Convert 123 456 789 012 345 568 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 123 456 789 012 345 568(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
123 456 789 012 345 568 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. 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;
  • 123 456 789 012 345 568 ÷ 2 = 61 728 394 506 172 784 + 0;
  • 61 728 394 506 172 784 ÷ 2 = 30 864 197 253 086 392 + 0;
  • 30 864 197 253 086 392 ÷ 2 = 15 432 098 626 543 196 + 0;
  • 15 432 098 626 543 196 ÷ 2 = 7 716 049 313 271 598 + 0;
  • 7 716 049 313 271 598 ÷ 2 = 3 858 024 656 635 799 + 0;
  • 3 858 024 656 635 799 ÷ 2 = 1 929 012 328 317 899 + 1;
  • 1 929 012 328 317 899 ÷ 2 = 964 506 164 158 949 + 1;
  • 964 506 164 158 949 ÷ 2 = 482 253 082 079 474 + 1;
  • 482 253 082 079 474 ÷ 2 = 241 126 541 039 737 + 0;
  • 241 126 541 039 737 ÷ 2 = 120 563 270 519 868 + 1;
  • 120 563 270 519 868 ÷ 2 = 60 281 635 259 934 + 0;
  • 60 281 635 259 934 ÷ 2 = 30 140 817 629 967 + 0;
  • 30 140 817 629 967 ÷ 2 = 15 070 408 814 983 + 1;
  • 15 070 408 814 983 ÷ 2 = 7 535 204 407 491 + 1;
  • 7 535 204 407 491 ÷ 2 = 3 767 602 203 745 + 1;
  • 3 767 602 203 745 ÷ 2 = 1 883 801 101 872 + 1;
  • 1 883 801 101 872 ÷ 2 = 941 900 550 936 + 0;
  • 941 900 550 936 ÷ 2 = 470 950 275 468 + 0;
  • 470 950 275 468 ÷ 2 = 235 475 137 734 + 0;
  • 235 475 137 734 ÷ 2 = 117 737 568 867 + 0;
  • 117 737 568 867 ÷ 2 = 58 868 784 433 + 1;
  • 58 868 784 433 ÷ 2 = 29 434 392 216 + 1;
  • 29 434 392 216 ÷ 2 = 14 717 196 108 + 0;
  • 14 717 196 108 ÷ 2 = 7 358 598 054 + 0;
  • 7 358 598 054 ÷ 2 = 3 679 299 027 + 0;
  • 3 679 299 027 ÷ 2 = 1 839 649 513 + 1;
  • 1 839 649 513 ÷ 2 = 919 824 756 + 1;
  • 919 824 756 ÷ 2 = 459 912 378 + 0;
  • 459 912 378 ÷ 2 = 229 956 189 + 0;
  • 229 956 189 ÷ 2 = 114 978 094 + 1;
  • 114 978 094 ÷ 2 = 57 489 047 + 0;
  • 57 489 047 ÷ 2 = 28 744 523 + 1;
  • 28 744 523 ÷ 2 = 14 372 261 + 1;
  • 14 372 261 ÷ 2 = 7 186 130 + 1;
  • 7 186 130 ÷ 2 = 3 593 065 + 0;
  • 3 593 065 ÷ 2 = 1 796 532 + 1;
  • 1 796 532 ÷ 2 = 898 266 + 0;
  • 898 266 ÷ 2 = 449 133 + 0;
  • 449 133 ÷ 2 = 224 566 + 1;
  • 224 566 ÷ 2 = 112 283 + 0;
  • 112 283 ÷ 2 = 56 141 + 1;
  • 56 141 ÷ 2 = 28 070 + 1;
  • 28 070 ÷ 2 = 14 035 + 0;
  • 14 035 ÷ 2 = 7 017 + 1;
  • 7 017 ÷ 2 = 3 508 + 1;
  • 3 508 ÷ 2 = 1 754 + 0;
  • 1 754 ÷ 2 = 877 + 0;
  • 877 ÷ 2 = 438 + 1;
  • 438 ÷ 2 = 219 + 0;
  • 219 ÷ 2 = 109 + 1;
  • 109 ÷ 2 = 54 + 1;
  • 54 ÷ 2 = 27 + 0;
  • 27 ÷ 2 = 13 + 1;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number:

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

123 456 789 012 345 568(10) = 1 1011 0110 1001 1011 0100 1011 1010 0110 0011 0000 1111 0010 1110 0000(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 57.

  • 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, 57,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


4. 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.


Decimal Number 123 456 789 012 345 568(10) converted to signed binary in two's complement representation:

123 456 789 012 345 568(10) = 0000 0001 1011 0110 1001 1011 0100 1011 1010 0110 0011 0000 1111 0010 1110 0000

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