Convert Decimal 1 000 000 100 009 935 to Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 000 000 100 009 935(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 000 000 100 009 935 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;
  • 1 000 000 100 009 935 ÷ 2 = 500 000 050 004 967 + 1;
  • 500 000 050 004 967 ÷ 2 = 250 000 025 002 483 + 1;
  • 250 000 025 002 483 ÷ 2 = 125 000 012 501 241 + 1;
  • 125 000 012 501 241 ÷ 2 = 62 500 006 250 620 + 1;
  • 62 500 006 250 620 ÷ 2 = 31 250 003 125 310 + 0;
  • 31 250 003 125 310 ÷ 2 = 15 625 001 562 655 + 0;
  • 15 625 001 562 655 ÷ 2 = 7 812 500 781 327 + 1;
  • 7 812 500 781 327 ÷ 2 = 3 906 250 390 663 + 1;
  • 3 906 250 390 663 ÷ 2 = 1 953 125 195 331 + 1;
  • 1 953 125 195 331 ÷ 2 = 976 562 597 665 + 1;
  • 976 562 597 665 ÷ 2 = 488 281 298 832 + 1;
  • 488 281 298 832 ÷ 2 = 244 140 649 416 + 0;
  • 244 140 649 416 ÷ 2 = 122 070 324 708 + 0;
  • 122 070 324 708 ÷ 2 = 61 035 162 354 + 0;
  • 61 035 162 354 ÷ 2 = 30 517 581 177 + 0;
  • 30 517 581 177 ÷ 2 = 15 258 790 588 + 1;
  • 15 258 790 588 ÷ 2 = 7 629 395 294 + 0;
  • 7 629 395 294 ÷ 2 = 3 814 697 647 + 0;
  • 3 814 697 647 ÷ 2 = 1 907 348 823 + 1;
  • 1 907 348 823 ÷ 2 = 953 674 411 + 1;
  • 953 674 411 ÷ 2 = 476 837 205 + 1;
  • 476 837 205 ÷ 2 = 238 418 602 + 1;
  • 238 418 602 ÷ 2 = 119 209 301 + 0;
  • 119 209 301 ÷ 2 = 59 604 650 + 1;
  • 59 604 650 ÷ 2 = 29 802 325 + 0;
  • 29 802 325 ÷ 2 = 14 901 162 + 1;
  • 14 901 162 ÷ 2 = 7 450 581 + 0;
  • 7 450 581 ÷ 2 = 3 725 290 + 1;
  • 3 725 290 ÷ 2 = 1 862 645 + 0;
  • 1 862 645 ÷ 2 = 931 322 + 1;
  • 931 322 ÷ 2 = 465 661 + 0;
  • 465 661 ÷ 2 = 232 830 + 1;
  • 232 830 ÷ 2 = 116 415 + 0;
  • 116 415 ÷ 2 = 58 207 + 1;
  • 58 207 ÷ 2 = 29 103 + 1;
  • 29 103 ÷ 2 = 14 551 + 1;
  • 14 551 ÷ 2 = 7 275 + 1;
  • 7 275 ÷ 2 = 3 637 + 1;
  • 3 637 ÷ 2 = 1 818 + 1;
  • 1 818 ÷ 2 = 909 + 0;
  • 909 ÷ 2 = 454 + 1;
  • 454 ÷ 2 = 227 + 0;
  • 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.

1 000 000 100 009 935(10) = 11 1000 1101 0111 1110 1010 1010 1011 1100 1000 0111 1100 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 1 000 000 100 009 935(10) converted to signed binary in one's complement representation:

1 000 000 100 009 935(10) = 0000 0000 0000 0011 1000 1101 0111 1110 1010 1010 1011 1100 1000 0111 1100 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