Convert 1 110 010 001 111 059 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 110 010 001 111 059(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 110 010 001 111 059 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;
  • 1 110 010 001 111 059 ÷ 2 = 555 005 000 555 529 + 1;
  • 555 005 000 555 529 ÷ 2 = 277 502 500 277 764 + 1;
  • 277 502 500 277 764 ÷ 2 = 138 751 250 138 882 + 0;
  • 138 751 250 138 882 ÷ 2 = 69 375 625 069 441 + 0;
  • 69 375 625 069 441 ÷ 2 = 34 687 812 534 720 + 1;
  • 34 687 812 534 720 ÷ 2 = 17 343 906 267 360 + 0;
  • 17 343 906 267 360 ÷ 2 = 8 671 953 133 680 + 0;
  • 8 671 953 133 680 ÷ 2 = 4 335 976 566 840 + 0;
  • 4 335 976 566 840 ÷ 2 = 2 167 988 283 420 + 0;
  • 2 167 988 283 420 ÷ 2 = 1 083 994 141 710 + 0;
  • 1 083 994 141 710 ÷ 2 = 541 997 070 855 + 0;
  • 541 997 070 855 ÷ 2 = 270 998 535 427 + 1;
  • 270 998 535 427 ÷ 2 = 135 499 267 713 + 1;
  • 135 499 267 713 ÷ 2 = 67 749 633 856 + 1;
  • 67 749 633 856 ÷ 2 = 33 874 816 928 + 0;
  • 33 874 816 928 ÷ 2 = 16 937 408 464 + 0;
  • 16 937 408 464 ÷ 2 = 8 468 704 232 + 0;
  • 8 468 704 232 ÷ 2 = 4 234 352 116 + 0;
  • 4 234 352 116 ÷ 2 = 2 117 176 058 + 0;
  • 2 117 176 058 ÷ 2 = 1 058 588 029 + 0;
  • 1 058 588 029 ÷ 2 = 529 294 014 + 1;
  • 529 294 014 ÷ 2 = 264 647 007 + 0;
  • 264 647 007 ÷ 2 = 132 323 503 + 1;
  • 132 323 503 ÷ 2 = 66 161 751 + 1;
  • 66 161 751 ÷ 2 = 33 080 875 + 1;
  • 33 080 875 ÷ 2 = 16 540 437 + 1;
  • 16 540 437 ÷ 2 = 8 270 218 + 1;
  • 8 270 218 ÷ 2 = 4 135 109 + 0;
  • 4 135 109 ÷ 2 = 2 067 554 + 1;
  • 2 067 554 ÷ 2 = 1 033 777 + 0;
  • 1 033 777 ÷ 2 = 516 888 + 1;
  • 516 888 ÷ 2 = 258 444 + 0;
  • 258 444 ÷ 2 = 129 222 + 0;
  • 129 222 ÷ 2 = 64 611 + 0;
  • 64 611 ÷ 2 = 32 305 + 1;
  • 32 305 ÷ 2 = 16 152 + 1;
  • 16 152 ÷ 2 = 8 076 + 0;
  • 8 076 ÷ 2 = 4 038 + 0;
  • 4 038 ÷ 2 = 2 019 + 0;
  • 2 019 ÷ 2 = 1 009 + 1;
  • 1 009 ÷ 2 = 504 + 1;
  • 504 ÷ 2 = 252 + 0;
  • 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.

1 110 010 001 111 059(10) = 11 1111 0001 1000 1100 0101 0111 1101 0000 0011 1000 0001 0011(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 1 110 010 001 111 059(10) converted to signed binary in one's complement representation:

1 110 010 001 111 059(10) = 0000 0000 0000 0011 1111 0001 1000 1100 0101 0111 1101 0000 0011 1000 0001 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