Convert 1 111 000 110 010 226 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 000 110 010 226(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 000 110 010 226 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;
  • 1 111 000 110 010 226 ÷ 2 = 555 500 055 005 113 + 0;
  • 555 500 055 005 113 ÷ 2 = 277 750 027 502 556 + 1;
  • 277 750 027 502 556 ÷ 2 = 138 875 013 751 278 + 0;
  • 138 875 013 751 278 ÷ 2 = 69 437 506 875 639 + 0;
  • 69 437 506 875 639 ÷ 2 = 34 718 753 437 819 + 1;
  • 34 718 753 437 819 ÷ 2 = 17 359 376 718 909 + 1;
  • 17 359 376 718 909 ÷ 2 = 8 679 688 359 454 + 1;
  • 8 679 688 359 454 ÷ 2 = 4 339 844 179 727 + 0;
  • 4 339 844 179 727 ÷ 2 = 2 169 922 089 863 + 1;
  • 2 169 922 089 863 ÷ 2 = 1 084 961 044 931 + 1;
  • 1 084 961 044 931 ÷ 2 = 542 480 522 465 + 1;
  • 542 480 522 465 ÷ 2 = 271 240 261 232 + 1;
  • 271 240 261 232 ÷ 2 = 135 620 130 616 + 0;
  • 135 620 130 616 ÷ 2 = 67 810 065 308 + 0;
  • 67 810 065 308 ÷ 2 = 33 905 032 654 + 0;
  • 33 905 032 654 ÷ 2 = 16 952 516 327 + 0;
  • 16 952 516 327 ÷ 2 = 8 476 258 163 + 1;
  • 8 476 258 163 ÷ 2 = 4 238 129 081 + 1;
  • 4 238 129 081 ÷ 2 = 2 119 064 540 + 1;
  • 2 119 064 540 ÷ 2 = 1 059 532 270 + 0;
  • 1 059 532 270 ÷ 2 = 529 766 135 + 0;
  • 529 766 135 ÷ 2 = 264 883 067 + 1;
  • 264 883 067 ÷ 2 = 132 441 533 + 1;
  • 132 441 533 ÷ 2 = 66 220 766 + 1;
  • 66 220 766 ÷ 2 = 33 110 383 + 0;
  • 33 110 383 ÷ 2 = 16 555 191 + 1;
  • 16 555 191 ÷ 2 = 8 277 595 + 1;
  • 8 277 595 ÷ 2 = 4 138 797 + 1;
  • 4 138 797 ÷ 2 = 2 069 398 + 1;
  • 2 069 398 ÷ 2 = 1 034 699 + 0;
  • 1 034 699 ÷ 2 = 517 349 + 1;
  • 517 349 ÷ 2 = 258 674 + 1;
  • 258 674 ÷ 2 = 129 337 + 0;
  • 129 337 ÷ 2 = 64 668 + 1;
  • 64 668 ÷ 2 = 32 334 + 0;
  • 32 334 ÷ 2 = 16 167 + 0;
  • 16 167 ÷ 2 = 8 083 + 1;
  • 8 083 ÷ 2 = 4 041 + 1;
  • 4 041 ÷ 2 = 2 020 + 1;
  • 2 020 ÷ 2 = 1 010 + 0;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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.

1 111 000 110 010 226(10) = 11 1111 0010 0111 0010 1101 1110 1110 0111 0000 1111 0111 0010(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 1 111 000 110 010 226(10) converted to signed binary in two's complement representation:

1 111 000 110 010 226(10) = 0000 0000 0000 0011 1111 0010 0111 0010 1101 1110 1110 0111 0000 1111 0111 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 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