Two's Complement: Integer ↗ Binary: -8 644 934 341 102 468 608 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 -8 644 934 341 102 468 608(10) converted and written as a signed binary in two's complement representation (base 2) = ?

1. Start with the positive version of the number:

|-8 644 934 341 102 468 608| = 8 644 934 341 102 468 608

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;
  • 8 644 934 341 102 468 608 ÷ 2 = 4 322 467 170 551 234 304 + 0;
  • 4 322 467 170 551 234 304 ÷ 2 = 2 161 233 585 275 617 152 + 0;
  • 2 161 233 585 275 617 152 ÷ 2 = 1 080 616 792 637 808 576 + 0;
  • 1 080 616 792 637 808 576 ÷ 2 = 540 308 396 318 904 288 + 0;
  • 540 308 396 318 904 288 ÷ 2 = 270 154 198 159 452 144 + 0;
  • 270 154 198 159 452 144 ÷ 2 = 135 077 099 079 726 072 + 0;
  • 135 077 099 079 726 072 ÷ 2 = 67 538 549 539 863 036 + 0;
  • 67 538 549 539 863 036 ÷ 2 = 33 769 274 769 931 518 + 0;
  • 33 769 274 769 931 518 ÷ 2 = 16 884 637 384 965 759 + 0;
  • 16 884 637 384 965 759 ÷ 2 = 8 442 318 692 482 879 + 1;
  • 8 442 318 692 482 879 ÷ 2 = 4 221 159 346 241 439 + 1;
  • 4 221 159 346 241 439 ÷ 2 = 2 110 579 673 120 719 + 1;
  • 2 110 579 673 120 719 ÷ 2 = 1 055 289 836 560 359 + 1;
  • 1 055 289 836 560 359 ÷ 2 = 527 644 918 280 179 + 1;
  • 527 644 918 280 179 ÷ 2 = 263 822 459 140 089 + 1;
  • 263 822 459 140 089 ÷ 2 = 131 911 229 570 044 + 1;
  • 131 911 229 570 044 ÷ 2 = 65 955 614 785 022 + 0;
  • 65 955 614 785 022 ÷ 2 = 32 977 807 392 511 + 0;
  • 32 977 807 392 511 ÷ 2 = 16 488 903 696 255 + 1;
  • 16 488 903 696 255 ÷ 2 = 8 244 451 848 127 + 1;
  • 8 244 451 848 127 ÷ 2 = 4 122 225 924 063 + 1;
  • 4 122 225 924 063 ÷ 2 = 2 061 112 962 031 + 1;
  • 2 061 112 962 031 ÷ 2 = 1 030 556 481 015 + 1;
  • 1 030 556 481 015 ÷ 2 = 515 278 240 507 + 1;
  • 515 278 240 507 ÷ 2 = 257 639 120 253 + 1;
  • 257 639 120 253 ÷ 2 = 128 819 560 126 + 1;
  • 128 819 560 126 ÷ 2 = 64 409 780 063 + 0;
  • 64 409 780 063 ÷ 2 = 32 204 890 031 + 1;
  • 32 204 890 031 ÷ 2 = 16 102 445 015 + 1;
  • 16 102 445 015 ÷ 2 = 8 051 222 507 + 1;
  • 8 051 222 507 ÷ 2 = 4 025 611 253 + 1;
  • 4 025 611 253 ÷ 2 = 2 012 805 626 + 1;
  • 2 012 805 626 ÷ 2 = 1 006 402 813 + 0;
  • 1 006 402 813 ÷ 2 = 503 201 406 + 1;
  • 503 201 406 ÷ 2 = 251 600 703 + 0;
  • 251 600 703 ÷ 2 = 125 800 351 + 1;
  • 125 800 351 ÷ 2 = 62 900 175 + 1;
  • 62 900 175 ÷ 2 = 31 450 087 + 1;
  • 31 450 087 ÷ 2 = 15 725 043 + 1;
  • 15 725 043 ÷ 2 = 7 862 521 + 1;
  • 7 862 521 ÷ 2 = 3 931 260 + 1;
  • 3 931 260 ÷ 2 = 1 965 630 + 0;
  • 1 965 630 ÷ 2 = 982 815 + 0;
  • 982 815 ÷ 2 = 491 407 + 1;
  • 491 407 ÷ 2 = 245 703 + 1;
  • 245 703 ÷ 2 = 122 851 + 1;
  • 122 851 ÷ 2 = 61 425 + 1;
  • 61 425 ÷ 2 = 30 712 + 1;
  • 30 712 ÷ 2 = 15 356 + 0;
  • 15 356 ÷ 2 = 7 678 + 0;
  • 7 678 ÷ 2 = 3 839 + 0;
  • 3 839 ÷ 2 = 1 919 + 1;
  • 1 919 ÷ 2 = 959 + 1;
  • 959 ÷ 2 = 479 + 1;
  • 479 ÷ 2 = 239 + 1;
  • 239 ÷ 2 = 119 + 1;
  • 119 ÷ 2 = 59 + 1;
  • 59 ÷ 2 = 29 + 1;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 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.


8 644 934 341 102 468 608(10) = 111 0111 1111 1000 1111 1001 1111 1010 1111 1011 1111 1100 1111 1110 0000 0000(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.


8 644 934 341 102 468 608(10) = 0111 0111 1111 1000 1111 1001 1111 1010 1111 1011 1111 1100 1111 1110 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.

!(0111 0111 1111 1000 1111 1001 1111 1010 1111 1011 1111 1100 1111 1110 0000 0000)

= 1000 1000 0000 0111 0000 0110 0000 0101 0000 0100 0000 0011 0000 0001 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

1000 1000 0000 0111 0000 0110 0000 0101 0000 0100 0000 0011 0000 0001 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):

-8 644 934 341 102 468 608 =

1000 1000 0000 0111 0000 0110 0000 0101 0000 0100 0000 0011 0000 0001 1111 1111 + 1


Number -8 644 934 341 102 468 608(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:

-8 644 934 341 102 468 608(10) = 1000 1000 0000 0111 0000 0110 0000 0101 0000 0100 0000 0011 0000 0010 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 -8 644 934 341 102 468 609 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 -8 644 934 341 102 468 607 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