Convert 1 001 101 009 999 167 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 001 101 009 999 167(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 001 101 009 999 167 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;
  • 1 001 101 009 999 167 ÷ 2 = 500 550 504 999 583 + 1;
  • 500 550 504 999 583 ÷ 2 = 250 275 252 499 791 + 1;
  • 250 275 252 499 791 ÷ 2 = 125 137 626 249 895 + 1;
  • 125 137 626 249 895 ÷ 2 = 62 568 813 124 947 + 1;
  • 62 568 813 124 947 ÷ 2 = 31 284 406 562 473 + 1;
  • 31 284 406 562 473 ÷ 2 = 15 642 203 281 236 + 1;
  • 15 642 203 281 236 ÷ 2 = 7 821 101 640 618 + 0;
  • 7 821 101 640 618 ÷ 2 = 3 910 550 820 309 + 0;
  • 3 910 550 820 309 ÷ 2 = 1 955 275 410 154 + 1;
  • 1 955 275 410 154 ÷ 2 = 977 637 705 077 + 0;
  • 977 637 705 077 ÷ 2 = 488 818 852 538 + 1;
  • 488 818 852 538 ÷ 2 = 244 409 426 269 + 0;
  • 244 409 426 269 ÷ 2 = 122 204 713 134 + 1;
  • 122 204 713 134 ÷ 2 = 61 102 356 567 + 0;
  • 61 102 356 567 ÷ 2 = 30 551 178 283 + 1;
  • 30 551 178 283 ÷ 2 = 15 275 589 141 + 1;
  • 15 275 589 141 ÷ 2 = 7 637 794 570 + 1;
  • 7 637 794 570 ÷ 2 = 3 818 897 285 + 0;
  • 3 818 897 285 ÷ 2 = 1 909 448 642 + 1;
  • 1 909 448 642 ÷ 2 = 954 724 321 + 0;
  • 954 724 321 ÷ 2 = 477 362 160 + 1;
  • 477 362 160 ÷ 2 = 238 681 080 + 0;
  • 238 681 080 ÷ 2 = 119 340 540 + 0;
  • 119 340 540 ÷ 2 = 59 670 270 + 0;
  • 59 670 270 ÷ 2 = 29 835 135 + 0;
  • 29 835 135 ÷ 2 = 14 917 567 + 1;
  • 14 917 567 ÷ 2 = 7 458 783 + 1;
  • 7 458 783 ÷ 2 = 3 729 391 + 1;
  • 3 729 391 ÷ 2 = 1 864 695 + 1;
  • 1 864 695 ÷ 2 = 932 347 + 1;
  • 932 347 ÷ 2 = 466 173 + 1;
  • 466 173 ÷ 2 = 233 086 + 1;
  • 233 086 ÷ 2 = 116 543 + 0;
  • 116 543 ÷ 2 = 58 271 + 1;
  • 58 271 ÷ 2 = 29 135 + 1;
  • 29 135 ÷ 2 = 14 567 + 1;
  • 14 567 ÷ 2 = 7 283 + 1;
  • 7 283 ÷ 2 = 3 641 + 1;
  • 3 641 ÷ 2 = 1 820 + 1;
  • 1 820 ÷ 2 = 910 + 0;
  • 910 ÷ 2 = 455 + 0;
  • 455 ÷ 2 = 227 + 1;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 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.

1 001 101 009 999 167(10) = 11 1000 1110 0111 1110 1111 1110 0001 0101 1101 0101 0011 1111(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 1 001 101 009 999 167(10) converted to signed binary in one's complement representation:

1 001 101 009 999 167(10) = 0000 0000 0000 0011 1000 1110 0111 1110 1111 1110 0001 0101 1101 0101 0011 1111

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