Convert 1 111 111 111 011 048 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):
1 111 111 111 011 048(10)
to a signed binary two's complement representation

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;
  • 1 111 111 111 011 048 ÷ 2 = 555 555 555 505 524 + 0;
  • 555 555 555 505 524 ÷ 2 = 277 777 777 752 762 + 0;
  • 277 777 777 752 762 ÷ 2 = 138 888 888 876 381 + 0;
  • 138 888 888 876 381 ÷ 2 = 69 444 444 438 190 + 1;
  • 69 444 444 438 190 ÷ 2 = 34 722 222 219 095 + 0;
  • 34 722 222 219 095 ÷ 2 = 17 361 111 109 547 + 1;
  • 17 361 111 109 547 ÷ 2 = 8 680 555 554 773 + 1;
  • 8 680 555 554 773 ÷ 2 = 4 340 277 777 386 + 1;
  • 4 340 277 777 386 ÷ 2 = 2 170 138 888 693 + 0;
  • 2 170 138 888 693 ÷ 2 = 1 085 069 444 346 + 1;
  • 1 085 069 444 346 ÷ 2 = 542 534 722 173 + 0;
  • 542 534 722 173 ÷ 2 = 271 267 361 086 + 1;
  • 271 267 361 086 ÷ 2 = 135 633 680 543 + 0;
  • 135 633 680 543 ÷ 2 = 67 816 840 271 + 1;
  • 67 816 840 271 ÷ 2 = 33 908 420 135 + 1;
  • 33 908 420 135 ÷ 2 = 16 954 210 067 + 1;
  • 16 954 210 067 ÷ 2 = 8 477 105 033 + 1;
  • 8 477 105 033 ÷ 2 = 4 238 552 516 + 1;
  • 4 238 552 516 ÷ 2 = 2 119 276 258 + 0;
  • 2 119 276 258 ÷ 2 = 1 059 638 129 + 0;
  • 1 059 638 129 ÷ 2 = 529 819 064 + 1;
  • 529 819 064 ÷ 2 = 264 909 532 + 0;
  • 264 909 532 ÷ 2 = 132 454 766 + 0;
  • 132 454 766 ÷ 2 = 66 227 383 + 0;
  • 66 227 383 ÷ 2 = 33 113 691 + 1;
  • 33 113 691 ÷ 2 = 16 556 845 + 1;
  • 16 556 845 ÷ 2 = 8 278 422 + 1;
  • 8 278 422 ÷ 2 = 4 139 211 + 0;
  • 4 139 211 ÷ 2 = 2 069 605 + 1;
  • 2 069 605 ÷ 2 = 1 034 802 + 1;
  • 1 034 802 ÷ 2 = 517 401 + 0;
  • 517 401 ÷ 2 = 258 700 + 1;
  • 258 700 ÷ 2 = 129 350 + 0;
  • 129 350 ÷ 2 = 64 675 + 0;
  • 64 675 ÷ 2 = 32 337 + 1;
  • 32 337 ÷ 2 = 16 168 + 1;
  • 16 168 ÷ 2 = 8 084 + 0;
  • 8 084 ÷ 2 = 4 042 + 0;
  • 4 042 ÷ 2 = 2 021 + 0;
  • 2 021 ÷ 2 = 1 010 + 1;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 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.

1 111 111 111 011 048(10) = 11 1111 0010 1000 1100 1011 0111 0001 0011 1110 1010 1110 1000(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; ...

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, 50,


so that the first bit (leftmost) could be zero


(we deal with a positive number at this moment)


is: 64.


4. 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:

1 111 111 111 011 048(10) = 0000 0000 0000 0011 1111 0010 1000 1100 1011 0111 0001 0011 1110 1010 1110 1000


Conclusion:

Number 1 111 111 111 011 048, a signed integer, converted from decimal system (base 10) to a signed binary two's complement representation:

1 111 111 111 011 048(10) = 0000 0000 0000 0011 1111 0010 1000 1100 1011 0111 0001 0011 1110 1010 1110 1000

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


More operations of this kind:

1 111 111 111 011 047 = ? | 1 111 111 111 011 049 = ?


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

1,111,111,111,011,048 to signed binary two's complement = ? Jan 21 01:12 UTC (GMT)
524,272 to signed binary two's complement = ? Jan 21 01:11 UTC (GMT)
100,000,000 to signed binary two's complement = ? Jan 21 01:10 UTC (GMT)
-990 to signed binary two's complement = ? Jan 21 01:10 UTC (GMT)
-1,389 to signed binary two's complement = ? Jan 21 01:09 UTC (GMT)
-259,062 to signed binary two's complement = ? Jan 21 01:08 UTC (GMT)
-2,934,587,398 to signed binary two's complement = ? Jan 21 01:08 UTC (GMT)
1,169,827,092 to signed binary two's complement = ? Jan 21 01:07 UTC (GMT)
88,894 to signed binary two's complement = ? Jan 21 01:07 UTC (GMT)
12,995 to signed binary two's complement = ? Jan 21 01:06 UTC (GMT)
18,133 to signed binary two's complement = ? Jan 21 01:06 UTC (GMT)
-117 to signed binary two's complement = ? Jan 21 01:06 UTC (GMT)
-21 to signed binary two's complement = ? Jan 21 01:06 UTC (GMT)
All decimal integer numbers converted to signed 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