Convert 7 700 000 961 713 984 957 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 7 700 000 961 713 984 957(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
7 700 000 961 713 984 957 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;
  • 7 700 000 961 713 984 957 ÷ 2 = 3 850 000 480 856 992 478 + 1;
  • 3 850 000 480 856 992 478 ÷ 2 = 1 925 000 240 428 496 239 + 0;
  • 1 925 000 240 428 496 239 ÷ 2 = 962 500 120 214 248 119 + 1;
  • 962 500 120 214 248 119 ÷ 2 = 481 250 060 107 124 059 + 1;
  • 481 250 060 107 124 059 ÷ 2 = 240 625 030 053 562 029 + 1;
  • 240 625 030 053 562 029 ÷ 2 = 120 312 515 026 781 014 + 1;
  • 120 312 515 026 781 014 ÷ 2 = 60 156 257 513 390 507 + 0;
  • 60 156 257 513 390 507 ÷ 2 = 30 078 128 756 695 253 + 1;
  • 30 078 128 756 695 253 ÷ 2 = 15 039 064 378 347 626 + 1;
  • 15 039 064 378 347 626 ÷ 2 = 7 519 532 189 173 813 + 0;
  • 7 519 532 189 173 813 ÷ 2 = 3 759 766 094 586 906 + 1;
  • 3 759 766 094 586 906 ÷ 2 = 1 879 883 047 293 453 + 0;
  • 1 879 883 047 293 453 ÷ 2 = 939 941 523 646 726 + 1;
  • 939 941 523 646 726 ÷ 2 = 469 970 761 823 363 + 0;
  • 469 970 761 823 363 ÷ 2 = 234 985 380 911 681 + 1;
  • 234 985 380 911 681 ÷ 2 = 117 492 690 455 840 + 1;
  • 117 492 690 455 840 ÷ 2 = 58 746 345 227 920 + 0;
  • 58 746 345 227 920 ÷ 2 = 29 373 172 613 960 + 0;
  • 29 373 172 613 960 ÷ 2 = 14 686 586 306 980 + 0;
  • 14 686 586 306 980 ÷ 2 = 7 343 293 153 490 + 0;
  • 7 343 293 153 490 ÷ 2 = 3 671 646 576 745 + 0;
  • 3 671 646 576 745 ÷ 2 = 1 835 823 288 372 + 1;
  • 1 835 823 288 372 ÷ 2 = 917 911 644 186 + 0;
  • 917 911 644 186 ÷ 2 = 458 955 822 093 + 0;
  • 458 955 822 093 ÷ 2 = 229 477 911 046 + 1;
  • 229 477 911 046 ÷ 2 = 114 738 955 523 + 0;
  • 114 738 955 523 ÷ 2 = 57 369 477 761 + 1;
  • 57 369 477 761 ÷ 2 = 28 684 738 880 + 1;
  • 28 684 738 880 ÷ 2 = 14 342 369 440 + 0;
  • 14 342 369 440 ÷ 2 = 7 171 184 720 + 0;
  • 7 171 184 720 ÷ 2 = 3 585 592 360 + 0;
  • 3 585 592 360 ÷ 2 = 1 792 796 180 + 0;
  • 1 792 796 180 ÷ 2 = 896 398 090 + 0;
  • 896 398 090 ÷ 2 = 448 199 045 + 0;
  • 448 199 045 ÷ 2 = 224 099 522 + 1;
  • 224 099 522 ÷ 2 = 112 049 761 + 0;
  • 112 049 761 ÷ 2 = 56 024 880 + 1;
  • 56 024 880 ÷ 2 = 28 012 440 + 0;
  • 28 012 440 ÷ 2 = 14 006 220 + 0;
  • 14 006 220 ÷ 2 = 7 003 110 + 0;
  • 7 003 110 ÷ 2 = 3 501 555 + 0;
  • 3 501 555 ÷ 2 = 1 750 777 + 1;
  • 1 750 777 ÷ 2 = 875 388 + 1;
  • 875 388 ÷ 2 = 437 694 + 0;
  • 437 694 ÷ 2 = 218 847 + 0;
  • 218 847 ÷ 2 = 109 423 + 1;
  • 109 423 ÷ 2 = 54 711 + 1;
  • 54 711 ÷ 2 = 27 355 + 1;
  • 27 355 ÷ 2 = 13 677 + 1;
  • 13 677 ÷ 2 = 6 838 + 1;
  • 6 838 ÷ 2 = 3 419 + 0;
  • 3 419 ÷ 2 = 1 709 + 1;
  • 1 709 ÷ 2 = 854 + 1;
  • 854 ÷ 2 = 427 + 0;
  • 427 ÷ 2 = 213 + 1;
  • 213 ÷ 2 = 106 + 1;
  • 106 ÷ 2 = 53 + 0;
  • 53 ÷ 2 = 26 + 1;
  • 26 ÷ 2 = 13 + 0;
  • 13 ÷ 2 = 6 + 1;
  • 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.

7 700 000 961 713 984 957(10) = 110 1010 1101 1011 1110 0110 0001 0100 0000 1101 0010 0000 1101 0101 1011 1101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 63.

  • 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, 63,

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 7 700 000 961 713 984 957(10) converted to signed binary in two's complement representation:

7 700 000 961 713 984 957(10) = 0110 1010 1101 1011 1110 0110 0001 0100 0000 1101 0010 0000 1101 0101 1011 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