Two's Complement: Integer ↗ Binary: 1 100 110 010 110 022 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 1 100 110 010 110 022(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;
  • 1 100 110 010 110 022 ÷ 2 = 550 055 005 055 011 + 0;
  • 550 055 005 055 011 ÷ 2 = 275 027 502 527 505 + 1;
  • 275 027 502 527 505 ÷ 2 = 137 513 751 263 752 + 1;
  • 137 513 751 263 752 ÷ 2 = 68 756 875 631 876 + 0;
  • 68 756 875 631 876 ÷ 2 = 34 378 437 815 938 + 0;
  • 34 378 437 815 938 ÷ 2 = 17 189 218 907 969 + 0;
  • 17 189 218 907 969 ÷ 2 = 8 594 609 453 984 + 1;
  • 8 594 609 453 984 ÷ 2 = 4 297 304 726 992 + 0;
  • 4 297 304 726 992 ÷ 2 = 2 148 652 363 496 + 0;
  • 2 148 652 363 496 ÷ 2 = 1 074 326 181 748 + 0;
  • 1 074 326 181 748 ÷ 2 = 537 163 090 874 + 0;
  • 537 163 090 874 ÷ 2 = 268 581 545 437 + 0;
  • 268 581 545 437 ÷ 2 = 134 290 772 718 + 1;
  • 134 290 772 718 ÷ 2 = 67 145 386 359 + 0;
  • 67 145 386 359 ÷ 2 = 33 572 693 179 + 1;
  • 33 572 693 179 ÷ 2 = 16 786 346 589 + 1;
  • 16 786 346 589 ÷ 2 = 8 393 173 294 + 1;
  • 8 393 173 294 ÷ 2 = 4 196 586 647 + 0;
  • 4 196 586 647 ÷ 2 = 2 098 293 323 + 1;
  • 2 098 293 323 ÷ 2 = 1 049 146 661 + 1;
  • 1 049 146 661 ÷ 2 = 524 573 330 + 1;
  • 524 573 330 ÷ 2 = 262 286 665 + 0;
  • 262 286 665 ÷ 2 = 131 143 332 + 1;
  • 131 143 332 ÷ 2 = 65 571 666 + 0;
  • 65 571 666 ÷ 2 = 32 785 833 + 0;
  • 32 785 833 ÷ 2 = 16 392 916 + 1;
  • 16 392 916 ÷ 2 = 8 196 458 + 0;
  • 8 196 458 ÷ 2 = 4 098 229 + 0;
  • 4 098 229 ÷ 2 = 2 049 114 + 1;
  • 2 049 114 ÷ 2 = 1 024 557 + 0;
  • 1 024 557 ÷ 2 = 512 278 + 1;
  • 512 278 ÷ 2 = 256 139 + 0;
  • 256 139 ÷ 2 = 128 069 + 1;
  • 128 069 ÷ 2 = 64 034 + 1;
  • 64 034 ÷ 2 = 32 017 + 0;
  • 32 017 ÷ 2 = 16 008 + 1;
  • 16 008 ÷ 2 = 8 004 + 0;
  • 8 004 ÷ 2 = 4 002 + 0;
  • 4 002 ÷ 2 = 2 001 + 0;
  • 2 001 ÷ 2 = 1 000 + 1;
  • 1 000 ÷ 2 = 500 + 0;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 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.


1 100 110 010 110 022(10) = 11 1110 1000 1000 1011 0101 0010 0101 1101 1101 0000 0100 0110(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 1 100 110 010 110 022(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:

1 100 110 010 110 022(10) = 0000 0000 0000 0011 1110 1000 1000 1011 0101 0010 0101 1101 1101 0000 0100 0110

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 1 100 110 010 110 021 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 1 100 110 010 110 023 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