Convert -3 074 457 345 618 258 463 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -3 074 457 345 618 258 463(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-3 074 457 345 618 258 463 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. Start with the positive version of the number:

|-3 074 457 345 618 258 463| = 3 074 457 345 618 258 463

2. 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;
  • 3 074 457 345 618 258 463 ÷ 2 = 1 537 228 672 809 129 231 + 1;
  • 1 537 228 672 809 129 231 ÷ 2 = 768 614 336 404 564 615 + 1;
  • 768 614 336 404 564 615 ÷ 2 = 384 307 168 202 282 307 + 1;
  • 384 307 168 202 282 307 ÷ 2 = 192 153 584 101 141 153 + 1;
  • 192 153 584 101 141 153 ÷ 2 = 96 076 792 050 570 576 + 1;
  • 96 076 792 050 570 576 ÷ 2 = 48 038 396 025 285 288 + 0;
  • 48 038 396 025 285 288 ÷ 2 = 24 019 198 012 642 644 + 0;
  • 24 019 198 012 642 644 ÷ 2 = 12 009 599 006 321 322 + 0;
  • 12 009 599 006 321 322 ÷ 2 = 6 004 799 503 160 661 + 0;
  • 6 004 799 503 160 661 ÷ 2 = 3 002 399 751 580 330 + 1;
  • 3 002 399 751 580 330 ÷ 2 = 1 501 199 875 790 165 + 0;
  • 1 501 199 875 790 165 ÷ 2 = 750 599 937 895 082 + 1;
  • 750 599 937 895 082 ÷ 2 = 375 299 968 947 541 + 0;
  • 375 299 968 947 541 ÷ 2 = 187 649 984 473 770 + 1;
  • 187 649 984 473 770 ÷ 2 = 93 824 992 236 885 + 0;
  • 93 824 992 236 885 ÷ 2 = 46 912 496 118 442 + 1;
  • 46 912 496 118 442 ÷ 2 = 23 456 248 059 221 + 0;
  • 23 456 248 059 221 ÷ 2 = 11 728 124 029 610 + 1;
  • 11 728 124 029 610 ÷ 2 = 5 864 062 014 805 + 0;
  • 5 864 062 014 805 ÷ 2 = 2 932 031 007 402 + 1;
  • 2 932 031 007 402 ÷ 2 = 1 466 015 503 701 + 0;
  • 1 466 015 503 701 ÷ 2 = 733 007 751 850 + 1;
  • 733 007 751 850 ÷ 2 = 366 503 875 925 + 0;
  • 366 503 875 925 ÷ 2 = 183 251 937 962 + 1;
  • 183 251 937 962 ÷ 2 = 91 625 968 981 + 0;
  • 91 625 968 981 ÷ 2 = 45 812 984 490 + 1;
  • 45 812 984 490 ÷ 2 = 22 906 492 245 + 0;
  • 22 906 492 245 ÷ 2 = 11 453 246 122 + 1;
  • 11 453 246 122 ÷ 2 = 5 726 623 061 + 0;
  • 5 726 623 061 ÷ 2 = 2 863 311 530 + 1;
  • 2 863 311 530 ÷ 2 = 1 431 655 765 + 0;
  • 1 431 655 765 ÷ 2 = 715 827 882 + 1;
  • 715 827 882 ÷ 2 = 357 913 941 + 0;
  • 357 913 941 ÷ 2 = 178 956 970 + 1;
  • 178 956 970 ÷ 2 = 89 478 485 + 0;
  • 89 478 485 ÷ 2 = 44 739 242 + 1;
  • 44 739 242 ÷ 2 = 22 369 621 + 0;
  • 22 369 621 ÷ 2 = 11 184 810 + 1;
  • 11 184 810 ÷ 2 = 5 592 405 + 0;
  • 5 592 405 ÷ 2 = 2 796 202 + 1;
  • 2 796 202 ÷ 2 = 1 398 101 + 0;
  • 1 398 101 ÷ 2 = 699 050 + 1;
  • 699 050 ÷ 2 = 349 525 + 0;
  • 349 525 ÷ 2 = 174 762 + 1;
  • 174 762 ÷ 2 = 87 381 + 0;
  • 87 381 ÷ 2 = 43 690 + 1;
  • 43 690 ÷ 2 = 21 845 + 0;
  • 21 845 ÷ 2 = 10 922 + 1;
  • 10 922 ÷ 2 = 5 461 + 0;
  • 5 461 ÷ 2 = 2 730 + 1;
  • 2 730 ÷ 2 = 1 365 + 0;
  • 1 365 ÷ 2 = 682 + 1;
  • 682 ÷ 2 = 341 + 0;
  • 341 ÷ 2 = 170 + 1;
  • 170 ÷ 2 = 85 + 0;
  • 85 ÷ 2 = 42 + 1;
  • 42 ÷ 2 = 21 + 0;
  • 21 ÷ 2 = 10 + 1;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

3. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

3 074 457 345 618 258 463(10) = 10 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 0001 1111(2)

4. 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.


5. 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.


3 074 457 345 618 258 463(10) = 0010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 0001 1111

6. Get the negative integer number representation. Part 1:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation, replace all the bits on 0 with 1s and all the bits set on 1 with 0s.

Reverse the digits, flip the digits:

Replace the bits set on 0 with 1s and the bits set on 1 with 0s.

!(0010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 0001 1111)

= 1101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1110 0000


7. Get the negative integer number representation. Part 2:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in two's complement representation, add 1 to the number calculated above 1101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1110 0000 (to the signed binary in one's complement representation).

Binary addition carries on a value of 2:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 10 = 11
  • 1 + 11 = 100

Add 1 to the number calculated above
(to the signed binary number in one's complement representation):

-3 074 457 345 618 258 463 =

1101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1110 0000 + 1


Decimal Number -3 074 457 345 618 258 463(10) converted to signed binary in two's complement representation:

-3 074 457 345 618 258 463(10) = 1101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1110 0001

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