Convert 10 011 111 000 582 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 10 011 111 000 582(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
10 011 111 000 582 to a signed binary in one's (1'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.

Stop when you get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 10 011 111 000 582 ÷ 2 = 5 005 555 500 291 + 0;
  • 5 005 555 500 291 ÷ 2 = 2 502 777 750 145 + 1;
  • 2 502 777 750 145 ÷ 2 = 1 251 388 875 072 + 1;
  • 1 251 388 875 072 ÷ 2 = 625 694 437 536 + 0;
  • 625 694 437 536 ÷ 2 = 312 847 218 768 + 0;
  • 312 847 218 768 ÷ 2 = 156 423 609 384 + 0;
  • 156 423 609 384 ÷ 2 = 78 211 804 692 + 0;
  • 78 211 804 692 ÷ 2 = 39 105 902 346 + 0;
  • 39 105 902 346 ÷ 2 = 19 552 951 173 + 0;
  • 19 552 951 173 ÷ 2 = 9 776 475 586 + 1;
  • 9 776 475 586 ÷ 2 = 4 888 237 793 + 0;
  • 4 888 237 793 ÷ 2 = 2 444 118 896 + 1;
  • 2 444 118 896 ÷ 2 = 1 222 059 448 + 0;
  • 1 222 059 448 ÷ 2 = 611 029 724 + 0;
  • 611 029 724 ÷ 2 = 305 514 862 + 0;
  • 305 514 862 ÷ 2 = 152 757 431 + 0;
  • 152 757 431 ÷ 2 = 76 378 715 + 1;
  • 76 378 715 ÷ 2 = 38 189 357 + 1;
  • 38 189 357 ÷ 2 = 19 094 678 + 1;
  • 19 094 678 ÷ 2 = 9 547 339 + 0;
  • 9 547 339 ÷ 2 = 4 773 669 + 1;
  • 4 773 669 ÷ 2 = 2 386 834 + 1;
  • 2 386 834 ÷ 2 = 1 193 417 + 0;
  • 1 193 417 ÷ 2 = 596 708 + 1;
  • 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.

10 011 111 000 582(10) = 1001 0001 1010 1110 0100 1011 0111 0000 1010 0000 0110(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 10 011 111 000 582(10) converted to signed binary in one's complement representation:

10 011 111 000 582(10) = 0000 0000 0000 0000 0000 1001 0001 1010 1110 0100 1011 0111 0000 1010 0000 0110

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed integers from the decimal system to signed binary in one's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in one'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 that is to be converted to binary, keeping track of each remainder, until we get a quotient that is equal to 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, fill in '0' bits in front (to the left) of the base 2 number calculated above, up to the right length; this way the first bit (leftmost) will always 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 '1's and all '1' bits with '0's.

Example: convert the negative number -49 from the decimal system (base ten) to signed binary one's complement:

  • 1. Start with the positive version of the number: |-49| = 49
  • 2. Divide repeatedly 49 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 49 ÷ 2 = 24 + 1
    • 24 ÷ 2 = 12 + 0
    • 12 ÷ 2 = 6 + 0
    • 6 ÷ 2 = 3 + 0
    • 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:
    49(10) = 11 0001(2)
  • 4. The actual bit length of base 2 representation is 6, so the positive binary computer representation of a signed binary will take in this case 8 bits (the least power of 2 that is larger than 6) - add '0's in front of the base 2 number, up to the required length:
    49(10) = 0011 0001(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's:
    -49(10) = 1100 1110
  • Number -49(10), signed integer, converted from the decimal system (base 10) to signed binary in one's complement representation = 1100 1110