Convert 11 000 110 111 000 170 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 11 000 110 111 000 170(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
11 000 110 111 000 170 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;
  • 11 000 110 111 000 170 ÷ 2 = 5 500 055 055 500 085 + 0;
  • 5 500 055 055 500 085 ÷ 2 = 2 750 027 527 750 042 + 1;
  • 2 750 027 527 750 042 ÷ 2 = 1 375 013 763 875 021 + 0;
  • 1 375 013 763 875 021 ÷ 2 = 687 506 881 937 510 + 1;
  • 687 506 881 937 510 ÷ 2 = 343 753 440 968 755 + 0;
  • 343 753 440 968 755 ÷ 2 = 171 876 720 484 377 + 1;
  • 171 876 720 484 377 ÷ 2 = 85 938 360 242 188 + 1;
  • 85 938 360 242 188 ÷ 2 = 42 969 180 121 094 + 0;
  • 42 969 180 121 094 ÷ 2 = 21 484 590 060 547 + 0;
  • 21 484 590 060 547 ÷ 2 = 10 742 295 030 273 + 1;
  • 10 742 295 030 273 ÷ 2 = 5 371 147 515 136 + 1;
  • 5 371 147 515 136 ÷ 2 = 2 685 573 757 568 + 0;
  • 2 685 573 757 568 ÷ 2 = 1 342 786 878 784 + 0;
  • 1 342 786 878 784 ÷ 2 = 671 393 439 392 + 0;
  • 671 393 439 392 ÷ 2 = 335 696 719 696 + 0;
  • 335 696 719 696 ÷ 2 = 167 848 359 848 + 0;
  • 167 848 359 848 ÷ 2 = 83 924 179 924 + 0;
  • 83 924 179 924 ÷ 2 = 41 962 089 962 + 0;
  • 41 962 089 962 ÷ 2 = 20 981 044 981 + 0;
  • 20 981 044 981 ÷ 2 = 10 490 522 490 + 1;
  • 10 490 522 490 ÷ 2 = 5 245 261 245 + 0;
  • 5 245 261 245 ÷ 2 = 2 622 630 622 + 1;
  • 2 622 630 622 ÷ 2 = 1 311 315 311 + 0;
  • 1 311 315 311 ÷ 2 = 655 657 655 + 1;
  • 655 657 655 ÷ 2 = 327 828 827 + 1;
  • 327 828 827 ÷ 2 = 163 914 413 + 1;
  • 163 914 413 ÷ 2 = 81 957 206 + 1;
  • 81 957 206 ÷ 2 = 40 978 603 + 0;
  • 40 978 603 ÷ 2 = 20 489 301 + 1;
  • 20 489 301 ÷ 2 = 10 244 650 + 1;
  • 10 244 650 ÷ 2 = 5 122 325 + 0;
  • 5 122 325 ÷ 2 = 2 561 162 + 1;
  • 2 561 162 ÷ 2 = 1 280 581 + 0;
  • 1 280 581 ÷ 2 = 640 290 + 1;
  • 640 290 ÷ 2 = 320 145 + 0;
  • 320 145 ÷ 2 = 160 072 + 1;
  • 160 072 ÷ 2 = 80 036 + 0;
  • 80 036 ÷ 2 = 40 018 + 0;
  • 40 018 ÷ 2 = 20 009 + 0;
  • 20 009 ÷ 2 = 10 004 + 1;
  • 10 004 ÷ 2 = 5 002 + 0;
  • 5 002 ÷ 2 = 2 501 + 0;
  • 2 501 ÷ 2 = 1 250 + 1;
  • 1 250 ÷ 2 = 625 + 0;
  • 625 ÷ 2 = 312 + 1;
  • 312 ÷ 2 = 156 + 0;
  • 156 ÷ 2 = 78 + 0;
  • 78 ÷ 2 = 39 + 0;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 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.

11 000 110 111 000 170(10) = 10 0111 0001 0100 1000 1010 1011 0111 1010 1000 0000 0110 0110 1010(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 54.

  • 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, 54,

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 11 000 110 111 000 170(10) converted to signed binary in two's complement representation:

11 000 110 111 000 170(10) = 0000 0000 0010 0111 0001 0100 1000 1010 1011 0111 1010 1000 0000 0110 0110 1010

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