Convert 1 101 111 001 000 072 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 101 111 001 000 072(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 101 111 001 000 072 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;
  • 1 101 111 001 000 072 ÷ 2 = 550 555 500 500 036 + 0;
  • 550 555 500 500 036 ÷ 2 = 275 277 750 250 018 + 0;
  • 275 277 750 250 018 ÷ 2 = 137 638 875 125 009 + 0;
  • 137 638 875 125 009 ÷ 2 = 68 819 437 562 504 + 1;
  • 68 819 437 562 504 ÷ 2 = 34 409 718 781 252 + 0;
  • 34 409 718 781 252 ÷ 2 = 17 204 859 390 626 + 0;
  • 17 204 859 390 626 ÷ 2 = 8 602 429 695 313 + 0;
  • 8 602 429 695 313 ÷ 2 = 4 301 214 847 656 + 1;
  • 4 301 214 847 656 ÷ 2 = 2 150 607 423 828 + 0;
  • 2 150 607 423 828 ÷ 2 = 1 075 303 711 914 + 0;
  • 1 075 303 711 914 ÷ 2 = 537 651 855 957 + 0;
  • 537 651 855 957 ÷ 2 = 268 825 927 978 + 1;
  • 268 825 927 978 ÷ 2 = 134 412 963 989 + 0;
  • 134 412 963 989 ÷ 2 = 67 206 481 994 + 1;
  • 67 206 481 994 ÷ 2 = 33 603 240 997 + 0;
  • 33 603 240 997 ÷ 2 = 16 801 620 498 + 1;
  • 16 801 620 498 ÷ 2 = 8 400 810 249 + 0;
  • 8 400 810 249 ÷ 2 = 4 200 405 124 + 1;
  • 4 200 405 124 ÷ 2 = 2 100 202 562 + 0;
  • 2 100 202 562 ÷ 2 = 1 050 101 281 + 0;
  • 1 050 101 281 ÷ 2 = 525 050 640 + 1;
  • 525 050 640 ÷ 2 = 262 525 320 + 0;
  • 262 525 320 ÷ 2 = 131 262 660 + 0;
  • 131 262 660 ÷ 2 = 65 631 330 + 0;
  • 65 631 330 ÷ 2 = 32 815 665 + 0;
  • 32 815 665 ÷ 2 = 16 407 832 + 1;
  • 16 407 832 ÷ 2 = 8 203 916 + 0;
  • 8 203 916 ÷ 2 = 4 101 958 + 0;
  • 4 101 958 ÷ 2 = 2 050 979 + 0;
  • 2 050 979 ÷ 2 = 1 025 489 + 1;
  • 1 025 489 ÷ 2 = 512 744 + 1;
  • 512 744 ÷ 2 = 256 372 + 0;
  • 256 372 ÷ 2 = 128 186 + 0;
  • 128 186 ÷ 2 = 64 093 + 0;
  • 64 093 ÷ 2 = 32 046 + 1;
  • 32 046 ÷ 2 = 16 023 + 0;
  • 16 023 ÷ 2 = 8 011 + 1;
  • 8 011 ÷ 2 = 4 005 + 1;
  • 4 005 ÷ 2 = 2 002 + 1;
  • 2 002 ÷ 2 = 1 001 + 0;
  • 1 001 ÷ 2 = 500 + 1;
  • 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.

1 101 111 001 000 072(10) = 11 1110 1001 0111 0100 0110 0010 0001 0010 1010 1000 1000 1000(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.


Decimal Number 1 101 111 001 000 072(10) converted to signed binary in two's complement representation:

1 101 111 001 000 072(10) = 0000 0000 0000 0011 1110 1001 0111 0100 0110 0010 0001 0010 1010 1000 1000 1000

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