Convert 101 011 010 101 257 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 101 011 010 101 257(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
101 011 010 101 257 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;
  • 101 011 010 101 257 ÷ 2 = 50 505 505 050 628 + 1;
  • 50 505 505 050 628 ÷ 2 = 25 252 752 525 314 + 0;
  • 25 252 752 525 314 ÷ 2 = 12 626 376 262 657 + 0;
  • 12 626 376 262 657 ÷ 2 = 6 313 188 131 328 + 1;
  • 6 313 188 131 328 ÷ 2 = 3 156 594 065 664 + 0;
  • 3 156 594 065 664 ÷ 2 = 1 578 297 032 832 + 0;
  • 1 578 297 032 832 ÷ 2 = 789 148 516 416 + 0;
  • 789 148 516 416 ÷ 2 = 394 574 258 208 + 0;
  • 394 574 258 208 ÷ 2 = 197 287 129 104 + 0;
  • 197 287 129 104 ÷ 2 = 98 643 564 552 + 0;
  • 98 643 564 552 ÷ 2 = 49 321 782 276 + 0;
  • 49 321 782 276 ÷ 2 = 24 660 891 138 + 0;
  • 24 660 891 138 ÷ 2 = 12 330 445 569 + 0;
  • 12 330 445 569 ÷ 2 = 6 165 222 784 + 1;
  • 6 165 222 784 ÷ 2 = 3 082 611 392 + 0;
  • 3 082 611 392 ÷ 2 = 1 541 305 696 + 0;
  • 1 541 305 696 ÷ 2 = 770 652 848 + 0;
  • 770 652 848 ÷ 2 = 385 326 424 + 0;
  • 385 326 424 ÷ 2 = 192 663 212 + 0;
  • 192 663 212 ÷ 2 = 96 331 606 + 0;
  • 96 331 606 ÷ 2 = 48 165 803 + 0;
  • 48 165 803 ÷ 2 = 24 082 901 + 1;
  • 24 082 901 ÷ 2 = 12 041 450 + 1;
  • 12 041 450 ÷ 2 = 6 020 725 + 0;
  • 6 020 725 ÷ 2 = 3 010 362 + 1;
  • 3 010 362 ÷ 2 = 1 505 181 + 0;
  • 1 505 181 ÷ 2 = 752 590 + 1;
  • 752 590 ÷ 2 = 376 295 + 0;
  • 376 295 ÷ 2 = 188 147 + 1;
  • 188 147 ÷ 2 = 94 073 + 1;
  • 94 073 ÷ 2 = 47 036 + 1;
  • 47 036 ÷ 2 = 23 518 + 0;
  • 23 518 ÷ 2 = 11 759 + 0;
  • 11 759 ÷ 2 = 5 879 + 1;
  • 5 879 ÷ 2 = 2 939 + 1;
  • 2 939 ÷ 2 = 1 469 + 1;
  • 1 469 ÷ 2 = 734 + 1;
  • 734 ÷ 2 = 367 + 0;
  • 367 ÷ 2 = 183 + 1;
  • 183 ÷ 2 = 91 + 1;
  • 91 ÷ 2 = 45 + 1;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.

101 011 010 101 257(10) = 101 1011 1101 1110 0111 0101 0110 0000 0010 0000 0000 1001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 101 011 010 101 257(10) converted to signed binary in two's complement representation:

101 011 010 101 257(10) = 0000 0000 0000 0000 0101 1011 1101 1110 0111 0101 0110 0000 0010 0000 0000 1001

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