Convert 1 100 001 101 010 685 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 100 001 101 010 685(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 100 001 101 010 685 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;
  • 1 100 001 101 010 685 ÷ 2 = 550 000 550 505 342 + 1;
  • 550 000 550 505 342 ÷ 2 = 275 000 275 252 671 + 0;
  • 275 000 275 252 671 ÷ 2 = 137 500 137 626 335 + 1;
  • 137 500 137 626 335 ÷ 2 = 68 750 068 813 167 + 1;
  • 68 750 068 813 167 ÷ 2 = 34 375 034 406 583 + 1;
  • 34 375 034 406 583 ÷ 2 = 17 187 517 203 291 + 1;
  • 17 187 517 203 291 ÷ 2 = 8 593 758 601 645 + 1;
  • 8 593 758 601 645 ÷ 2 = 4 296 879 300 822 + 1;
  • 4 296 879 300 822 ÷ 2 = 2 148 439 650 411 + 0;
  • 2 148 439 650 411 ÷ 2 = 1 074 219 825 205 + 1;
  • 1 074 219 825 205 ÷ 2 = 537 109 912 602 + 1;
  • 537 109 912 602 ÷ 2 = 268 554 956 301 + 0;
  • 268 554 956 301 ÷ 2 = 134 277 478 150 + 1;
  • 134 277 478 150 ÷ 2 = 67 138 739 075 + 0;
  • 67 138 739 075 ÷ 2 = 33 569 369 537 + 1;
  • 33 569 369 537 ÷ 2 = 16 784 684 768 + 1;
  • 16 784 684 768 ÷ 2 = 8 392 342 384 + 0;
  • 8 392 342 384 ÷ 2 = 4 196 171 192 + 0;
  • 4 196 171 192 ÷ 2 = 2 098 085 596 + 0;
  • 2 098 085 596 ÷ 2 = 1 049 042 798 + 0;
  • 1 049 042 798 ÷ 2 = 524 521 399 + 0;
  • 524 521 399 ÷ 2 = 262 260 699 + 1;
  • 262 260 699 ÷ 2 = 131 130 349 + 1;
  • 131 130 349 ÷ 2 = 65 565 174 + 1;
  • 65 565 174 ÷ 2 = 32 782 587 + 0;
  • 32 782 587 ÷ 2 = 16 391 293 + 1;
  • 16 391 293 ÷ 2 = 8 195 646 + 1;
  • 8 195 646 ÷ 2 = 4 097 823 + 0;
  • 4 097 823 ÷ 2 = 2 048 911 + 1;
  • 2 048 911 ÷ 2 = 1 024 455 + 1;
  • 1 024 455 ÷ 2 = 512 227 + 1;
  • 512 227 ÷ 2 = 256 113 + 1;
  • 256 113 ÷ 2 = 128 056 + 1;
  • 128 056 ÷ 2 = 64 028 + 0;
  • 64 028 ÷ 2 = 32 014 + 0;
  • 32 014 ÷ 2 = 16 007 + 0;
  • 16 007 ÷ 2 = 8 003 + 1;
  • 8 003 ÷ 2 = 4 001 + 1;
  • 4 001 ÷ 2 = 2 000 + 1;
  • 2 000 ÷ 2 = 1 000 + 0;
  • 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.

1 100 001 101 010 685(10) = 11 1110 1000 0111 0001 1111 0110 1110 0000 1101 0110 1111 1101(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 1 100 001 101 010 685(10) converted to signed binary in two's complement representation:

1 100 001 101 010 685(10) = 0000 0000 0000 0011 1110 1000 0111 0001 1111 0110 1110 0000 1101 0110 1111 1101

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