Convert Decimal 1 010 101 101 010 165 to Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 010 101 101 010 165(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 010 101 101 010 165 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;
  • 1 010 101 101 010 165 ÷ 2 = 505 050 550 505 082 + 1;
  • 505 050 550 505 082 ÷ 2 = 252 525 275 252 541 + 0;
  • 252 525 275 252 541 ÷ 2 = 126 262 637 626 270 + 1;
  • 126 262 637 626 270 ÷ 2 = 63 131 318 813 135 + 0;
  • 63 131 318 813 135 ÷ 2 = 31 565 659 406 567 + 1;
  • 31 565 659 406 567 ÷ 2 = 15 782 829 703 283 + 1;
  • 15 782 829 703 283 ÷ 2 = 7 891 414 851 641 + 1;
  • 7 891 414 851 641 ÷ 2 = 3 945 707 425 820 + 1;
  • 3 945 707 425 820 ÷ 2 = 1 972 853 712 910 + 0;
  • 1 972 853 712 910 ÷ 2 = 986 426 856 455 + 0;
  • 986 426 856 455 ÷ 2 = 493 213 428 227 + 1;
  • 493 213 428 227 ÷ 2 = 246 606 714 113 + 1;
  • 246 606 714 113 ÷ 2 = 123 303 357 056 + 1;
  • 123 303 357 056 ÷ 2 = 61 651 678 528 + 0;
  • 61 651 678 528 ÷ 2 = 30 825 839 264 + 0;
  • 30 825 839 264 ÷ 2 = 15 412 919 632 + 0;
  • 15 412 919 632 ÷ 2 = 7 706 459 816 + 0;
  • 7 706 459 816 ÷ 2 = 3 853 229 908 + 0;
  • 3 853 229 908 ÷ 2 = 1 926 614 954 + 0;
  • 1 926 614 954 ÷ 2 = 963 307 477 + 0;
  • 963 307 477 ÷ 2 = 481 653 738 + 1;
  • 481 653 738 ÷ 2 = 240 826 869 + 0;
  • 240 826 869 ÷ 2 = 120 413 434 + 1;
  • 120 413 434 ÷ 2 = 60 206 717 + 0;
  • 60 206 717 ÷ 2 = 30 103 358 + 1;
  • 30 103 358 ÷ 2 = 15 051 679 + 0;
  • 15 051 679 ÷ 2 = 7 525 839 + 1;
  • 7 525 839 ÷ 2 = 3 762 919 + 1;
  • 3 762 919 ÷ 2 = 1 881 459 + 1;
  • 1 881 459 ÷ 2 = 940 729 + 1;
  • 940 729 ÷ 2 = 470 364 + 1;
  • 470 364 ÷ 2 = 235 182 + 0;
  • 235 182 ÷ 2 = 117 591 + 0;
  • 117 591 ÷ 2 = 58 795 + 1;
  • 58 795 ÷ 2 = 29 397 + 1;
  • 29 397 ÷ 2 = 14 698 + 1;
  • 14 698 ÷ 2 = 7 349 + 0;
  • 7 349 ÷ 2 = 3 674 + 1;
  • 3 674 ÷ 2 = 1 837 + 0;
  • 1 837 ÷ 2 = 918 + 1;
  • 918 ÷ 2 = 459 + 0;
  • 459 ÷ 2 = 229 + 1;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 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.

1 010 101 101 010 165(10) = 11 1001 0110 1010 1110 0111 1101 0101 0000 0001 1100 1111 0101(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 1 010 101 101 010 165(10) converted to signed binary in one's complement representation:

1 010 101 101 010 165(10) = 0000 0000 0000 0011 1001 0110 1010 1110 0111 1101 0101 0000 0001 1100 1111 0101

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