Two's Complement: Integer ↗ Binary: -4 611 686 017 823 408 172 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 -4 611 686 017 823 408 172(10) converted and written as a signed binary in two's complement representation (base 2) = ?

1. Start with the positive version of the number:

|-4 611 686 017 823 408 172| = 4 611 686 017 823 408 172

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;
  • 4 611 686 017 823 408 172 ÷ 2 = 2 305 843 008 911 704 086 + 0;
  • 2 305 843 008 911 704 086 ÷ 2 = 1 152 921 504 455 852 043 + 0;
  • 1 152 921 504 455 852 043 ÷ 2 = 576 460 752 227 926 021 + 1;
  • 576 460 752 227 926 021 ÷ 2 = 288 230 376 113 963 010 + 1;
  • 288 230 376 113 963 010 ÷ 2 = 144 115 188 056 981 505 + 0;
  • 144 115 188 056 981 505 ÷ 2 = 72 057 594 028 490 752 + 1;
  • 72 057 594 028 490 752 ÷ 2 = 36 028 797 014 245 376 + 0;
  • 36 028 797 014 245 376 ÷ 2 = 18 014 398 507 122 688 + 0;
  • 18 014 398 507 122 688 ÷ 2 = 9 007 199 253 561 344 + 0;
  • 9 007 199 253 561 344 ÷ 2 = 4 503 599 626 780 672 + 0;
  • 4 503 599 626 780 672 ÷ 2 = 2 251 799 813 390 336 + 0;
  • 2 251 799 813 390 336 ÷ 2 = 1 125 899 906 695 168 + 0;
  • 1 125 899 906 695 168 ÷ 2 = 562 949 953 347 584 + 0;
  • 562 949 953 347 584 ÷ 2 = 281 474 976 673 792 + 0;
  • 281 474 976 673 792 ÷ 2 = 140 737 488 336 896 + 0;
  • 140 737 488 336 896 ÷ 2 = 70 368 744 168 448 + 0;
  • 70 368 744 168 448 ÷ 2 = 35 184 372 084 224 + 0;
  • 35 184 372 084 224 ÷ 2 = 17 592 186 042 112 + 0;
  • 17 592 186 042 112 ÷ 2 = 8 796 093 021 056 + 0;
  • 8 796 093 021 056 ÷ 2 = 4 398 046 510 528 + 0;
  • 4 398 046 510 528 ÷ 2 = 2 199 023 255 264 + 0;
  • 2 199 023 255 264 ÷ 2 = 1 099 511 627 632 + 0;
  • 1 099 511 627 632 ÷ 2 = 549 755 813 816 + 0;
  • 549 755 813 816 ÷ 2 = 274 877 906 908 + 0;
  • 274 877 906 908 ÷ 2 = 137 438 953 454 + 0;
  • 137 438 953 454 ÷ 2 = 68 719 476 727 + 0;
  • 68 719 476 727 ÷ 2 = 34 359 738 363 + 1;
  • 34 359 738 363 ÷ 2 = 17 179 869 181 + 1;
  • 17 179 869 181 ÷ 2 = 8 589 934 590 + 1;
  • 8 589 934 590 ÷ 2 = 4 294 967 295 + 0;
  • 4 294 967 295 ÷ 2 = 2 147 483 647 + 1;
  • 2 147 483 647 ÷ 2 = 1 073 741 823 + 1;
  • 1 073 741 823 ÷ 2 = 536 870 911 + 1;
  • 536 870 911 ÷ 2 = 268 435 455 + 1;
  • 268 435 455 ÷ 2 = 134 217 727 + 1;
  • 134 217 727 ÷ 2 = 67 108 863 + 1;
  • 67 108 863 ÷ 2 = 33 554 431 + 1;
  • 33 554 431 ÷ 2 = 16 777 215 + 1;
  • 16 777 215 ÷ 2 = 8 388 607 + 1;
  • 8 388 607 ÷ 2 = 4 194 303 + 1;
  • 4 194 303 ÷ 2 = 2 097 151 + 1;
  • 2 097 151 ÷ 2 = 1 048 575 + 1;
  • 1 048 575 ÷ 2 = 524 287 + 1;
  • 524 287 ÷ 2 = 262 143 + 1;
  • 262 143 ÷ 2 = 131 071 + 1;
  • 131 071 ÷ 2 = 65 535 + 1;
  • 65 535 ÷ 2 = 32 767 + 1;
  • 32 767 ÷ 2 = 16 383 + 1;
  • 16 383 ÷ 2 = 8 191 + 1;
  • 8 191 ÷ 2 = 4 095 + 1;
  • 4 095 ÷ 2 = 2 047 + 1;
  • 2 047 ÷ 2 = 1 023 + 1;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 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:

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


4 611 686 017 823 408 172(10) = 11 1111 1111 1111 1111 1111 1111 1111 1101 1100 0000 0000 0000 0000 0010 1100(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.


4 611 686 017 823 408 172(10) = 0011 1111 1111 1111 1111 1111 1111 1111 1101 1100 0000 0000 0000 0000 0010 1100


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.

!(0011 1111 1111 1111 1111 1111 1111 1111 1101 1100 0000 0000 0000 0000 0010 1100)

= 1100 0000 0000 0000 0000 0000 0000 0000 0010 0011 1111 1111 1111 1111 1101 0011


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

1100 0000 0000 0000 0000 0000 0000 0000 0010 0011 1111 1111 1111 1111 1101 0011

(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):

-4 611 686 017 823 408 172 =

1100 0000 0000 0000 0000 0000 0000 0000 0010 0011 1111 1111 1111 1111 1101 0011 + 1


Number -4 611 686 017 823 408 172(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:

-4 611 686 017 823 408 172(10) = 1100 0000 0000 0000 0000 0000 0000 0000 0010 0011 1111 1111 1111 1111 1101 0100

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 -4 611 686 017 823 408 173 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 -4 611 686 017 823 408 171 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