Convert 101 101 111 111 135 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 101 101 111 111 135(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
101 101 111 111 135 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;
  • 101 101 111 111 135 ÷ 2 = 50 550 555 555 567 + 1;
  • 50 550 555 555 567 ÷ 2 = 25 275 277 777 783 + 1;
  • 25 275 277 777 783 ÷ 2 = 12 637 638 888 891 + 1;
  • 12 637 638 888 891 ÷ 2 = 6 318 819 444 445 + 1;
  • 6 318 819 444 445 ÷ 2 = 3 159 409 722 222 + 1;
  • 3 159 409 722 222 ÷ 2 = 1 579 704 861 111 + 0;
  • 1 579 704 861 111 ÷ 2 = 789 852 430 555 + 1;
  • 789 852 430 555 ÷ 2 = 394 926 215 277 + 1;
  • 394 926 215 277 ÷ 2 = 197 463 107 638 + 1;
  • 197 463 107 638 ÷ 2 = 98 731 553 819 + 0;
  • 98 731 553 819 ÷ 2 = 49 365 776 909 + 1;
  • 49 365 776 909 ÷ 2 = 24 682 888 454 + 1;
  • 24 682 888 454 ÷ 2 = 12 341 444 227 + 0;
  • 12 341 444 227 ÷ 2 = 6 170 722 113 + 1;
  • 6 170 722 113 ÷ 2 = 3 085 361 056 + 1;
  • 3 085 361 056 ÷ 2 = 1 542 680 528 + 0;
  • 1 542 680 528 ÷ 2 = 771 340 264 + 0;
  • 771 340 264 ÷ 2 = 385 670 132 + 0;
  • 385 670 132 ÷ 2 = 192 835 066 + 0;
  • 192 835 066 ÷ 2 = 96 417 533 + 0;
  • 96 417 533 ÷ 2 = 48 208 766 + 1;
  • 48 208 766 ÷ 2 = 24 104 383 + 0;
  • 24 104 383 ÷ 2 = 12 052 191 + 1;
  • 12 052 191 ÷ 2 = 6 026 095 + 1;
  • 6 026 095 ÷ 2 = 3 013 047 + 1;
  • 3 013 047 ÷ 2 = 1 506 523 + 1;
  • 1 506 523 ÷ 2 = 753 261 + 1;
  • 753 261 ÷ 2 = 376 630 + 1;
  • 376 630 ÷ 2 = 188 315 + 0;
  • 188 315 ÷ 2 = 94 157 + 1;
  • 94 157 ÷ 2 = 47 078 + 1;
  • 47 078 ÷ 2 = 23 539 + 0;
  • 23 539 ÷ 2 = 11 769 + 1;
  • 11 769 ÷ 2 = 5 884 + 1;
  • 5 884 ÷ 2 = 2 942 + 0;
  • 2 942 ÷ 2 = 1 471 + 0;
  • 1 471 ÷ 2 = 735 + 1;
  • 735 ÷ 2 = 367 + 1;
  • 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.

101 101 111 111 135(10) = 101 1011 1111 0011 0110 1111 1101 0000 0110 1101 1101 1111(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 101 101 111 111 135(10) converted to signed binary in two's complement representation:

101 101 111 111 135(10) = 0000 0000 0000 0000 0101 1011 1111 0011 0110 1111 1101 0000 0110 1101 1101 1111

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