Convert 45 931 421 713 987 958 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 45 931 421 713 987 958(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
45 931 421 713 987 958 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;
  • 45 931 421 713 987 958 ÷ 2 = 22 965 710 856 993 979 + 0;
  • 22 965 710 856 993 979 ÷ 2 = 11 482 855 428 496 989 + 1;
  • 11 482 855 428 496 989 ÷ 2 = 5 741 427 714 248 494 + 1;
  • 5 741 427 714 248 494 ÷ 2 = 2 870 713 857 124 247 + 0;
  • 2 870 713 857 124 247 ÷ 2 = 1 435 356 928 562 123 + 1;
  • 1 435 356 928 562 123 ÷ 2 = 717 678 464 281 061 + 1;
  • 717 678 464 281 061 ÷ 2 = 358 839 232 140 530 + 1;
  • 358 839 232 140 530 ÷ 2 = 179 419 616 070 265 + 0;
  • 179 419 616 070 265 ÷ 2 = 89 709 808 035 132 + 1;
  • 89 709 808 035 132 ÷ 2 = 44 854 904 017 566 + 0;
  • 44 854 904 017 566 ÷ 2 = 22 427 452 008 783 + 0;
  • 22 427 452 008 783 ÷ 2 = 11 213 726 004 391 + 1;
  • 11 213 726 004 391 ÷ 2 = 5 606 863 002 195 + 1;
  • 5 606 863 002 195 ÷ 2 = 2 803 431 501 097 + 1;
  • 2 803 431 501 097 ÷ 2 = 1 401 715 750 548 + 1;
  • 1 401 715 750 548 ÷ 2 = 700 857 875 274 + 0;
  • 700 857 875 274 ÷ 2 = 350 428 937 637 + 0;
  • 350 428 937 637 ÷ 2 = 175 214 468 818 + 1;
  • 175 214 468 818 ÷ 2 = 87 607 234 409 + 0;
  • 87 607 234 409 ÷ 2 = 43 803 617 204 + 1;
  • 43 803 617 204 ÷ 2 = 21 901 808 602 + 0;
  • 21 901 808 602 ÷ 2 = 10 950 904 301 + 0;
  • 10 950 904 301 ÷ 2 = 5 475 452 150 + 1;
  • 5 475 452 150 ÷ 2 = 2 737 726 075 + 0;
  • 2 737 726 075 ÷ 2 = 1 368 863 037 + 1;
  • 1 368 863 037 ÷ 2 = 684 431 518 + 1;
  • 684 431 518 ÷ 2 = 342 215 759 + 0;
  • 342 215 759 ÷ 2 = 171 107 879 + 1;
  • 171 107 879 ÷ 2 = 85 553 939 + 1;
  • 85 553 939 ÷ 2 = 42 776 969 + 1;
  • 42 776 969 ÷ 2 = 21 388 484 + 1;
  • 21 388 484 ÷ 2 = 10 694 242 + 0;
  • 10 694 242 ÷ 2 = 5 347 121 + 0;
  • 5 347 121 ÷ 2 = 2 673 560 + 1;
  • 2 673 560 ÷ 2 = 1 336 780 + 0;
  • 1 336 780 ÷ 2 = 668 390 + 0;
  • 668 390 ÷ 2 = 334 195 + 0;
  • 334 195 ÷ 2 = 167 097 + 1;
  • 167 097 ÷ 2 = 83 548 + 1;
  • 83 548 ÷ 2 = 41 774 + 0;
  • 41 774 ÷ 2 = 20 887 + 0;
  • 20 887 ÷ 2 = 10 443 + 1;
  • 10 443 ÷ 2 = 5 221 + 1;
  • 5 221 ÷ 2 = 2 610 + 1;
  • 2 610 ÷ 2 = 1 305 + 0;
  • 1 305 ÷ 2 = 652 + 1;
  • 652 ÷ 2 = 326 + 0;
  • 326 ÷ 2 = 163 + 0;
  • 163 ÷ 2 = 81 + 1;
  • 81 ÷ 2 = 40 + 1;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 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.

45 931 421 713 987 958(10) = 1010 0011 0010 1110 0110 0010 0111 1011 0100 1010 0111 1001 0111 0110(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 45 931 421 713 987 958(10) converted to signed binary in two's complement representation:

45 931 421 713 987 958(10) = 0000 0000 1010 0011 0010 1110 0110 0010 0111 1011 0100 1010 0111 1001 0111 0110

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