Convert -10 001 000 011 415 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -10 001 000 011 415(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-10 001 000 011 415 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. Start with the positive version of the number:

|-10 001 000 011 415| = 10 001 000 011 415

2. 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;
  • 10 001 000 011 415 ÷ 2 = 5 000 500 005 707 + 1;
  • 5 000 500 005 707 ÷ 2 = 2 500 250 002 853 + 1;
  • 2 500 250 002 853 ÷ 2 = 1 250 125 001 426 + 1;
  • 1 250 125 001 426 ÷ 2 = 625 062 500 713 + 0;
  • 625 062 500 713 ÷ 2 = 312 531 250 356 + 1;
  • 312 531 250 356 ÷ 2 = 156 265 625 178 + 0;
  • 156 265 625 178 ÷ 2 = 78 132 812 589 + 0;
  • 78 132 812 589 ÷ 2 = 39 066 406 294 + 1;
  • 39 066 406 294 ÷ 2 = 19 533 203 147 + 0;
  • 19 533 203 147 ÷ 2 = 9 766 601 573 + 1;
  • 9 766 601 573 ÷ 2 = 4 883 300 786 + 1;
  • 4 883 300 786 ÷ 2 = 2 441 650 393 + 0;
  • 2 441 650 393 ÷ 2 = 1 220 825 196 + 1;
  • 1 220 825 196 ÷ 2 = 610 412 598 + 0;
  • 610 412 598 ÷ 2 = 305 206 299 + 0;
  • 305 206 299 ÷ 2 = 152 603 149 + 1;
  • 152 603 149 ÷ 2 = 76 301 574 + 1;
  • 76 301 574 ÷ 2 = 38 150 787 + 0;
  • 38 150 787 ÷ 2 = 19 075 393 + 1;
  • 19 075 393 ÷ 2 = 9 537 696 + 1;
  • 9 537 696 ÷ 2 = 4 768 848 + 0;
  • 4 768 848 ÷ 2 = 2 384 424 + 0;
  • 2 384 424 ÷ 2 = 1 192 212 + 0;
  • 1 192 212 ÷ 2 = 596 106 + 0;
  • 596 106 ÷ 2 = 298 053 + 0;
  • 298 053 ÷ 2 = 149 026 + 1;
  • 149 026 ÷ 2 = 74 513 + 0;
  • 74 513 ÷ 2 = 37 256 + 1;
  • 37 256 ÷ 2 = 18 628 + 0;
  • 18 628 ÷ 2 = 9 314 + 0;
  • 9 314 ÷ 2 = 4 657 + 0;
  • 4 657 ÷ 2 = 2 328 + 1;
  • 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;

3. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

10 001 000 011 415(10) = 1001 0001 1000 1000 1010 0000 1101 1001 0110 1001 0111(2)

4. 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.


5. 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.


10 001 000 011 415(10) = 0000 0000 0000 0000 0000 1001 0001 1000 1000 1010 0000 1101 1001 0110 1001 0111

6. Get the negative integer number representation. Part 1:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation, replace all the bits on 0 with 1s and all the bits set on 1 with 0s.

Reverse the digits, flip the digits:

Replace the bits set on 0 with 1s and the bits set on 1 with 0s.

!(0000 0000 0000 0000 0000 1001 0001 1000 1000 1010 0000 1101 1001 0110 1001 0111)

= 1111 1111 1111 1111 1111 0110 1110 0111 0111 0101 1111 0010 0110 1001 0110 1000


7. Get the negative integer number representation. Part 2:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in two's complement representation, add 1 to the number calculated above 1111 1111 1111 1111 1111 0110 1110 0111 0111 0101 1111 0010 0110 1001 0110 1000 (to the signed binary in one's complement representation).

Binary addition carries on a value of 2:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 10 = 11
  • 1 + 11 = 100

Add 1 to the number calculated above
(to the signed binary number in one's complement representation):

-10 001 000 011 415 =

1111 1111 1111 1111 1111 0110 1110 0111 0111 0101 1111 0010 0110 1001 0110 1000 + 1


Decimal Number -10 001 000 011 415(10) converted to signed binary in two's complement representation:

-10 001 000 011 415(10) = 1111 1111 1111 1111 1111 0110 1110 0111 0111 0101 1111 0010 0110 1001 0110 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