Convert 1 100 111 100 008 645 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 100 111 100 008 645(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 100 111 100 008 645 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;
  • 1 100 111 100 008 645 ÷ 2 = 550 055 550 004 322 + 1;
  • 550 055 550 004 322 ÷ 2 = 275 027 775 002 161 + 0;
  • 275 027 775 002 161 ÷ 2 = 137 513 887 501 080 + 1;
  • 137 513 887 501 080 ÷ 2 = 68 756 943 750 540 + 0;
  • 68 756 943 750 540 ÷ 2 = 34 378 471 875 270 + 0;
  • 34 378 471 875 270 ÷ 2 = 17 189 235 937 635 + 0;
  • 17 189 235 937 635 ÷ 2 = 8 594 617 968 817 + 1;
  • 8 594 617 968 817 ÷ 2 = 4 297 308 984 408 + 1;
  • 4 297 308 984 408 ÷ 2 = 2 148 654 492 204 + 0;
  • 2 148 654 492 204 ÷ 2 = 1 074 327 246 102 + 0;
  • 1 074 327 246 102 ÷ 2 = 537 163 623 051 + 0;
  • 537 163 623 051 ÷ 2 = 268 581 811 525 + 1;
  • 268 581 811 525 ÷ 2 = 134 290 905 762 + 1;
  • 134 290 905 762 ÷ 2 = 67 145 452 881 + 0;
  • 67 145 452 881 ÷ 2 = 33 572 726 440 + 1;
  • 33 572 726 440 ÷ 2 = 16 786 363 220 + 0;
  • 16 786 363 220 ÷ 2 = 8 393 181 610 + 0;
  • 8 393 181 610 ÷ 2 = 4 196 590 805 + 0;
  • 4 196 590 805 ÷ 2 = 2 098 295 402 + 1;
  • 2 098 295 402 ÷ 2 = 1 049 147 701 + 0;
  • 1 049 147 701 ÷ 2 = 524 573 850 + 1;
  • 524 573 850 ÷ 2 = 262 286 925 + 0;
  • 262 286 925 ÷ 2 = 131 143 462 + 1;
  • 131 143 462 ÷ 2 = 65 571 731 + 0;
  • 65 571 731 ÷ 2 = 32 785 865 + 1;
  • 32 785 865 ÷ 2 = 16 392 932 + 1;
  • 16 392 932 ÷ 2 = 8 196 466 + 0;
  • 8 196 466 ÷ 2 = 4 098 233 + 0;
  • 4 098 233 ÷ 2 = 2 049 116 + 1;
  • 2 049 116 ÷ 2 = 1 024 558 + 0;
  • 1 024 558 ÷ 2 = 512 279 + 0;
  • 512 279 ÷ 2 = 256 139 + 1;
  • 256 139 ÷ 2 = 128 069 + 1;
  • 128 069 ÷ 2 = 64 034 + 1;
  • 64 034 ÷ 2 = 32 017 + 0;
  • 32 017 ÷ 2 = 16 008 + 1;
  • 16 008 ÷ 2 = 8 004 + 0;
  • 8 004 ÷ 2 = 4 002 + 0;
  • 4 002 ÷ 2 = 2 001 + 0;
  • 2 001 ÷ 2 = 1 000 + 1;
  • 1 000 ÷ 2 = 500 + 0;
  • 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.

1 100 111 100 008 645(10) = 11 1110 1000 1000 1011 1001 0011 0101 0100 0101 1000 1100 0101(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 1 100 111 100 008 645(10) converted to signed binary in two's complement representation:

1 100 111 100 008 645(10) = 0000 0000 0000 0011 1110 1000 1000 1011 1001 0011 0101 0100 0101 1000 1100 0101

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