Convert 101 100 101 111 010 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 101 100 101 111 010(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
101 100 101 111 010 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;
  • 101 100 101 111 010 ÷ 2 = 50 550 050 555 505 + 0;
  • 50 550 050 555 505 ÷ 2 = 25 275 025 277 752 + 1;
  • 25 275 025 277 752 ÷ 2 = 12 637 512 638 876 + 0;
  • 12 637 512 638 876 ÷ 2 = 6 318 756 319 438 + 0;
  • 6 318 756 319 438 ÷ 2 = 3 159 378 159 719 + 0;
  • 3 159 378 159 719 ÷ 2 = 1 579 689 079 859 + 1;
  • 1 579 689 079 859 ÷ 2 = 789 844 539 929 + 1;
  • 789 844 539 929 ÷ 2 = 394 922 269 964 + 1;
  • 394 922 269 964 ÷ 2 = 197 461 134 982 + 0;
  • 197 461 134 982 ÷ 2 = 98 730 567 491 + 0;
  • 98 730 567 491 ÷ 2 = 49 365 283 745 + 1;
  • 49 365 283 745 ÷ 2 = 24 682 641 872 + 1;
  • 24 682 641 872 ÷ 2 = 12 341 320 936 + 0;
  • 12 341 320 936 ÷ 2 = 6 170 660 468 + 0;
  • 6 170 660 468 ÷ 2 = 3 085 330 234 + 0;
  • 3 085 330 234 ÷ 2 = 1 542 665 117 + 0;
  • 1 542 665 117 ÷ 2 = 771 332 558 + 1;
  • 771 332 558 ÷ 2 = 385 666 279 + 0;
  • 385 666 279 ÷ 2 = 192 833 139 + 1;
  • 192 833 139 ÷ 2 = 96 416 569 + 1;
  • 96 416 569 ÷ 2 = 48 208 284 + 1;
  • 48 208 284 ÷ 2 = 24 104 142 + 0;
  • 24 104 142 ÷ 2 = 12 052 071 + 0;
  • 12 052 071 ÷ 2 = 6 026 035 + 1;
  • 6 026 035 ÷ 2 = 3 013 017 + 1;
  • 3 013 017 ÷ 2 = 1 506 508 + 1;
  • 1 506 508 ÷ 2 = 753 254 + 0;
  • 753 254 ÷ 2 = 376 627 + 0;
  • 376 627 ÷ 2 = 188 313 + 1;
  • 188 313 ÷ 2 = 94 156 + 1;
  • 94 156 ÷ 2 = 47 078 + 0;
  • 47 078 ÷ 2 = 23 539 + 0;
  • 23 539 ÷ 2 = 11 769 + 1;
  • 11 769 ÷ 2 = 5 884 + 1;
  • 5 884 ÷ 2 = 2 942 + 0;
  • 2 942 ÷ 2 = 1 471 + 0;
  • 1 471 ÷ 2 = 735 + 1;
  • 735 ÷ 2 = 367 + 1;
  • 367 ÷ 2 = 183 + 1;
  • 183 ÷ 2 = 91 + 1;
  • 91 ÷ 2 = 45 + 1;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.

101 100 101 111 010(10) = 101 1011 1111 0011 0011 0011 1001 1101 0000 1100 1110 0010(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.


Decimal Number 101 100 101 111 010(10) converted to signed binary in one's complement representation:

101 100 101 111 010(10) = 0000 0000 0000 0000 0101 1011 1111 0011 0011 0011 1001 1101 0000 1100 1110 0010

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