One's Complement: Integer -> Binary: 111 001 100 111 059 Convert the Integer Number to a Signed Binary in One's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number 111 001 100 111 059(10) converted and written as a signed binary in one's complement representation (base 2) = ?

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;
  • 111 001 100 111 059 ÷ 2 = 55 500 550 055 529 + 1;
  • 55 500 550 055 529 ÷ 2 = 27 750 275 027 764 + 1;
  • 27 750 275 027 764 ÷ 2 = 13 875 137 513 882 + 0;
  • 13 875 137 513 882 ÷ 2 = 6 937 568 756 941 + 0;
  • 6 937 568 756 941 ÷ 2 = 3 468 784 378 470 + 1;
  • 3 468 784 378 470 ÷ 2 = 1 734 392 189 235 + 0;
  • 1 734 392 189 235 ÷ 2 = 867 196 094 617 + 1;
  • 867 196 094 617 ÷ 2 = 433 598 047 308 + 1;
  • 433 598 047 308 ÷ 2 = 216 799 023 654 + 0;
  • 216 799 023 654 ÷ 2 = 108 399 511 827 + 0;
  • 108 399 511 827 ÷ 2 = 54 199 755 913 + 1;
  • 54 199 755 913 ÷ 2 = 27 099 877 956 + 1;
  • 27 099 877 956 ÷ 2 = 13 549 938 978 + 0;
  • 13 549 938 978 ÷ 2 = 6 774 969 489 + 0;
  • 6 774 969 489 ÷ 2 = 3 387 484 744 + 1;
  • 3 387 484 744 ÷ 2 = 1 693 742 372 + 0;
  • 1 693 742 372 ÷ 2 = 846 871 186 + 0;
  • 846 871 186 ÷ 2 = 423 435 593 + 0;
  • 423 435 593 ÷ 2 = 211 717 796 + 1;
  • 211 717 796 ÷ 2 = 105 858 898 + 0;
  • 105 858 898 ÷ 2 = 52 929 449 + 0;
  • 52 929 449 ÷ 2 = 26 464 724 + 1;
  • 26 464 724 ÷ 2 = 13 232 362 + 0;
  • 13 232 362 ÷ 2 = 6 616 181 + 0;
  • 6 616 181 ÷ 2 = 3 308 090 + 1;
  • 3 308 090 ÷ 2 = 1 654 045 + 0;
  • 1 654 045 ÷ 2 = 827 022 + 1;
  • 827 022 ÷ 2 = 413 511 + 0;
  • 413 511 ÷ 2 = 206 755 + 1;
  • 206 755 ÷ 2 = 103 377 + 1;
  • 103 377 ÷ 2 = 51 688 + 1;
  • 51 688 ÷ 2 = 25 844 + 0;
  • 25 844 ÷ 2 = 12 922 + 0;
  • 12 922 ÷ 2 = 6 461 + 0;
  • 6 461 ÷ 2 = 3 230 + 1;
  • 3 230 ÷ 2 = 1 615 + 0;
  • 1 615 ÷ 2 = 807 + 1;
  • 807 ÷ 2 = 403 + 1;
  • 403 ÷ 2 = 201 + 1;
  • 201 ÷ 2 = 100 + 1;
  • 100 ÷ 2 = 50 + 0;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 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.


111 001 100 111 059(10) = 110 0100 1111 0100 0111 0101 0010 0100 0100 1100 1101 0011(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 47.


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, 47,

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.


Number 111 001 100 111 059(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:

111 001 100 111 059(10) = 0000 0000 0000 0000 0110 0100 1111 0100 0111 0101 0010 0100 0100 1100 1101 0011

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