Convert -2 146 233 054 to a signed binary in two's complement representation, from a signed integer number in base 10 decimal system

How to convert a signed integer in decimal system (in base 10):
-2 146 233 054(10)
to a signed binary two's complement representation

1. Start with the positive version of the number:

|-2 146 233 054| = 2 146 233 054

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;
  • 2 146 233 054 ÷ 2 = 1 073 116 527 + 0;
  • 1 073 116 527 ÷ 2 = 536 558 263 + 1;
  • 536 558 263 ÷ 2 = 268 279 131 + 1;
  • 268 279 131 ÷ 2 = 134 139 565 + 1;
  • 134 139 565 ÷ 2 = 67 069 782 + 1;
  • 67 069 782 ÷ 2 = 33 534 891 + 0;
  • 33 534 891 ÷ 2 = 16 767 445 + 1;
  • 16 767 445 ÷ 2 = 8 383 722 + 1;
  • 8 383 722 ÷ 2 = 4 191 861 + 0;
  • 4 191 861 ÷ 2 = 2 095 930 + 1;
  • 2 095 930 ÷ 2 = 1 047 965 + 0;
  • 1 047 965 ÷ 2 = 523 982 + 1;
  • 523 982 ÷ 2 = 261 991 + 0;
  • 261 991 ÷ 2 = 130 995 + 1;
  • 130 995 ÷ 2 = 65 497 + 1;
  • 65 497 ÷ 2 = 32 748 + 1;
  • 32 748 ÷ 2 = 16 374 + 0;
  • 16 374 ÷ 2 = 8 187 + 0;
  • 8 187 ÷ 2 = 4 093 + 1;
  • 4 093 ÷ 2 = 2 046 + 1;
  • 2 046 ÷ 2 = 1 023 + 0;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 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:

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

2 146 233 054(10) = 111 1111 1110 1100 1110 1010 1101 1110(2)


4. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 31.

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; ...

First bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.

The least number that is:


a power of 2


and is larger than the actual length, 31,


so that the first bit (leftmost) could be zero


(we deal with a positive number at this moment)


is: 32.


5. Positive binary computer representation on 32 bits (4 Bytes):

If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32:

2 146 233 054(10) = 0111 1111 1110 1100 1110 1010 1101 1110


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

To get the negative integer number representation on 32 bits (4 Bytes),


signed binary one's complement,


replace all the bits on 0 with 1s


and all the bits set on 1 with 0s


(reverse the digits, flip the digits)


!(0111 1111 1110 1100 1110 1010 1101 1110) =


1000 0000 0001 0011 0001 0101 0010 0001


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

To get the negative integer number representation on 32 bits (4 Bytes),


signed binary two's complement,


add 1 to the number calculated above


1000 0000 0001 0011 0001 0101 0010 0001 + 1 =


1000 0000 0001 0011 0001 0101 0010 0010


Conclusion:

Number -2 146 233 054, a signed integer, converted from decimal system (base 10) to a signed binary two's complement representation:

-2 146 233 054(10) = 1000 0000 0001 0011 0001 0101 0010 0010

Spaces used to group digits: for binary, by 4; for decimal, by 3.


More operations of this kind:

-2 146 233 055 = ? | -2 146 233 053 = ?


Convert signed integer numbers from the decimal system (base ten) to signed binary two's complement representation

How to convert a base 10 signed integer number to signed binary in two's complement representation:

1) Divide the positive version of number repeatedly by 2, keeping track of each remainder, till getting a quotient that is equal to 0.

2) Construct the base 2 representation by taking the previously calculated remainders starting from the last remainder up to the first one, in that order.

3) Construct the positive binary computer representation so that the first bit is 0.

4) Only if the initial number is negative, switch all the bits from 0 to 1 and from 1 to 0 (reversing the digits).

5) Only if the initial number is negative, add 1 to the number at the previous point.

Latest signed integers converted from decimal system to binary two's complement representation

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