Convert 2 740 293 748 741 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 2 740 293 748 741(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
2 740 293 748 741 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;
  • 2 740 293 748 741 ÷ 2 = 1 370 146 874 370 + 1;
  • 1 370 146 874 370 ÷ 2 = 685 073 437 185 + 0;
  • 685 073 437 185 ÷ 2 = 342 536 718 592 + 1;
  • 342 536 718 592 ÷ 2 = 171 268 359 296 + 0;
  • 171 268 359 296 ÷ 2 = 85 634 179 648 + 0;
  • 85 634 179 648 ÷ 2 = 42 817 089 824 + 0;
  • 42 817 089 824 ÷ 2 = 21 408 544 912 + 0;
  • 21 408 544 912 ÷ 2 = 10 704 272 456 + 0;
  • 10 704 272 456 ÷ 2 = 5 352 136 228 + 0;
  • 5 352 136 228 ÷ 2 = 2 676 068 114 + 0;
  • 2 676 068 114 ÷ 2 = 1 338 034 057 + 0;
  • 1 338 034 057 ÷ 2 = 669 017 028 + 1;
  • 669 017 028 ÷ 2 = 334 508 514 + 0;
  • 334 508 514 ÷ 2 = 167 254 257 + 0;
  • 167 254 257 ÷ 2 = 83 627 128 + 1;
  • 83 627 128 ÷ 2 = 41 813 564 + 0;
  • 41 813 564 ÷ 2 = 20 906 782 + 0;
  • 20 906 782 ÷ 2 = 10 453 391 + 0;
  • 10 453 391 ÷ 2 = 5 226 695 + 1;
  • 5 226 695 ÷ 2 = 2 613 347 + 1;
  • 2 613 347 ÷ 2 = 1 306 673 + 1;
  • 1 306 673 ÷ 2 = 653 336 + 1;
  • 653 336 ÷ 2 = 326 668 + 0;
  • 326 668 ÷ 2 = 163 334 + 0;
  • 163 334 ÷ 2 = 81 667 + 0;
  • 81 667 ÷ 2 = 40 833 + 1;
  • 40 833 ÷ 2 = 20 416 + 1;
  • 20 416 ÷ 2 = 10 208 + 0;
  • 10 208 ÷ 2 = 5 104 + 0;
  • 5 104 ÷ 2 = 2 552 + 0;
  • 2 552 ÷ 2 = 1 276 + 0;
  • 1 276 ÷ 2 = 638 + 0;
  • 638 ÷ 2 = 319 + 0;
  • 319 ÷ 2 = 159 + 1;
  • 159 ÷ 2 = 79 + 1;
  • 79 ÷ 2 = 39 + 1;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 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.

2 740 293 748 741(10) = 10 0111 1110 0000 0110 0011 1100 0100 1000 0000 0101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 42.

  • 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, 42,

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 2 740 293 748 741(10) converted to signed binary in two's complement representation:

2 740 293 748 741(10) = 0000 0000 0000 0000 0000 0010 0111 1110 0000 0110 0011 1100 0100 1000 0000 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