Convert 3 333 333 333 333 333 641 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 3 333 333 333 333 333 641(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
3 333 333 333 333 333 641 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;
  • 3 333 333 333 333 333 641 ÷ 2 = 1 666 666 666 666 666 820 + 1;
  • 1 666 666 666 666 666 820 ÷ 2 = 833 333 333 333 333 410 + 0;
  • 833 333 333 333 333 410 ÷ 2 = 416 666 666 666 666 705 + 0;
  • 416 666 666 666 666 705 ÷ 2 = 208 333 333 333 333 352 + 1;
  • 208 333 333 333 333 352 ÷ 2 = 104 166 666 666 666 676 + 0;
  • 104 166 666 666 666 676 ÷ 2 = 52 083 333 333 333 338 + 0;
  • 52 083 333 333 333 338 ÷ 2 = 26 041 666 666 666 669 + 0;
  • 26 041 666 666 666 669 ÷ 2 = 13 020 833 333 333 334 + 1;
  • 13 020 833 333 333 334 ÷ 2 = 6 510 416 666 666 667 + 0;
  • 6 510 416 666 666 667 ÷ 2 = 3 255 208 333 333 333 + 1;
  • 3 255 208 333 333 333 ÷ 2 = 1 627 604 166 666 666 + 1;
  • 1 627 604 166 666 666 ÷ 2 = 813 802 083 333 333 + 0;
  • 813 802 083 333 333 ÷ 2 = 406 901 041 666 666 + 1;
  • 406 901 041 666 666 ÷ 2 = 203 450 520 833 333 + 0;
  • 203 450 520 833 333 ÷ 2 = 101 725 260 416 666 + 1;
  • 101 725 260 416 666 ÷ 2 = 50 862 630 208 333 + 0;
  • 50 862 630 208 333 ÷ 2 = 25 431 315 104 166 + 1;
  • 25 431 315 104 166 ÷ 2 = 12 715 657 552 083 + 0;
  • 12 715 657 552 083 ÷ 2 = 6 357 828 776 041 + 1;
  • 6 357 828 776 041 ÷ 2 = 3 178 914 388 020 + 1;
  • 3 178 914 388 020 ÷ 2 = 1 589 457 194 010 + 0;
  • 1 589 457 194 010 ÷ 2 = 794 728 597 005 + 0;
  • 794 728 597 005 ÷ 2 = 397 364 298 502 + 1;
  • 397 364 298 502 ÷ 2 = 198 682 149 251 + 0;
  • 198 682 149 251 ÷ 2 = 99 341 074 625 + 1;
  • 99 341 074 625 ÷ 2 = 49 670 537 312 + 1;
  • 49 670 537 312 ÷ 2 = 24 835 268 656 + 0;
  • 24 835 268 656 ÷ 2 = 12 417 634 328 + 0;
  • 12 417 634 328 ÷ 2 = 6 208 817 164 + 0;
  • 6 208 817 164 ÷ 2 = 3 104 408 582 + 0;
  • 3 104 408 582 ÷ 2 = 1 552 204 291 + 0;
  • 1 552 204 291 ÷ 2 = 776 102 145 + 1;
  • 776 102 145 ÷ 2 = 388 051 072 + 1;
  • 388 051 072 ÷ 2 = 194 025 536 + 0;
  • 194 025 536 ÷ 2 = 97 012 768 + 0;
  • 97 012 768 ÷ 2 = 48 506 384 + 0;
  • 48 506 384 ÷ 2 = 24 253 192 + 0;
  • 24 253 192 ÷ 2 = 12 126 596 + 0;
  • 12 126 596 ÷ 2 = 6 063 298 + 0;
  • 6 063 298 ÷ 2 = 3 031 649 + 0;
  • 3 031 649 ÷ 2 = 1 515 824 + 1;
  • 1 515 824 ÷ 2 = 757 912 + 0;
  • 757 912 ÷ 2 = 378 956 + 0;
  • 378 956 ÷ 2 = 189 478 + 0;
  • 189 478 ÷ 2 = 94 739 + 0;
  • 94 739 ÷ 2 = 47 369 + 1;
  • 47 369 ÷ 2 = 23 684 + 1;
  • 23 684 ÷ 2 = 11 842 + 0;
  • 11 842 ÷ 2 = 5 921 + 0;
  • 5 921 ÷ 2 = 2 960 + 1;
  • 2 960 ÷ 2 = 1 480 + 0;
  • 1 480 ÷ 2 = 740 + 0;
  • 740 ÷ 2 = 370 + 0;
  • 370 ÷ 2 = 185 + 0;
  • 185 ÷ 2 = 92 + 1;
  • 92 ÷ 2 = 46 + 0;
  • 46 ÷ 2 = 23 + 0;
  • 23 ÷ 2 = 11 + 1;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.

3 333 333 333 333 333 641(10) = 10 1110 0100 0010 0110 0001 0000 0001 1000 0011 0100 1101 0101 0110 1000 1001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 62.

  • 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, 62,

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 3 333 333 333 333 333 641(10) converted to signed binary in two's complement representation:

3 333 333 333 333 333 641(10) = 0010 1110 0100 0010 0110 0001 0000 0001 1000 0011 0100 1101 0101 0110 1000 1001

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