Convert 10 011 101 000 855 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 011 101 000 855(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 011 101 000 855 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;
  • 10 011 101 000 855 ÷ 2 = 5 005 550 500 427 + 1;
  • 5 005 550 500 427 ÷ 2 = 2 502 775 250 213 + 1;
  • 2 502 775 250 213 ÷ 2 = 1 251 387 625 106 + 1;
  • 1 251 387 625 106 ÷ 2 = 625 693 812 553 + 0;
  • 625 693 812 553 ÷ 2 = 312 846 906 276 + 1;
  • 312 846 906 276 ÷ 2 = 156 423 453 138 + 0;
  • 156 423 453 138 ÷ 2 = 78 211 726 569 + 0;
  • 78 211 726 569 ÷ 2 = 39 105 863 284 + 1;
  • 39 105 863 284 ÷ 2 = 19 552 931 642 + 0;
  • 19 552 931 642 ÷ 2 = 9 776 465 821 + 0;
  • 9 776 465 821 ÷ 2 = 4 888 232 910 + 1;
  • 4 888 232 910 ÷ 2 = 2 444 116 455 + 0;
  • 2 444 116 455 ÷ 2 = 1 222 058 227 + 1;
  • 1 222 058 227 ÷ 2 = 611 029 113 + 1;
  • 611 029 113 ÷ 2 = 305 514 556 + 1;
  • 305 514 556 ÷ 2 = 152 757 278 + 0;
  • 152 757 278 ÷ 2 = 76 378 639 + 0;
  • 76 378 639 ÷ 2 = 38 189 319 + 1;
  • 38 189 319 ÷ 2 = 19 094 659 + 1;
  • 19 094 659 ÷ 2 = 9 547 329 + 1;
  • 9 547 329 ÷ 2 = 4 773 664 + 1;
  • 4 773 664 ÷ 2 = 2 386 832 + 0;
  • 2 386 832 ÷ 2 = 1 193 416 + 0;
  • 1 193 416 ÷ 2 = 596 708 + 0;
  • 596 708 ÷ 2 = 298 354 + 0;
  • 298 354 ÷ 2 = 149 177 + 0;
  • 149 177 ÷ 2 = 74 588 + 1;
  • 74 588 ÷ 2 = 37 294 + 0;
  • 37 294 ÷ 2 = 18 647 + 0;
  • 18 647 ÷ 2 = 9 323 + 1;
  • 9 323 ÷ 2 = 4 661 + 1;
  • 4 661 ÷ 2 = 2 330 + 1;
  • 2 330 ÷ 2 = 1 165 + 0;
  • 1 165 ÷ 2 = 582 + 1;
  • 582 ÷ 2 = 291 + 0;
  • 291 ÷ 2 = 145 + 1;
  • 145 ÷ 2 = 72 + 1;
  • 72 ÷ 2 = 36 + 0;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.

10 011 101 000 855(10) = 1001 0001 1010 1110 0100 0001 1110 0111 0100 1001 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 44.

  • 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, 44,

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 10 011 101 000 855(10) converted to signed binary in two's complement representation:

10 011 101 000 855(10) = 0000 0000 0000 0000 0000 1001 0001 1010 1110 0100 0001 1110 0111 0100 1001 0111

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