Two's Complement: Integer ↗ Binary: 10 111 101 100 058 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 10 111 101 100 058(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;
  • 10 111 101 100 058 ÷ 2 = 5 055 550 550 029 + 0;
  • 5 055 550 550 029 ÷ 2 = 2 527 775 275 014 + 1;
  • 2 527 775 275 014 ÷ 2 = 1 263 887 637 507 + 0;
  • 1 263 887 637 507 ÷ 2 = 631 943 818 753 + 1;
  • 631 943 818 753 ÷ 2 = 315 971 909 376 + 1;
  • 315 971 909 376 ÷ 2 = 157 985 954 688 + 0;
  • 157 985 954 688 ÷ 2 = 78 992 977 344 + 0;
  • 78 992 977 344 ÷ 2 = 39 496 488 672 + 0;
  • 39 496 488 672 ÷ 2 = 19 748 244 336 + 0;
  • 19 748 244 336 ÷ 2 = 9 874 122 168 + 0;
  • 9 874 122 168 ÷ 2 = 4 937 061 084 + 0;
  • 4 937 061 084 ÷ 2 = 2 468 530 542 + 0;
  • 2 468 530 542 ÷ 2 = 1 234 265 271 + 0;
  • 1 234 265 271 ÷ 2 = 617 132 635 + 1;
  • 617 132 635 ÷ 2 = 308 566 317 + 1;
  • 308 566 317 ÷ 2 = 154 283 158 + 1;
  • 154 283 158 ÷ 2 = 77 141 579 + 0;
  • 77 141 579 ÷ 2 = 38 570 789 + 1;
  • 38 570 789 ÷ 2 = 19 285 394 + 1;
  • 19 285 394 ÷ 2 = 9 642 697 + 0;
  • 9 642 697 ÷ 2 = 4 821 348 + 1;
  • 4 821 348 ÷ 2 = 2 410 674 + 0;
  • 2 410 674 ÷ 2 = 1 205 337 + 0;
  • 1 205 337 ÷ 2 = 602 668 + 1;
  • 602 668 ÷ 2 = 301 334 + 0;
  • 301 334 ÷ 2 = 150 667 + 0;
  • 150 667 ÷ 2 = 75 333 + 1;
  • 75 333 ÷ 2 = 37 666 + 1;
  • 37 666 ÷ 2 = 18 833 + 0;
  • 18 833 ÷ 2 = 9 416 + 1;
  • 9 416 ÷ 2 = 4 708 + 0;
  • 4 708 ÷ 2 = 2 354 + 0;
  • 2 354 ÷ 2 = 1 177 + 0;
  • 1 177 ÷ 2 = 588 + 1;
  • 588 ÷ 2 = 294 + 0;
  • 294 ÷ 2 = 147 + 0;
  • 147 ÷ 2 = 73 + 1;
  • 73 ÷ 2 = 36 + 1;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.


10 111 101 100 058(10) = 1001 0011 0010 0010 1100 1001 0110 1110 0000 0001 1010(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 44.

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, 44,

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 10 111 101 100 058(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:

10 111 101 100 058(10) = 0000 0000 0000 0000 0000 1001 0011 0010 0010 1100 1001 0110 1110 0000 0001 1010

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 10 111 101 100 057 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 10 111 101 100 059 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