Convert 10 011 110 001 000 680 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 011 110 001 000 680(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 011 110 001 000 680 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;
  • 10 011 110 001 000 680 ÷ 2 = 5 005 555 000 500 340 + 0;
  • 5 005 555 000 500 340 ÷ 2 = 2 502 777 500 250 170 + 0;
  • 2 502 777 500 250 170 ÷ 2 = 1 251 388 750 125 085 + 0;
  • 1 251 388 750 125 085 ÷ 2 = 625 694 375 062 542 + 1;
  • 625 694 375 062 542 ÷ 2 = 312 847 187 531 271 + 0;
  • 312 847 187 531 271 ÷ 2 = 156 423 593 765 635 + 1;
  • 156 423 593 765 635 ÷ 2 = 78 211 796 882 817 + 1;
  • 78 211 796 882 817 ÷ 2 = 39 105 898 441 408 + 1;
  • 39 105 898 441 408 ÷ 2 = 19 552 949 220 704 + 0;
  • 19 552 949 220 704 ÷ 2 = 9 776 474 610 352 + 0;
  • 9 776 474 610 352 ÷ 2 = 4 888 237 305 176 + 0;
  • 4 888 237 305 176 ÷ 2 = 2 444 118 652 588 + 0;
  • 2 444 118 652 588 ÷ 2 = 1 222 059 326 294 + 0;
  • 1 222 059 326 294 ÷ 2 = 611 029 663 147 + 0;
  • 611 029 663 147 ÷ 2 = 305 514 831 573 + 1;
  • 305 514 831 573 ÷ 2 = 152 757 415 786 + 1;
  • 152 757 415 786 ÷ 2 = 76 378 707 893 + 0;
  • 76 378 707 893 ÷ 2 = 38 189 353 946 + 1;
  • 38 189 353 946 ÷ 2 = 19 094 676 973 + 0;
  • 19 094 676 973 ÷ 2 = 9 547 338 486 + 1;
  • 9 547 338 486 ÷ 2 = 4 773 669 243 + 0;
  • 4 773 669 243 ÷ 2 = 2 386 834 621 + 1;
  • 2 386 834 621 ÷ 2 = 1 193 417 310 + 1;
  • 1 193 417 310 ÷ 2 = 596 708 655 + 0;
  • 596 708 655 ÷ 2 = 298 354 327 + 1;
  • 298 354 327 ÷ 2 = 149 177 163 + 1;
  • 149 177 163 ÷ 2 = 74 588 581 + 1;
  • 74 588 581 ÷ 2 = 37 294 290 + 1;
  • 37 294 290 ÷ 2 = 18 647 145 + 0;
  • 18 647 145 ÷ 2 = 9 323 572 + 1;
  • 9 323 572 ÷ 2 = 4 661 786 + 0;
  • 4 661 786 ÷ 2 = 2 330 893 + 0;
  • 2 330 893 ÷ 2 = 1 165 446 + 1;
  • 1 165 446 ÷ 2 = 582 723 + 0;
  • 582 723 ÷ 2 = 291 361 + 1;
  • 291 361 ÷ 2 = 145 680 + 1;
  • 145 680 ÷ 2 = 72 840 + 0;
  • 72 840 ÷ 2 = 36 420 + 0;
  • 36 420 ÷ 2 = 18 210 + 0;
  • 18 210 ÷ 2 = 9 105 + 0;
  • 9 105 ÷ 2 = 4 552 + 1;
  • 4 552 ÷ 2 = 2 276 + 0;
  • 2 276 ÷ 2 = 1 138 + 0;
  • 1 138 ÷ 2 = 569 + 0;
  • 569 ÷ 2 = 284 + 1;
  • 284 ÷ 2 = 142 + 0;
  • 142 ÷ 2 = 71 + 0;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 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.

10 011 110 001 000 680(10) = 10 0011 1001 0001 0000 1101 0010 1111 0110 1010 1100 0000 1110 1000(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 10 011 110 001 000 680(10) converted to signed binary in two's complement representation:

10 011 110 001 000 680(10) = 0000 0000 0010 0011 1001 0001 0000 1101 0010 1111 0110 1010 1100 0000 1110 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