Convert 1 100 101 011 111 126 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 100 101 011 111 126(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 100 101 011 111 126 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;
  • 1 100 101 011 111 126 ÷ 2 = 550 050 505 555 563 + 0;
  • 550 050 505 555 563 ÷ 2 = 275 025 252 777 781 + 1;
  • 275 025 252 777 781 ÷ 2 = 137 512 626 388 890 + 1;
  • 137 512 626 388 890 ÷ 2 = 68 756 313 194 445 + 0;
  • 68 756 313 194 445 ÷ 2 = 34 378 156 597 222 + 1;
  • 34 378 156 597 222 ÷ 2 = 17 189 078 298 611 + 0;
  • 17 189 078 298 611 ÷ 2 = 8 594 539 149 305 + 1;
  • 8 594 539 149 305 ÷ 2 = 4 297 269 574 652 + 1;
  • 4 297 269 574 652 ÷ 2 = 2 148 634 787 326 + 0;
  • 2 148 634 787 326 ÷ 2 = 1 074 317 393 663 + 0;
  • 1 074 317 393 663 ÷ 2 = 537 158 696 831 + 1;
  • 537 158 696 831 ÷ 2 = 268 579 348 415 + 1;
  • 268 579 348 415 ÷ 2 = 134 289 674 207 + 1;
  • 134 289 674 207 ÷ 2 = 67 144 837 103 + 1;
  • 67 144 837 103 ÷ 2 = 33 572 418 551 + 1;
  • 33 572 418 551 ÷ 2 = 16 786 209 275 + 1;
  • 16 786 209 275 ÷ 2 = 8 393 104 637 + 1;
  • 8 393 104 637 ÷ 2 = 4 196 552 318 + 1;
  • 4 196 552 318 ÷ 2 = 2 098 276 159 + 0;
  • 2 098 276 159 ÷ 2 = 1 049 138 079 + 1;
  • 1 049 138 079 ÷ 2 = 524 569 039 + 1;
  • 524 569 039 ÷ 2 = 262 284 519 + 1;
  • 262 284 519 ÷ 2 = 131 142 259 + 1;
  • 131 142 259 ÷ 2 = 65 571 129 + 1;
  • 65 571 129 ÷ 2 = 32 785 564 + 1;
  • 32 785 564 ÷ 2 = 16 392 782 + 0;
  • 16 392 782 ÷ 2 = 8 196 391 + 0;
  • 8 196 391 ÷ 2 = 4 098 195 + 1;
  • 4 098 195 ÷ 2 = 2 049 097 + 1;
  • 2 049 097 ÷ 2 = 1 024 548 + 1;
  • 1 024 548 ÷ 2 = 512 274 + 0;
  • 512 274 ÷ 2 = 256 137 + 0;
  • 256 137 ÷ 2 = 128 068 + 1;
  • 128 068 ÷ 2 = 64 034 + 0;
  • 64 034 ÷ 2 = 32 017 + 0;
  • 32 017 ÷ 2 = 16 008 + 1;
  • 16 008 ÷ 2 = 8 004 + 0;
  • 8 004 ÷ 2 = 4 002 + 0;
  • 4 002 ÷ 2 = 2 001 + 0;
  • 2 001 ÷ 2 = 1 000 + 1;
  • 1 000 ÷ 2 = 500 + 0;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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.

1 100 101 011 111 126(10) = 11 1110 1000 1000 1001 0011 1001 1111 1011 1111 1100 1101 0110(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 1 100 101 011 111 126(10) converted to signed binary in two's complement representation:

1 100 101 011 111 126(10) = 0000 0000 0000 0011 1110 1000 1000 1001 0011 1001 1111 1011 1111 1100 1101 0110

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