Convert 1 111 000 011 100 501 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 000 011 100 501(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 000 011 100 501 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;
  • 1 111 000 011 100 501 ÷ 2 = 555 500 005 550 250 + 1;
  • 555 500 005 550 250 ÷ 2 = 277 750 002 775 125 + 0;
  • 277 750 002 775 125 ÷ 2 = 138 875 001 387 562 + 1;
  • 138 875 001 387 562 ÷ 2 = 69 437 500 693 781 + 0;
  • 69 437 500 693 781 ÷ 2 = 34 718 750 346 890 + 1;
  • 34 718 750 346 890 ÷ 2 = 17 359 375 173 445 + 0;
  • 17 359 375 173 445 ÷ 2 = 8 679 687 586 722 + 1;
  • 8 679 687 586 722 ÷ 2 = 4 339 843 793 361 + 0;
  • 4 339 843 793 361 ÷ 2 = 2 169 921 896 680 + 1;
  • 2 169 921 896 680 ÷ 2 = 1 084 960 948 340 + 0;
  • 1 084 960 948 340 ÷ 2 = 542 480 474 170 + 0;
  • 542 480 474 170 ÷ 2 = 271 240 237 085 + 0;
  • 271 240 237 085 ÷ 2 = 135 620 118 542 + 1;
  • 135 620 118 542 ÷ 2 = 67 810 059 271 + 0;
  • 67 810 059 271 ÷ 2 = 33 905 029 635 + 1;
  • 33 905 029 635 ÷ 2 = 16 952 514 817 + 1;
  • 16 952 514 817 ÷ 2 = 8 476 257 408 + 1;
  • 8 476 257 408 ÷ 2 = 4 238 128 704 + 0;
  • 4 238 128 704 ÷ 2 = 2 119 064 352 + 0;
  • 2 119 064 352 ÷ 2 = 1 059 532 176 + 0;
  • 1 059 532 176 ÷ 2 = 529 766 088 + 0;
  • 529 766 088 ÷ 2 = 264 883 044 + 0;
  • 264 883 044 ÷ 2 = 132 441 522 + 0;
  • 132 441 522 ÷ 2 = 66 220 761 + 0;
  • 66 220 761 ÷ 2 = 33 110 380 + 1;
  • 33 110 380 ÷ 2 = 16 555 190 + 0;
  • 16 555 190 ÷ 2 = 8 277 595 + 0;
  • 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.

1 111 000 011 100 501(10) = 11 1111 0010 0111 0010 1101 1001 0000 0001 1101 0001 0101 0101(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 1 111 000 011 100 501(10) converted to signed binary in two's complement representation:

1 111 000 011 100 501(10) = 0000 0000 0000 0011 1111 0010 0111 0010 1101 1001 0000 0001 1101 0001 0101 0101

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