Two's Complement: Integer ↗ Binary: -7 484 539 350 057 490 745 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 -7 484 539 350 057 490 745(10) converted and written as a signed binary in two's complement representation (base 2) = ?

1. Start with the positive version of the number:

|-7 484 539 350 057 490 745| = 7 484 539 350 057 490 745

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;
  • 7 484 539 350 057 490 745 ÷ 2 = 3 742 269 675 028 745 372 + 1;
  • 3 742 269 675 028 745 372 ÷ 2 = 1 871 134 837 514 372 686 + 0;
  • 1 871 134 837 514 372 686 ÷ 2 = 935 567 418 757 186 343 + 0;
  • 935 567 418 757 186 343 ÷ 2 = 467 783 709 378 593 171 + 1;
  • 467 783 709 378 593 171 ÷ 2 = 233 891 854 689 296 585 + 1;
  • 233 891 854 689 296 585 ÷ 2 = 116 945 927 344 648 292 + 1;
  • 116 945 927 344 648 292 ÷ 2 = 58 472 963 672 324 146 + 0;
  • 58 472 963 672 324 146 ÷ 2 = 29 236 481 836 162 073 + 0;
  • 29 236 481 836 162 073 ÷ 2 = 14 618 240 918 081 036 + 1;
  • 14 618 240 918 081 036 ÷ 2 = 7 309 120 459 040 518 + 0;
  • 7 309 120 459 040 518 ÷ 2 = 3 654 560 229 520 259 + 0;
  • 3 654 560 229 520 259 ÷ 2 = 1 827 280 114 760 129 + 1;
  • 1 827 280 114 760 129 ÷ 2 = 913 640 057 380 064 + 1;
  • 913 640 057 380 064 ÷ 2 = 456 820 028 690 032 + 0;
  • 456 820 028 690 032 ÷ 2 = 228 410 014 345 016 + 0;
  • 228 410 014 345 016 ÷ 2 = 114 205 007 172 508 + 0;
  • 114 205 007 172 508 ÷ 2 = 57 102 503 586 254 + 0;
  • 57 102 503 586 254 ÷ 2 = 28 551 251 793 127 + 0;
  • 28 551 251 793 127 ÷ 2 = 14 275 625 896 563 + 1;
  • 14 275 625 896 563 ÷ 2 = 7 137 812 948 281 + 1;
  • 7 137 812 948 281 ÷ 2 = 3 568 906 474 140 + 1;
  • 3 568 906 474 140 ÷ 2 = 1 784 453 237 070 + 0;
  • 1 784 453 237 070 ÷ 2 = 892 226 618 535 + 0;
  • 892 226 618 535 ÷ 2 = 446 113 309 267 + 1;
  • 446 113 309 267 ÷ 2 = 223 056 654 633 + 1;
  • 223 056 654 633 ÷ 2 = 111 528 327 316 + 1;
  • 111 528 327 316 ÷ 2 = 55 764 163 658 + 0;
  • 55 764 163 658 ÷ 2 = 27 882 081 829 + 0;
  • 27 882 081 829 ÷ 2 = 13 941 040 914 + 1;
  • 13 941 040 914 ÷ 2 = 6 970 520 457 + 0;
  • 6 970 520 457 ÷ 2 = 3 485 260 228 + 1;
  • 3 485 260 228 ÷ 2 = 1 742 630 114 + 0;
  • 1 742 630 114 ÷ 2 = 871 315 057 + 0;
  • 871 315 057 ÷ 2 = 435 657 528 + 1;
  • 435 657 528 ÷ 2 = 217 828 764 + 0;
  • 217 828 764 ÷ 2 = 108 914 382 + 0;
  • 108 914 382 ÷ 2 = 54 457 191 + 0;
  • 54 457 191 ÷ 2 = 27 228 595 + 1;
  • 27 228 595 ÷ 2 = 13 614 297 + 1;
  • 13 614 297 ÷ 2 = 6 807 148 + 1;
  • 6 807 148 ÷ 2 = 3 403 574 + 0;
  • 3 403 574 ÷ 2 = 1 701 787 + 0;
  • 1 701 787 ÷ 2 = 850 893 + 1;
  • 850 893 ÷ 2 = 425 446 + 1;
  • 425 446 ÷ 2 = 212 723 + 0;
  • 212 723 ÷ 2 = 106 361 + 1;
  • 106 361 ÷ 2 = 53 180 + 1;
  • 53 180 ÷ 2 = 26 590 + 0;
  • 26 590 ÷ 2 = 13 295 + 0;
  • 13 295 ÷ 2 = 6 647 + 1;
  • 6 647 ÷ 2 = 3 323 + 1;
  • 3 323 ÷ 2 = 1 661 + 1;
  • 1 661 ÷ 2 = 830 + 1;
  • 830 ÷ 2 = 415 + 0;
  • 415 ÷ 2 = 207 + 1;
  • 207 ÷ 2 = 103 + 1;
  • 103 ÷ 2 = 51 + 1;
  • 51 ÷ 2 = 25 + 1;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 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.


7 484 539 350 057 490 745(10) = 110 0111 1101 1110 0110 1100 1110 0010 0101 0011 1001 1100 0001 1001 0011 1001(2)


4. 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.


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.


7 484 539 350 057 490 745(10) = 0110 0111 1101 1110 0110 1100 1110 0010 0101 0011 1001 1100 0001 1001 0011 1001


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.

!(0110 0111 1101 1110 0110 1100 1110 0010 0101 0011 1001 1100 0001 1001 0011 1001)

= 1001 1000 0010 0001 1001 0011 0001 1101 1010 1100 0110 0011 1110 0110 1100 0110


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

1001 1000 0010 0001 1001 0011 0001 1101 1010 1100 0110 0011 1110 0110 1100 0110

(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):

-7 484 539 350 057 490 745 =

1001 1000 0010 0001 1001 0011 0001 1101 1010 1100 0110 0011 1110 0110 1100 0110 + 1


Number -7 484 539 350 057 490 745(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:

-7 484 539 350 057 490 745(10) = 1001 1000 0010 0001 1001 0011 0001 1101 1010 1100 0110 0011 1110 0110 1100 0111

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 -7 484 539 350 057 490 746 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 -7 484 539 350 057 490 744 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