Convert -7 563 680 054 008 462 334 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -7 563 680 054 008 462 334(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-7 563 680 054 008 462 334 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:

|-7 563 680 054 008 462 334| = 7 563 680 054 008 462 334

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;
  • 7 563 680 054 008 462 334 ÷ 2 = 3 781 840 027 004 231 167 + 0;
  • 3 781 840 027 004 231 167 ÷ 2 = 1 890 920 013 502 115 583 + 1;
  • 1 890 920 013 502 115 583 ÷ 2 = 945 460 006 751 057 791 + 1;
  • 945 460 006 751 057 791 ÷ 2 = 472 730 003 375 528 895 + 1;
  • 472 730 003 375 528 895 ÷ 2 = 236 365 001 687 764 447 + 1;
  • 236 365 001 687 764 447 ÷ 2 = 118 182 500 843 882 223 + 1;
  • 118 182 500 843 882 223 ÷ 2 = 59 091 250 421 941 111 + 1;
  • 59 091 250 421 941 111 ÷ 2 = 29 545 625 210 970 555 + 1;
  • 29 545 625 210 970 555 ÷ 2 = 14 772 812 605 485 277 + 1;
  • 14 772 812 605 485 277 ÷ 2 = 7 386 406 302 742 638 + 1;
  • 7 386 406 302 742 638 ÷ 2 = 3 693 203 151 371 319 + 0;
  • 3 693 203 151 371 319 ÷ 2 = 1 846 601 575 685 659 + 1;
  • 1 846 601 575 685 659 ÷ 2 = 923 300 787 842 829 + 1;
  • 923 300 787 842 829 ÷ 2 = 461 650 393 921 414 + 1;
  • 461 650 393 921 414 ÷ 2 = 230 825 196 960 707 + 0;
  • 230 825 196 960 707 ÷ 2 = 115 412 598 480 353 + 1;
  • 115 412 598 480 353 ÷ 2 = 57 706 299 240 176 + 1;
  • 57 706 299 240 176 ÷ 2 = 28 853 149 620 088 + 0;
  • 28 853 149 620 088 ÷ 2 = 14 426 574 810 044 + 0;
  • 14 426 574 810 044 ÷ 2 = 7 213 287 405 022 + 0;
  • 7 213 287 405 022 ÷ 2 = 3 606 643 702 511 + 0;
  • 3 606 643 702 511 ÷ 2 = 1 803 321 851 255 + 1;
  • 1 803 321 851 255 ÷ 2 = 901 660 925 627 + 1;
  • 901 660 925 627 ÷ 2 = 450 830 462 813 + 1;
  • 450 830 462 813 ÷ 2 = 225 415 231 406 + 1;
  • 225 415 231 406 ÷ 2 = 112 707 615 703 + 0;
  • 112 707 615 703 ÷ 2 = 56 353 807 851 + 1;
  • 56 353 807 851 ÷ 2 = 28 176 903 925 + 1;
  • 28 176 903 925 ÷ 2 = 14 088 451 962 + 1;
  • 14 088 451 962 ÷ 2 = 7 044 225 981 + 0;
  • 7 044 225 981 ÷ 2 = 3 522 112 990 + 1;
  • 3 522 112 990 ÷ 2 = 1 761 056 495 + 0;
  • 1 761 056 495 ÷ 2 = 880 528 247 + 1;
  • 880 528 247 ÷ 2 = 440 264 123 + 1;
  • 440 264 123 ÷ 2 = 220 132 061 + 1;
  • 220 132 061 ÷ 2 = 110 066 030 + 1;
  • 110 066 030 ÷ 2 = 55 033 015 + 0;
  • 55 033 015 ÷ 2 = 27 516 507 + 1;
  • 27 516 507 ÷ 2 = 13 758 253 + 1;
  • 13 758 253 ÷ 2 = 6 879 126 + 1;
  • 6 879 126 ÷ 2 = 3 439 563 + 0;
  • 3 439 563 ÷ 2 = 1 719 781 + 1;
  • 1 719 781 ÷ 2 = 859 890 + 1;
  • 859 890 ÷ 2 = 429 945 + 0;
  • 429 945 ÷ 2 = 214 972 + 1;
  • 214 972 ÷ 2 = 107 486 + 0;
  • 107 486 ÷ 2 = 53 743 + 0;
  • 53 743 ÷ 2 = 26 871 + 1;
  • 26 871 ÷ 2 = 13 435 + 1;
  • 13 435 ÷ 2 = 6 717 + 1;
  • 6 717 ÷ 2 = 3 358 + 1;
  • 3 358 ÷ 2 = 1 679 + 0;
  • 1 679 ÷ 2 = 839 + 1;
  • 839 ÷ 2 = 419 + 1;
  • 419 ÷ 2 = 209 + 1;
  • 209 ÷ 2 = 104 + 1;
  • 104 ÷ 2 = 52 + 0;
  • 52 ÷ 2 = 26 + 0;
  • 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.

7 563 680 054 008 462 334(10) = 110 1000 1111 0111 1001 0110 1110 1111 0101 1101 1110 0001 1011 1011 1111 1110(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.


7 563 680 054 008 462 334(10) = 0110 1000 1111 0111 1001 0110 1110 1111 0101 1101 1110 0001 1011 1011 1111 1110

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 1000 1111 0111 1001 0110 1110 1111 0101 1101 1110 0001 1011 1011 1111 1110)

= 1001 0111 0000 1000 0110 1001 0001 0000 1010 0010 0001 1110 0100 0100 0000 0001


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 0111 0000 1000 0110 1001 0001 0000 1010 0010 0001 1110 0100 0100 0000 0001 (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):

-7 563 680 054 008 462 334 =

1001 0111 0000 1000 0110 1001 0001 0000 1010 0010 0001 1110 0100 0100 0000 0001 + 1


Decimal Number -7 563 680 054 008 462 334(10) converted to signed binary in two's complement representation:

-7 563 680 054 008 462 334(10) = 1001 0111 0000 1000 0110 1001 0001 0000 1010 0010 0001 1110 0100 0100 0000 0010

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