Convert 1 101 110 111 099 638 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 101 110 111 099 638(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 101 110 111 099 638 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;
  • 1 101 110 111 099 638 ÷ 2 = 550 555 055 549 819 + 0;
  • 550 555 055 549 819 ÷ 2 = 275 277 527 774 909 + 1;
  • 275 277 527 774 909 ÷ 2 = 137 638 763 887 454 + 1;
  • 137 638 763 887 454 ÷ 2 = 68 819 381 943 727 + 0;
  • 68 819 381 943 727 ÷ 2 = 34 409 690 971 863 + 1;
  • 34 409 690 971 863 ÷ 2 = 17 204 845 485 931 + 1;
  • 17 204 845 485 931 ÷ 2 = 8 602 422 742 965 + 1;
  • 8 602 422 742 965 ÷ 2 = 4 301 211 371 482 + 1;
  • 4 301 211 371 482 ÷ 2 = 2 150 605 685 741 + 0;
  • 2 150 605 685 741 ÷ 2 = 1 075 302 842 870 + 1;
  • 1 075 302 842 870 ÷ 2 = 537 651 421 435 + 0;
  • 537 651 421 435 ÷ 2 = 268 825 710 717 + 1;
  • 268 825 710 717 ÷ 2 = 134 412 855 358 + 1;
  • 134 412 855 358 ÷ 2 = 67 206 427 679 + 0;
  • 67 206 427 679 ÷ 2 = 33 603 213 839 + 1;
  • 33 603 213 839 ÷ 2 = 16 801 606 919 + 1;
  • 16 801 606 919 ÷ 2 = 8 400 803 459 + 1;
  • 8 400 803 459 ÷ 2 = 4 200 401 729 + 1;
  • 4 200 401 729 ÷ 2 = 2 100 200 864 + 1;
  • 2 100 200 864 ÷ 2 = 1 050 100 432 + 0;
  • 1 050 100 432 ÷ 2 = 525 050 216 + 0;
  • 525 050 216 ÷ 2 = 262 525 108 + 0;
  • 262 525 108 ÷ 2 = 131 262 554 + 0;
  • 131 262 554 ÷ 2 = 65 631 277 + 0;
  • 65 631 277 ÷ 2 = 32 815 638 + 1;
  • 32 815 638 ÷ 2 = 16 407 819 + 0;
  • 16 407 819 ÷ 2 = 8 203 909 + 1;
  • 8 203 909 ÷ 2 = 4 101 954 + 1;
  • 4 101 954 ÷ 2 = 2 050 977 + 0;
  • 2 050 977 ÷ 2 = 1 025 488 + 1;
  • 1 025 488 ÷ 2 = 512 744 + 0;
  • 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.

1 101 110 111 099 638(10) = 11 1110 1001 0111 0100 0010 1101 0000 0111 1101 1010 1111 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.


Decimal Number 1 101 110 111 099 638(10) converted to signed binary in two's complement representation:

1 101 110 111 099 638(10) = 0000 0000 0000 0011 1110 1001 0111 0100 0010 1101 0000 0111 1101 1010 1111 0110

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