Convert 1 001 010 000 010 316 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 001 010 000 010 316(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 001 010 000 010 316 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;
  • 1 001 010 000 010 316 ÷ 2 = 500 505 000 005 158 + 0;
  • 500 505 000 005 158 ÷ 2 = 250 252 500 002 579 + 0;
  • 250 252 500 002 579 ÷ 2 = 125 126 250 001 289 + 1;
  • 125 126 250 001 289 ÷ 2 = 62 563 125 000 644 + 1;
  • 62 563 125 000 644 ÷ 2 = 31 281 562 500 322 + 0;
  • 31 281 562 500 322 ÷ 2 = 15 640 781 250 161 + 0;
  • 15 640 781 250 161 ÷ 2 = 7 820 390 625 080 + 1;
  • 7 820 390 625 080 ÷ 2 = 3 910 195 312 540 + 0;
  • 3 910 195 312 540 ÷ 2 = 1 955 097 656 270 + 0;
  • 1 955 097 656 270 ÷ 2 = 977 548 828 135 + 0;
  • 977 548 828 135 ÷ 2 = 488 774 414 067 + 1;
  • 488 774 414 067 ÷ 2 = 244 387 207 033 + 1;
  • 244 387 207 033 ÷ 2 = 122 193 603 516 + 1;
  • 122 193 603 516 ÷ 2 = 61 096 801 758 + 0;
  • 61 096 801 758 ÷ 2 = 30 548 400 879 + 0;
  • 30 548 400 879 ÷ 2 = 15 274 200 439 + 1;
  • 15 274 200 439 ÷ 2 = 7 637 100 219 + 1;
  • 7 637 100 219 ÷ 2 = 3 818 550 109 + 1;
  • 3 818 550 109 ÷ 2 = 1 909 275 054 + 1;
  • 1 909 275 054 ÷ 2 = 954 637 527 + 0;
  • 954 637 527 ÷ 2 = 477 318 763 + 1;
  • 477 318 763 ÷ 2 = 238 659 381 + 1;
  • 238 659 381 ÷ 2 = 119 329 690 + 1;
  • 119 329 690 ÷ 2 = 59 664 845 + 0;
  • 59 664 845 ÷ 2 = 29 832 422 + 1;
  • 29 832 422 ÷ 2 = 14 916 211 + 0;
  • 14 916 211 ÷ 2 = 7 458 105 + 1;
  • 7 458 105 ÷ 2 = 3 729 052 + 1;
  • 3 729 052 ÷ 2 = 1 864 526 + 0;
  • 1 864 526 ÷ 2 = 932 263 + 0;
  • 932 263 ÷ 2 = 466 131 + 1;
  • 466 131 ÷ 2 = 233 065 + 1;
  • 233 065 ÷ 2 = 116 532 + 1;
  • 116 532 ÷ 2 = 58 266 + 0;
  • 58 266 ÷ 2 = 29 133 + 0;
  • 29 133 ÷ 2 = 14 566 + 1;
  • 14 566 ÷ 2 = 7 283 + 0;
  • 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.

1 001 010 000 010 316(10) = 11 1000 1110 0110 1001 1100 1101 0111 0111 1001 1100 0100 1100(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 1 001 010 000 010 316(10) converted to signed binary in two's complement representation:

1 001 010 000 010 316(10) = 0000 0000 0000 0011 1000 1110 0110 1001 1100 1101 0111 0111 1001 1100 0100 1100

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