Convert 10 000 100 011 151 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 000 100 011 151(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 000 100 011 151 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;
  • 10 000 100 011 151 ÷ 2 = 5 000 050 005 575 + 1;
  • 5 000 050 005 575 ÷ 2 = 2 500 025 002 787 + 1;
  • 2 500 025 002 787 ÷ 2 = 1 250 012 501 393 + 1;
  • 1 250 012 501 393 ÷ 2 = 625 006 250 696 + 1;
  • 625 006 250 696 ÷ 2 = 312 503 125 348 + 0;
  • 312 503 125 348 ÷ 2 = 156 251 562 674 + 0;
  • 156 251 562 674 ÷ 2 = 78 125 781 337 + 0;
  • 78 125 781 337 ÷ 2 = 39 062 890 668 + 1;
  • 39 062 890 668 ÷ 2 = 19 531 445 334 + 0;
  • 19 531 445 334 ÷ 2 = 9 765 722 667 + 0;
  • 9 765 722 667 ÷ 2 = 4 882 861 333 + 1;
  • 4 882 861 333 ÷ 2 = 2 441 430 666 + 1;
  • 2 441 430 666 ÷ 2 = 1 220 715 333 + 0;
  • 1 220 715 333 ÷ 2 = 610 357 666 + 1;
  • 610 357 666 ÷ 2 = 305 178 833 + 0;
  • 305 178 833 ÷ 2 = 152 589 416 + 1;
  • 152 589 416 ÷ 2 = 76 294 708 + 0;
  • 76 294 708 ÷ 2 = 38 147 354 + 0;
  • 38 147 354 ÷ 2 = 19 073 677 + 0;
  • 19 073 677 ÷ 2 = 9 536 838 + 1;
  • 9 536 838 ÷ 2 = 4 768 419 + 0;
  • 4 768 419 ÷ 2 = 2 384 209 + 1;
  • 2 384 209 ÷ 2 = 1 192 104 + 1;
  • 1 192 104 ÷ 2 = 596 052 + 0;
  • 596 052 ÷ 2 = 298 026 + 0;
  • 298 026 ÷ 2 = 149 013 + 0;
  • 149 013 ÷ 2 = 74 506 + 1;
  • 74 506 ÷ 2 = 37 253 + 0;
  • 37 253 ÷ 2 = 18 626 + 1;
  • 18 626 ÷ 2 = 9 313 + 0;
  • 9 313 ÷ 2 = 4 656 + 1;
  • 4 656 ÷ 2 = 2 328 + 0;
  • 2 328 ÷ 2 = 1 164 + 0;
  • 1 164 ÷ 2 = 582 + 0;
  • 582 ÷ 2 = 291 + 0;
  • 291 ÷ 2 = 145 + 1;
  • 145 ÷ 2 = 72 + 1;
  • 72 ÷ 2 = 36 + 0;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 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.

10 000 100 011 151(10) = 1001 0001 1000 0101 0100 0110 1000 1010 1100 1000 1111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 44.

  • 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, 44,

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 10 000 100 011 151(10) converted to signed binary in two's complement representation:

10 000 100 011 151(10) = 0000 0000 0000 0000 0000 1001 0001 1000 0101 0100 0110 1000 1010 1100 1000 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