Convert 1 010 101 010 100 916 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 010 101 010 100 916(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 010 101 010 100 916 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;
  • 1 010 101 010 100 916 ÷ 2 = 505 050 505 050 458 + 0;
  • 505 050 505 050 458 ÷ 2 = 252 525 252 525 229 + 0;
  • 252 525 252 525 229 ÷ 2 = 126 262 626 262 614 + 1;
  • 126 262 626 262 614 ÷ 2 = 63 131 313 131 307 + 0;
  • 63 131 313 131 307 ÷ 2 = 31 565 656 565 653 + 1;
  • 31 565 656 565 653 ÷ 2 = 15 782 828 282 826 + 1;
  • 15 782 828 282 826 ÷ 2 = 7 891 414 141 413 + 0;
  • 7 891 414 141 413 ÷ 2 = 3 945 707 070 706 + 1;
  • 3 945 707 070 706 ÷ 2 = 1 972 853 535 353 + 0;
  • 1 972 853 535 353 ÷ 2 = 986 426 767 676 + 1;
  • 986 426 767 676 ÷ 2 = 493 213 383 838 + 0;
  • 493 213 383 838 ÷ 2 = 246 606 691 919 + 0;
  • 246 606 691 919 ÷ 2 = 123 303 345 959 + 1;
  • 123 303 345 959 ÷ 2 = 61 651 672 979 + 1;
  • 61 651 672 979 ÷ 2 = 30 825 836 489 + 1;
  • 30 825 836 489 ÷ 2 = 15 412 918 244 + 1;
  • 15 412 918 244 ÷ 2 = 7 706 459 122 + 0;
  • 7 706 459 122 ÷ 2 = 3 853 229 561 + 0;
  • 3 853 229 561 ÷ 2 = 1 926 614 780 + 1;
  • 1 926 614 780 ÷ 2 = 963 307 390 + 0;
  • 963 307 390 ÷ 2 = 481 653 695 + 0;
  • 481 653 695 ÷ 2 = 240 826 847 + 1;
  • 240 826 847 ÷ 2 = 120 413 423 + 1;
  • 120 413 423 ÷ 2 = 60 206 711 + 1;
  • 60 206 711 ÷ 2 = 30 103 355 + 1;
  • 30 103 355 ÷ 2 = 15 051 677 + 1;
  • 15 051 677 ÷ 2 = 7 525 838 + 1;
  • 7 525 838 ÷ 2 = 3 762 919 + 0;
  • 3 762 919 ÷ 2 = 1 881 459 + 1;
  • 1 881 459 ÷ 2 = 940 729 + 1;
  • 940 729 ÷ 2 = 470 364 + 1;
  • 470 364 ÷ 2 = 235 182 + 0;
  • 235 182 ÷ 2 = 117 591 + 0;
  • 117 591 ÷ 2 = 58 795 + 1;
  • 58 795 ÷ 2 = 29 397 + 1;
  • 29 397 ÷ 2 = 14 698 + 1;
  • 14 698 ÷ 2 = 7 349 + 0;
  • 7 349 ÷ 2 = 3 674 + 1;
  • 3 674 ÷ 2 = 1 837 + 0;
  • 1 837 ÷ 2 = 918 + 1;
  • 918 ÷ 2 = 459 + 0;
  • 459 ÷ 2 = 229 + 1;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 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.

1 010 101 010 100 916(10) = 11 1001 0110 1010 1110 0111 0111 1110 0100 1111 0010 1011 0100(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 1 010 101 010 100 916(10) converted to signed binary in two's complement representation:

1 010 101 010 100 916(10) = 0000 0000 0000 0011 1001 0110 1010 1110 0111 0111 1110 0100 1111 0010 1011 0100

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