Convert Decimal 1 001 100 099 999 957 to Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 001 100 099 999 957(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 001 100 099 999 957 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;
  • 1 001 100 099 999 957 ÷ 2 = 500 550 049 999 978 + 1;
  • 500 550 049 999 978 ÷ 2 = 250 275 024 999 989 + 0;
  • 250 275 024 999 989 ÷ 2 = 125 137 512 499 994 + 1;
  • 125 137 512 499 994 ÷ 2 = 62 568 756 249 997 + 0;
  • 62 568 756 249 997 ÷ 2 = 31 284 378 124 998 + 1;
  • 31 284 378 124 998 ÷ 2 = 15 642 189 062 499 + 0;
  • 15 642 189 062 499 ÷ 2 = 7 821 094 531 249 + 1;
  • 7 821 094 531 249 ÷ 2 = 3 910 547 265 624 + 1;
  • 3 910 547 265 624 ÷ 2 = 1 955 273 632 812 + 0;
  • 1 955 273 632 812 ÷ 2 = 977 636 816 406 + 0;
  • 977 636 816 406 ÷ 2 = 488 818 408 203 + 0;
  • 488 818 408 203 ÷ 2 = 244 409 204 101 + 1;
  • 244 409 204 101 ÷ 2 = 122 204 602 050 + 1;
  • 122 204 602 050 ÷ 2 = 61 102 301 025 + 0;
  • 61 102 301 025 ÷ 2 = 30 551 150 512 + 1;
  • 30 551 150 512 ÷ 2 = 15 275 575 256 + 0;
  • 15 275 575 256 ÷ 2 = 7 637 787 628 + 0;
  • 7 637 787 628 ÷ 2 = 3 818 893 814 + 0;
  • 3 818 893 814 ÷ 2 = 1 909 446 907 + 0;
  • 1 909 446 907 ÷ 2 = 954 723 453 + 1;
  • 954 723 453 ÷ 2 = 477 361 726 + 1;
  • 477 361 726 ÷ 2 = 238 680 863 + 0;
  • 238 680 863 ÷ 2 = 119 340 431 + 1;
  • 119 340 431 ÷ 2 = 59 670 215 + 1;
  • 59 670 215 ÷ 2 = 29 835 107 + 1;
  • 29 835 107 ÷ 2 = 14 917 553 + 1;
  • 14 917 553 ÷ 2 = 7 458 776 + 1;
  • 7 458 776 ÷ 2 = 3 729 388 + 0;
  • 3 729 388 ÷ 2 = 1 864 694 + 0;
  • 1 864 694 ÷ 2 = 932 347 + 0;
  • 932 347 ÷ 2 = 466 173 + 1;
  • 466 173 ÷ 2 = 233 086 + 1;
  • 233 086 ÷ 2 = 116 543 + 0;
  • 116 543 ÷ 2 = 58 271 + 1;
  • 58 271 ÷ 2 = 29 135 + 1;
  • 29 135 ÷ 2 = 14 567 + 1;
  • 14 567 ÷ 2 = 7 283 + 1;
  • 7 283 ÷ 2 = 3 641 + 1;
  • 3 641 ÷ 2 = 1 820 + 1;
  • 1 820 ÷ 2 = 910 + 0;
  • 910 ÷ 2 = 455 + 0;
  • 455 ÷ 2 = 227 + 1;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.

1 001 100 099 999 957(10) = 11 1000 1110 0111 1110 1100 0111 1101 1000 0101 1000 1101 0101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

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 1 001 100 099 999 957(10) converted to signed binary in two's complement representation:

1 001 100 099 999 957(10) = 0000 0000 0000 0011 1000 1110 0111 1110 1100 0111 1101 1000 0101 1000 1101 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