Two's Complement: Integer ↗ Binary: 111 101 111 023 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 111 101 111 023(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;
  • 111 101 111 023 ÷ 2 = 55 550 555 511 + 1;
  • 55 550 555 511 ÷ 2 = 27 775 277 755 + 1;
  • 27 775 277 755 ÷ 2 = 13 887 638 877 + 1;
  • 13 887 638 877 ÷ 2 = 6 943 819 438 + 1;
  • 6 943 819 438 ÷ 2 = 3 471 909 719 + 0;
  • 3 471 909 719 ÷ 2 = 1 735 954 859 + 1;
  • 1 735 954 859 ÷ 2 = 867 977 429 + 1;
  • 867 977 429 ÷ 2 = 433 988 714 + 1;
  • 433 988 714 ÷ 2 = 216 994 357 + 0;
  • 216 994 357 ÷ 2 = 108 497 178 + 1;
  • 108 497 178 ÷ 2 = 54 248 589 + 0;
  • 54 248 589 ÷ 2 = 27 124 294 + 1;
  • 27 124 294 ÷ 2 = 13 562 147 + 0;
  • 13 562 147 ÷ 2 = 6 781 073 + 1;
  • 6 781 073 ÷ 2 = 3 390 536 + 1;
  • 3 390 536 ÷ 2 = 1 695 268 + 0;
  • 1 695 268 ÷ 2 = 847 634 + 0;
  • 847 634 ÷ 2 = 423 817 + 0;
  • 423 817 ÷ 2 = 211 908 + 1;
  • 211 908 ÷ 2 = 105 954 + 0;
  • 105 954 ÷ 2 = 52 977 + 0;
  • 52 977 ÷ 2 = 26 488 + 1;
  • 26 488 ÷ 2 = 13 244 + 0;
  • 13 244 ÷ 2 = 6 622 + 0;
  • 6 622 ÷ 2 = 3 311 + 0;
  • 3 311 ÷ 2 = 1 655 + 1;
  • 1 655 ÷ 2 = 827 + 1;
  • 827 ÷ 2 = 413 + 1;
  • 413 ÷ 2 = 206 + 1;
  • 206 ÷ 2 = 103 + 0;
  • 103 ÷ 2 = 51 + 1;
  • 51 ÷ 2 = 25 + 1;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.


111 101 111 023(10) = 1 1001 1101 1110 0010 0100 0110 1010 1110 1111(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 37.

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, 37,

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 111 101 111 023(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:

111 101 111 023(10) = 0000 0000 0000 0000 0000 0000 0001 1001 1101 1110 0010 0100 0110 1010 1110 1111

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 111 101 111 022 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 111 101 111 024 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