Two's Complement: Integer ↗ Binary: 1 011 010 110 011 173 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number 1 011 010 110 011 173(10) converted and written as a signed binary in two's complement representation (base 2) = ?

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;
  • 1 011 010 110 011 173 ÷ 2 = 505 505 055 005 586 + 1;
  • 505 505 055 005 586 ÷ 2 = 252 752 527 502 793 + 0;
  • 252 752 527 502 793 ÷ 2 = 126 376 263 751 396 + 1;
  • 126 376 263 751 396 ÷ 2 = 63 188 131 875 698 + 0;
  • 63 188 131 875 698 ÷ 2 = 31 594 065 937 849 + 0;
  • 31 594 065 937 849 ÷ 2 = 15 797 032 968 924 + 1;
  • 15 797 032 968 924 ÷ 2 = 7 898 516 484 462 + 0;
  • 7 898 516 484 462 ÷ 2 = 3 949 258 242 231 + 0;
  • 3 949 258 242 231 ÷ 2 = 1 974 629 121 115 + 1;
  • 1 974 629 121 115 ÷ 2 = 987 314 560 557 + 1;
  • 987 314 560 557 ÷ 2 = 493 657 280 278 + 1;
  • 493 657 280 278 ÷ 2 = 246 828 640 139 + 0;
  • 246 828 640 139 ÷ 2 = 123 414 320 069 + 1;
  • 123 414 320 069 ÷ 2 = 61 707 160 034 + 1;
  • 61 707 160 034 ÷ 2 = 30 853 580 017 + 0;
  • 30 853 580 017 ÷ 2 = 15 426 790 008 + 1;
  • 15 426 790 008 ÷ 2 = 7 713 395 004 + 0;
  • 7 713 395 004 ÷ 2 = 3 856 697 502 + 0;
  • 3 856 697 502 ÷ 2 = 1 928 348 751 + 0;
  • 1 928 348 751 ÷ 2 = 964 174 375 + 1;
  • 964 174 375 ÷ 2 = 482 087 187 + 1;
  • 482 087 187 ÷ 2 = 241 043 593 + 1;
  • 241 043 593 ÷ 2 = 120 521 796 + 1;
  • 120 521 796 ÷ 2 = 60 260 898 + 0;
  • 60 260 898 ÷ 2 = 30 130 449 + 0;
  • 30 130 449 ÷ 2 = 15 065 224 + 1;
  • 15 065 224 ÷ 2 = 7 532 612 + 0;
  • 7 532 612 ÷ 2 = 3 766 306 + 0;
  • 3 766 306 ÷ 2 = 1 883 153 + 0;
  • 1 883 153 ÷ 2 = 941 576 + 1;
  • 941 576 ÷ 2 = 470 788 + 0;
  • 470 788 ÷ 2 = 235 394 + 0;
  • 235 394 ÷ 2 = 117 697 + 0;
  • 117 697 ÷ 2 = 58 848 + 1;
  • 58 848 ÷ 2 = 29 424 + 0;
  • 29 424 ÷ 2 = 14 712 + 0;
  • 14 712 ÷ 2 = 7 356 + 0;
  • 7 356 ÷ 2 = 3 678 + 0;
  • 3 678 ÷ 2 = 1 839 + 0;
  • 1 839 ÷ 2 = 919 + 1;
  • 919 ÷ 2 = 459 + 1;
  • 459 ÷ 2 = 229 + 1;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 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.


1 011 010 110 011 173(10) = 11 1001 0111 1000 0010 0010 0010 0111 1000 1011 0111 0010 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.


Number 1 011 010 110 011 173(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:

1 011 010 110 011 173(10) = 0000 0000 0000 0011 1001 0111 1000 0010 0010 0010 0111 1000 1011 0111 0010 0101

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

More operations on integers written as signed binary numbers in two's complement representation:

» Signed integer number 1 011 010 110 011 172 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

» Signed integer number 1 011 010 110 011 174 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

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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