 ÷ 2 = 20 463 + 0;
  • 20 463 ÷ 2 = 10 231 + 1;
  • 10 231 ÷ 2 = 5 115 + 1;
  • 5 115 ÷ 2 = 2 557 + 1;
  • 2 557 ÷ 2 = 1 278 + 1;
  • 1 278 ÷ 2 = 639 + 0;
  • 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;

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

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

44 999 401 713 989 277(10) = 1001 1111 1101 1110 1011 0111 1010 1010 0100 0100 0111 0110 1001 1101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 56.

  • 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, 56,

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 44 999 401 713 989 277(10) converted to signed binary in two's complement representation:

44 999 401 713 989 277(10) = 0000 0000 1001 1111 1101 1110 1011 0111 1010 1010 0100 0100 0111 0110 1001 1101

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