Convert 5 436 006 977 639 628 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 5 436 006 977 639 628(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
5 436 006 977 639 628 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;
  • 5 436 006 977 639 628 ÷ 2 = 2 718 003 488 819 814 + 0;
  • 2 718 003 488 819 814 ÷ 2 = 1 359 001 744 409 907 + 0;
  • 1 359 001 744 409 907 ÷ 2 = 679 500 872 204 953 + 1;
  • 679 500 872 204 953 ÷ 2 = 339 750 436 102 476 + 1;
  • 339 750 436 102 476 ÷ 2 = 169 875 218 051 238 + 0;
  • 169 875 218 051 238 ÷ 2 = 84 937 609 025 619 + 0;
  • 84 937 609 025 619 ÷ 2 = 42 468 804 512 809 + 1;
  • 42 468 804 512 809 ÷ 2 = 21 234 402 256 404 + 1;
  • 21 234 402 256 404 ÷ 2 = 10 617 201 128 202 + 0;
  • 10 617 201 128 202 ÷ 2 = 5 308 600 564 101 + 0;
  • 5 308 600 564 101 ÷ 2 = 2 654 300 282 050 + 1;
  • 2 654 300 282 050 ÷ 2 = 1 327 150 141 025 + 0;
  • 1 327 150 141 025 ÷ 2 = 663 575 070 512 + 1;
  • 663 575 070 512 ÷ 2 = 331 787 535 256 + 0;
  • 331 787 535 256 ÷ 2 = 165 893 767 628 + 0;
  • 165 893 767 628 ÷ 2 = 82 946 883 814 + 0;
  • 82 946 883 814 ÷ 2 = 41 473 441 907 + 0;
  • 41 473 441 907 ÷ 2 = 20 736 720 953 + 1;
  • 20 736 720 953 ÷ 2 = 10 368 360 476 + 1;
  • 10 368 360 476 ÷ 2 = 5 184 180 238 + 0;
  • 5 184 180 238 ÷ 2 = 2 592 090 119 + 0;
  • 2 592 090 119 ÷ 2 = 1 296 045 059 + 1;
  • 1 296 045 059 ÷ 2 = 648 022 529 + 1;
  • 648 022 529 ÷ 2 = 324 011 264 + 1;
  • 324 011 264 ÷ 2 = 162 005 632 + 0;
  • 162 005 632 ÷ 2 = 81 002 816 + 0;
  • 81 002 816 ÷ 2 = 40 501 408 + 0;
  • 40 501 408 ÷ 2 = 20 250 704 + 0;
  • 20 250 704 ÷ 2 = 10 125 352 + 0;
  • 10 125 352 ÷ 2 = 5 062 676 + 0;
  • 5 062 676 ÷ 2 = 2 531 338 + 0;
  • 2 531 338 ÷ 2 = 1 265 669 + 0;
  • 1 265 669 ÷ 2 = 632 834 + 1;
  • 632 834 ÷ 2 = 316 417 + 0;
  • 316 417 ÷ 2 = 158 208 + 1;
  • 158 208 ÷ 2 = 79 104 + 0;
  • 79 104 ÷ 2 = 39 552 + 0;
  • 39 552 ÷ 2 = 19 776 + 0;
  • 19 776 ÷ 2 = 9 888 + 0;
  • 9 888 ÷ 2 = 4 944 + 0;
  • 4 944 ÷ 2 = 2 472 + 0;
  • 2 472 ÷ 2 = 1 236 + 0;
  • 1 236 ÷ 2 = 618 + 0;
  • 618 ÷ 2 = 309 + 0;
  • 309 ÷ 2 = 154 + 1;
  • 154 ÷ 2 = 77 + 0;
  • 77 ÷ 2 = 38 + 1;
  • 38 ÷ 2 = 19 + 0;
  • 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.

5 436 006 977 639 628(10) = 1 0011 0101 0000 0000 0101 0000 0000 1110 0110 0001 0100 1100 1100(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 53.

  • 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, 53,

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 5 436 006 977 639 628(10) converted to signed binary in two's complement representation:

5 436 006 977 639 628(10) = 0000 0000 0001 0011 0101 0000 0000 0101 0000 0000 1110 0110 0001 0100 1100 1100

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