Convert 100 111 010 010 986 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 100 111 010 010 986(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
100 111 010 010 986 to a signed binary in two's (2'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.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 100 111 010 010 986 ÷ 2 = 50 055 505 005 493 + 0;
  • 50 055 505 005 493 ÷ 2 = 25 027 752 502 746 + 1;
  • 25 027 752 502 746 ÷ 2 = 12 513 876 251 373 + 0;
  • 12 513 876 251 373 ÷ 2 = 6 256 938 125 686 + 1;
  • 6 256 938 125 686 ÷ 2 = 3 128 469 062 843 + 0;
  • 3 128 469 062 843 ÷ 2 = 1 564 234 531 421 + 1;
  • 1 564 234 531 421 ÷ 2 = 782 117 265 710 + 1;
  • 782 117 265 710 ÷ 2 = 391 058 632 855 + 0;
  • 391 058 632 855 ÷ 2 = 195 529 316 427 + 1;
  • 195 529 316 427 ÷ 2 = 97 764 658 213 + 1;
  • 97 764 658 213 ÷ 2 = 48 882 329 106 + 1;
  • 48 882 329 106 ÷ 2 = 24 441 164 553 + 0;
  • 24 441 164 553 ÷ 2 = 12 220 582 276 + 1;
  • 12 220 582 276 ÷ 2 = 6 110 291 138 + 0;
  • 6 110 291 138 ÷ 2 = 3 055 145 569 + 0;
  • 3 055 145 569 ÷ 2 = 1 527 572 784 + 1;
  • 1 527 572 784 ÷ 2 = 763 786 392 + 0;
  • 763 786 392 ÷ 2 = 381 893 196 + 0;
  • 381 893 196 ÷ 2 = 190 946 598 + 0;
  • 190 946 598 ÷ 2 = 95 473 299 + 0;
  • 95 473 299 ÷ 2 = 47 736 649 + 1;
  • 47 736 649 ÷ 2 = 23 868 324 + 1;
  • 23 868 324 ÷ 2 = 11 934 162 + 0;
  • 11 934 162 ÷ 2 = 5 967 081 + 0;
  • 5 967 081 ÷ 2 = 2 983 540 + 1;
  • 2 983 540 ÷ 2 = 1 491 770 + 0;
  • 1 491 770 ÷ 2 = 745 885 + 0;
  • 745 885 ÷ 2 = 372 942 + 1;
  • 372 942 ÷ 2 = 186 471 + 0;
  • 186 471 ÷ 2 = 93 235 + 1;
  • 93 235 ÷ 2 = 46 617 + 1;
  • 46 617 ÷ 2 = 23 308 + 1;
  • 23 308 ÷ 2 = 11 654 + 0;
  • 11 654 ÷ 2 = 5 827 + 0;
  • 5 827 ÷ 2 = 2 913 + 1;
  • 2 913 ÷ 2 = 1 456 + 1;
  • 1 456 ÷ 2 = 728 + 0;
  • 728 ÷ 2 = 364 + 0;
  • 364 ÷ 2 = 182 + 0;
  • 182 ÷ 2 = 91 + 0;
  • 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.

100 111 010 010 986(10) = 101 1011 0000 1100 1110 1001 0011 0000 1001 0111 0110 1010(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 100 111 010 010 986(10) converted to signed binary in two's complement representation:

100 111 010 010 986(10) = 0000 0000 0000 0000 0101 1011 0000 1100 1110 1001 0011 0000 1001 0111 0110 1010

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed integers from decimal system to signed binary in two's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in two'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, keeping track of each remainder, until we get a quotient that is 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, add extra bits on 0 in front (to the left) of the base 2 number above, up to the required length, so that the first bit (the leftmost) will 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 1s and all 1 bits with 0s (reversing the digits).
  • 6. To get the negative integer number, in signed binary two's complement representation, add 1 to the number above.

Example: convert the negative number -60 from the decimal system (base ten) to signed binary in two's complement:

  • 1. Start with the positive version of the number: |-60| = 60
  • 2. Divide repeatedly 60 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 60 ÷ 2 = 30 + 0
    • 30 ÷ 2 = 15 + 0
    • 15 ÷ 2 = 7 + 1
    • 7 ÷ 2 = 3 + 1
    • 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:
    60(10) = 11 1100(2)
  • 4. Bit length of base 2 representation number is 6, so the positive binary computer representation of a signed binary will take in this particular case 8 bits (the least power of 2 larger than 6) - add extra 0 digits in front of the base 2 number, up to the required length:
    60(10) = 0011 1100(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all the 0 bits with 1s and all 1 bits with 0s (reversing the digits):
    !(0011 1100) = 1100 0011
  • 6. To get the negative integer number, signed binary in two's complement representation, add 1 to the number above:
    -60(10) = 1100 0011 + 1 = 1100 0100
  • Number -60(10), signed integer, converted from decimal system (base 10) to signed binary two's complement representation = 1100 0100