Two's Complement: Integer ↗ Binary: -1 942 713 270 595 850 240 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number -1 942 713 270 595 850 240(10) converted and written as a signed binary in two's complement representation (base 2) = ?

1. Start with the positive version of the number:

|-1 942 713 270 595 850 240| = 1 942 713 270 595 850 240

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;
  • 1 942 713 270 595 850 240 ÷ 2 = 971 356 635 297 925 120 + 0;
  • 971 356 635 297 925 120 ÷ 2 = 485 678 317 648 962 560 + 0;
  • 485 678 317 648 962 560 ÷ 2 = 242 839 158 824 481 280 + 0;
  • 242 839 158 824 481 280 ÷ 2 = 121 419 579 412 240 640 + 0;
  • 121 419 579 412 240 640 ÷ 2 = 60 709 789 706 120 320 + 0;
  • 60 709 789 706 120 320 ÷ 2 = 30 354 894 853 060 160 + 0;
  • 30 354 894 853 060 160 ÷ 2 = 15 177 447 426 530 080 + 0;
  • 15 177 447 426 530 080 ÷ 2 = 7 588 723 713 265 040 + 0;
  • 7 588 723 713 265 040 ÷ 2 = 3 794 361 856 632 520 + 0;
  • 3 794 361 856 632 520 ÷ 2 = 1 897 180 928 316 260 + 0;
  • 1 897 180 928 316 260 ÷ 2 = 948 590 464 158 130 + 0;
  • 948 590 464 158 130 ÷ 2 = 474 295 232 079 065 + 0;
  • 474 295 232 079 065 ÷ 2 = 237 147 616 039 532 + 1;
  • 237 147 616 039 532 ÷ 2 = 118 573 808 019 766 + 0;
  • 118 573 808 019 766 ÷ 2 = 59 286 904 009 883 + 0;
  • 59 286 904 009 883 ÷ 2 = 29 643 452 004 941 + 1;
  • 29 643 452 004 941 ÷ 2 = 14 821 726 002 470 + 1;
  • 14 821 726 002 470 ÷ 2 = 7 410 863 001 235 + 0;
  • 7 410 863 001 235 ÷ 2 = 3 705 431 500 617 + 1;
  • 3 705 431 500 617 ÷ 2 = 1 852 715 750 308 + 1;
  • 1 852 715 750 308 ÷ 2 = 926 357 875 154 + 0;
  • 926 357 875 154 ÷ 2 = 463 178 937 577 + 0;
  • 463 178 937 577 ÷ 2 = 231 589 468 788 + 1;
  • 231 589 468 788 ÷ 2 = 115 794 734 394 + 0;
  • 115 794 734 394 ÷ 2 = 57 897 367 197 + 0;
  • 57 897 367 197 ÷ 2 = 28 948 683 598 + 1;
  • 28 948 683 598 ÷ 2 = 14 474 341 799 + 0;
  • 14 474 341 799 ÷ 2 = 7 237 170 899 + 1;
  • 7 237 170 899 ÷ 2 = 3 618 585 449 + 1;
  • 3 618 585 449 ÷ 2 = 1 809 292 724 + 1;
  • 1 809 292 724 ÷ 2 = 904 646 362 + 0;
  • 904 646 362 ÷ 2 = 452 323 181 + 0;
  • 452 323 181 ÷ 2 = 226 161 590 + 1;
  • 226 161 590 ÷ 2 = 113 080 795 + 0;
  • 113 080 795 ÷ 2 = 56 540 397 + 1;
  • 56 540 397 ÷ 2 = 28 270 198 + 1;
  • 28 270 198 ÷ 2 = 14 135 099 + 0;
  • 14 135 099 ÷ 2 = 7 067 549 + 1;
  • 7 067 549 ÷ 2 = 3 533 774 + 1;
  • 3 533 774 ÷ 2 = 1 766 887 + 0;
  • 1 766 887 ÷ 2 = 883 443 + 1;
  • 883 443 ÷ 2 = 441 721 + 1;
  • 441 721 ÷ 2 = 220 860 + 1;
  • 220 860 ÷ 2 = 110 430 + 0;
  • 110 430 ÷ 2 = 55 215 + 0;
  • 55 215 ÷ 2 = 27 607 + 1;
  • 27 607 ÷ 2 = 13 803 + 1;
  • 13 803 ÷ 2 = 6 901 + 1;
  • 6 901 ÷ 2 = 3 450 + 1;
  • 3 450 ÷ 2 = 1 725 + 0;
  • 1 725 ÷ 2 = 862 + 1;
  • 862 ÷ 2 = 431 + 0;
  • 431 ÷ 2 = 215 + 1;
  • 215 ÷ 2 = 107 + 1;
  • 107 ÷ 2 = 53 + 1;
  • 53 ÷ 2 = 26 + 1;
  • 26 ÷ 2 = 13 + 0;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 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.


1 942 713 270 595 850 240(10) = 1 1010 1111 0101 1110 0111 0110 1101 0011 1010 0100 1101 1001 0000 0000 0000(2)


4. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 61.

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, 61,

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.


1 942 713 270 595 850 240(10) = 0001 1010 1111 0101 1110 0111 0110 1101 0011 1010 0100 1101 1001 0000 0000 0000


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.

!(0001 1010 1111 0101 1110 0111 0110 1101 0011 1010 0100 1101 1001 0000 0000 0000)

= 1110 0101 0000 1010 0001 1000 1001 0010 1100 0101 1011 0010 0110 1111 1111 1111


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

1110 0101 0000 1010 0001 1000 1001 0010 1100 0101 1011 0010 0110 1111 1111 1111

(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):

-1 942 713 270 595 850 240 =

1110 0101 0000 1010 0001 1000 1001 0010 1100 0101 1011 0010 0110 1111 1111 1111 + 1


Number -1 942 713 270 595 850 240(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:

-1 942 713 270 595 850 240(10) = 1110 0101 0000 1010 0001 1000 1001 0010 1100 0101 1011 0010 0111 0000 0000 0000

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

More operations on integers written as signed binary numbers in two's complement representation:

» Signed integer number -1 942 713 270 595 850 241 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

» Signed integer number -1 942 713 270 595 850 239 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

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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