Convert 1 111 110 011 110 623 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 110 011 110 623(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 110 011 110 623 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;
  • 1 111 110 011 110 623 ÷ 2 = 555 555 005 555 311 + 1;
  • 555 555 005 555 311 ÷ 2 = 277 777 502 777 655 + 1;
  • 277 777 502 777 655 ÷ 2 = 138 888 751 388 827 + 1;
  • 138 888 751 388 827 ÷ 2 = 69 444 375 694 413 + 1;
  • 69 444 375 694 413 ÷ 2 = 34 722 187 847 206 + 1;
  • 34 722 187 847 206 ÷ 2 = 17 361 093 923 603 + 0;
  • 17 361 093 923 603 ÷ 2 = 8 680 546 961 801 + 1;
  • 8 680 546 961 801 ÷ 2 = 4 340 273 480 900 + 1;
  • 4 340 273 480 900 ÷ 2 = 2 170 136 740 450 + 0;
  • 2 170 136 740 450 ÷ 2 = 1 085 068 370 225 + 0;
  • 1 085 068 370 225 ÷ 2 = 542 534 185 112 + 1;
  • 542 534 185 112 ÷ 2 = 271 267 092 556 + 0;
  • 271 267 092 556 ÷ 2 = 135 633 546 278 + 0;
  • 135 633 546 278 ÷ 2 = 67 816 773 139 + 0;
  • 67 816 773 139 ÷ 2 = 33 908 386 569 + 1;
  • 33 908 386 569 ÷ 2 = 16 954 193 284 + 1;
  • 16 954 193 284 ÷ 2 = 8 477 096 642 + 0;
  • 8 477 096 642 ÷ 2 = 4 238 548 321 + 0;
  • 4 238 548 321 ÷ 2 = 2 119 274 160 + 1;
  • 2 119 274 160 ÷ 2 = 1 059 637 080 + 0;
  • 1 059 637 080 ÷ 2 = 529 818 540 + 0;
  • 529 818 540 ÷ 2 = 264 909 270 + 0;
  • 264 909 270 ÷ 2 = 132 454 635 + 0;
  • 132 454 635 ÷ 2 = 66 227 317 + 1;
  • 66 227 317 ÷ 2 = 33 113 658 + 1;
  • 33 113 658 ÷ 2 = 16 556 829 + 0;
  • 16 556 829 ÷ 2 = 8 278 414 + 1;
  • 8 278 414 ÷ 2 = 4 139 207 + 0;
  • 4 139 207 ÷ 2 = 2 069 603 + 1;
  • 2 069 603 ÷ 2 = 1 034 801 + 1;
  • 1 034 801 ÷ 2 = 517 400 + 1;
  • 517 400 ÷ 2 = 258 700 + 0;
  • 258 700 ÷ 2 = 129 350 + 0;
  • 129 350 ÷ 2 = 64 675 + 0;
  • 64 675 ÷ 2 = 32 337 + 1;
  • 32 337 ÷ 2 = 16 168 + 1;
  • 16 168 ÷ 2 = 8 084 + 0;
  • 8 084 ÷ 2 = 4 042 + 0;
  • 4 042 ÷ 2 = 2 021 + 0;
  • 2 021 ÷ 2 = 1 010 + 1;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 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.

1 111 110 011 110 623(10) = 11 1111 0010 1000 1100 0111 0101 1000 0100 1100 0100 1101 1111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

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 1 111 110 011 110 623(10) converted to signed binary in two's complement representation:

1 111 110 011 110 623(10) = 0000 0000 0000 0011 1111 0010 1000 1100 0111 0101 1000 0100 1100 0100 1101 1111

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