Convert -4 413 527 634 823 086 096 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -4 413 527 634 823 086 096(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-4 413 527 634 823 086 096 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:

|-4 413 527 634 823 086 096| = 4 413 527 634 823 086 096

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;
  • 4 413 527 634 823 086 096 ÷ 2 = 2 206 763 817 411 543 048 + 0;
  • 2 206 763 817 411 543 048 ÷ 2 = 1 103 381 908 705 771 524 + 0;
  • 1 103 381 908 705 771 524 ÷ 2 = 551 690 954 352 885 762 + 0;
  • 551 690 954 352 885 762 ÷ 2 = 275 845 477 176 442 881 + 0;
  • 275 845 477 176 442 881 ÷ 2 = 137 922 738 588 221 440 + 1;
  • 137 922 738 588 221 440 ÷ 2 = 68 961 369 294 110 720 + 0;
  • 68 961 369 294 110 720 ÷ 2 = 34 480 684 647 055 360 + 0;
  • 34 480 684 647 055 360 ÷ 2 = 17 240 342 323 527 680 + 0;
  • 17 240 342 323 527 680 ÷ 2 = 8 620 171 161 763 840 + 0;
  • 8 620 171 161 763 840 ÷ 2 = 4 310 085 580 881 920 + 0;
  • 4 310 085 580 881 920 ÷ 2 = 2 155 042 790 440 960 + 0;
  • 2 155 042 790 440 960 ÷ 2 = 1 077 521 395 220 480 + 0;
  • 1 077 521 395 220 480 ÷ 2 = 538 760 697 610 240 + 0;
  • 538 760 697 610 240 ÷ 2 = 269 380 348 805 120 + 0;
  • 269 380 348 805 120 ÷ 2 = 134 690 174 402 560 + 0;
  • 134 690 174 402 560 ÷ 2 = 67 345 087 201 280 + 0;
  • 67 345 087 201 280 ÷ 2 = 33 672 543 600 640 + 0;
  • 33 672 543 600 640 ÷ 2 = 16 836 271 800 320 + 0;
  • 16 836 271 800 320 ÷ 2 = 8 418 135 900 160 + 0;
  • 8 418 135 900 160 ÷ 2 = 4 209 067 950 080 + 0;
  • 4 209 067 950 080 ÷ 2 = 2 104 533 975 040 + 0;
  • 2 104 533 975 040 ÷ 2 = 1 052 266 987 520 + 0;
  • 1 052 266 987 520 ÷ 2 = 526 133 493 760 + 0;
  • 526 133 493 760 ÷ 2 = 263 066 746 880 + 0;
  • 263 066 746 880 ÷ 2 = 131 533 373 440 + 0;
  • 131 533 373 440 ÷ 2 = 65 766 686 720 + 0;
  • 65 766 686 720 ÷ 2 = 32 883 343 360 + 0;
  • 32 883 343 360 ÷ 2 = 16 441 671 680 + 0;
  • 16 441 671 680 ÷ 2 = 8 220 835 840 + 0;
  • 8 220 835 840 ÷ 2 = 4 110 417 920 + 0;
  • 4 110 417 920 ÷ 2 = 2 055 208 960 + 0;
  • 2 055 208 960 ÷ 2 = 1 027 604 480 + 0;
  • 1 027 604 480 ÷ 2 = 513 802 240 + 0;
  • 513 802 240 ÷ 2 = 256 901 120 + 0;
  • 256 901 120 ÷ 2 = 128 450 560 + 0;
  • 128 450 560 ÷ 2 = 64 225 280 + 0;
  • 64 225 280 ÷ 2 = 32 112 640 + 0;
  • 32 112 640 ÷ 2 = 16 056 320 + 0;
  • 16 056 320 ÷ 2 = 8 028 160 + 0;
  • 8 028 160 ÷ 2 = 4 014 080 + 0;
  • 4 014 080 ÷ 2 = 2 007 040 + 0;
  • 2 007 040 ÷ 2 = 1 003 520 + 0;
  • 1 003 520 ÷ 2 = 501 760 + 0;
  • 501 760 ÷ 2 = 250 880 + 0;
  • 250 880 ÷ 2 = 125 440 + 0;
  • 125 440 ÷ 2 = 62 720 + 0;
  • 62 720 ÷ 2 = 31 360 + 0;
  • 31 360 ÷ 2 = 15 680 + 0;
  • 15 680 ÷ 2 = 7 840 + 0;
  • 7 840 ÷ 2 = 3 920 + 0;
  • 3 920 ÷ 2 = 1 960 + 0;
  • 1 960 ÷ 2 = 980 + 0;
  • 980 ÷ 2 = 490 + 0;
  • 490 ÷ 2 = 245 + 0;
  • 245 ÷ 2 = 122 + 1;
  • 122 ÷ 2 = 61 + 0;
  • 61 ÷ 2 = 30 + 1;
  • 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:

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

4 413 527 634 823 086 096(10) = 11 1101 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000(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.


4 413 527 634 823 086 096(10) = 0011 1101 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000

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.

!(0011 1101 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000)

= 1100 0010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 1111


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 1100 0010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 1111 (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):

-4 413 527 634 823 086 096 =

1100 0010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 1111 + 1


Decimal Number -4 413 527 634 823 086 096(10) converted to signed binary in two's complement representation:

-4 413 527 634 823 086 096(10) = 1100 0010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000

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