Convert -2 980 208 352 340 350 888 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -2 980 208 352 340 350 888(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-2 980 208 352 340 350 888 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. Start with the positive version of the number:

|-2 980 208 352 340 350 888| = 2 980 208 352 340 350 888

2. 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;
  • 2 980 208 352 340 350 888 ÷ 2 = 1 490 104 176 170 175 444 + 0;
  • 1 490 104 176 170 175 444 ÷ 2 = 745 052 088 085 087 722 + 0;
  • 745 052 088 085 087 722 ÷ 2 = 372 526 044 042 543 861 + 0;
  • 372 526 044 042 543 861 ÷ 2 = 186 263 022 021 271 930 + 1;
  • 186 263 022 021 271 930 ÷ 2 = 93 131 511 010 635 965 + 0;
  • 93 131 511 010 635 965 ÷ 2 = 46 565 755 505 317 982 + 1;
  • 46 565 755 505 317 982 ÷ 2 = 23 282 877 752 658 991 + 0;
  • 23 282 877 752 658 991 ÷ 2 = 11 641 438 876 329 495 + 1;
  • 11 641 438 876 329 495 ÷ 2 = 5 820 719 438 164 747 + 1;
  • 5 820 719 438 164 747 ÷ 2 = 2 910 359 719 082 373 + 1;
  • 2 910 359 719 082 373 ÷ 2 = 1 455 179 859 541 186 + 1;
  • 1 455 179 859 541 186 ÷ 2 = 727 589 929 770 593 + 0;
  • 727 589 929 770 593 ÷ 2 = 363 794 964 885 296 + 1;
  • 363 794 964 885 296 ÷ 2 = 181 897 482 442 648 + 0;
  • 181 897 482 442 648 ÷ 2 = 90 948 741 221 324 + 0;
  • 90 948 741 221 324 ÷ 2 = 45 474 370 610 662 + 0;
  • 45 474 370 610 662 ÷ 2 = 22 737 185 305 331 + 0;
  • 22 737 185 305 331 ÷ 2 = 11 368 592 652 665 + 1;
  • 11 368 592 652 665 ÷ 2 = 5 684 296 326 332 + 1;
  • 5 684 296 326 332 ÷ 2 = 2 842 148 163 166 + 0;
  • 2 842 148 163 166 ÷ 2 = 1 421 074 081 583 + 0;
  • 1 421 074 081 583 ÷ 2 = 710 537 040 791 + 1;
  • 710 537 040 791 ÷ 2 = 355 268 520 395 + 1;
  • 355 268 520 395 ÷ 2 = 177 634 260 197 + 1;
  • 177 634 260 197 ÷ 2 = 88 817 130 098 + 1;
  • 88 817 130 098 ÷ 2 = 44 408 565 049 + 0;
  • 44 408 565 049 ÷ 2 = 22 204 282 524 + 1;
  • 22 204 282 524 ÷ 2 = 11 102 141 262 + 0;
  • 11 102 141 262 ÷ 2 = 5 551 070 631 + 0;
  • 5 551 070 631 ÷ 2 = 2 775 535 315 + 1;
  • 2 775 535 315 ÷ 2 = 1 387 767 657 + 1;
  • 1 387 767 657 ÷ 2 = 693 883 828 + 1;
  • 693 883 828 ÷ 2 = 346 941 914 + 0;
  • 346 941 914 ÷ 2 = 173 470 957 + 0;
  • 173 470 957 ÷ 2 = 86 735 478 + 1;
  • 86 735 478 ÷ 2 = 43 367 739 + 0;
  • 43 367 739 ÷ 2 = 21 683 869 + 1;
  • 21 683 869 ÷ 2 = 10 841 934 + 1;
  • 10 841 934 ÷ 2 = 5 420 967 + 0;
  • 5 420 967 ÷ 2 = 2 710 483 + 1;
  • 2 710 483 ÷ 2 = 1 355 241 + 1;
  • 1 355 241 ÷ 2 = 677 620 + 1;
  • 677 620 ÷ 2 = 338 810 + 0;
  • 338 810 ÷ 2 = 169 405 + 0;
  • 169 405 ÷ 2 = 84 702 + 1;
  • 84 702 ÷ 2 = 42 351 + 0;
  • 42 351 ÷ 2 = 21 175 + 1;
  • 21 175 ÷ 2 = 10 587 + 1;
  • 10 587 ÷ 2 = 5 293 + 1;
  • 5 293 ÷ 2 = 2 646 + 1;
  • 2 646 ÷ 2 = 1 323 + 0;
  • 1 323 ÷ 2 = 661 + 1;
  • 661 ÷ 2 = 330 + 1;
  • 330 ÷ 2 = 165 + 0;
  • 165 ÷ 2 = 82 + 1;
  • 82 ÷ 2 = 41 + 0;
  • 41 ÷ 2 = 20 + 1;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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

2 980 208 352 340 350 888(10) = 10 1001 0101 1011 1101 0011 1011 0100 1110 0101 1110 0110 0001 0111 1010 1000(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 62.

  • 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, 62,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


5. 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.


2 980 208 352 340 350 888(10) = 0010 1001 0101 1011 1101 0011 1011 0100 1110 0101 1110 0110 0001 0111 1010 1000

6. Get the negative integer number representation. Part 1:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation, replace all the bits on 0 with 1s and all the bits set on 1 with 0s.

Reverse the digits, flip the digits:

Replace the bits set on 0 with 1s and the bits set on 1 with 0s.

!(0010 1001 0101 1011 1101 0011 1011 0100 1110 0101 1110 0110 0001 0111 1010 1000)

= 1101 0110 1010 0100 0010 1100 0100 1011 0001 1010 0001 1001 1110 1000 0101 0111


7. Get the negative integer number representation. Part 2:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in two's complement representation, add 1 to the number calculated above 1101 0110 1010 0100 0010 1100 0100 1011 0001 1010 0001 1001 1110 1000 0101 0111 (to the signed binary in one's complement representation).

Binary addition carries on a value of 2:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 10 = 11
  • 1 + 11 = 100

Add 1 to the number calculated above
(to the signed binary number in one's complement representation):

-2 980 208 352 340 350 888 =

1101 0110 1010 0100 0010 1100 0100 1011 0001 1010 0001 1001 1110 1000 0101 0111 + 1


Decimal Number -2 980 208 352 340 350 888(10) converted to signed binary in two's complement representation:

-2 980 208 352 340 350 888(10) = 1101 0110 1010 0100 0010 1100 0100 1011 0001 1010 0001 1001 1110 1000 0101 1000

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