Convert -10 654 388 379 646 to signed binary, from a base 10 decimal system signed integer number

How to convert the signed integer in decimal system (in base 10):
-10 654 388 379 646(10)
to a signed binary

1. Start with the positive version of the number:

|-10 654 388 379 646| = 10 654 388 379 646

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;
  • 10 654 388 379 646 ÷ 2 = 5 327 194 189 823 + 0;
  • 5 327 194 189 823 ÷ 2 = 2 663 597 094 911 + 1;
  • 2 663 597 094 911 ÷ 2 = 1 331 798 547 455 + 1;
  • 1 331 798 547 455 ÷ 2 = 665 899 273 727 + 1;
  • 665 899 273 727 ÷ 2 = 332 949 636 863 + 1;
  • 332 949 636 863 ÷ 2 = 166 474 818 431 + 1;
  • 166 474 818 431 ÷ 2 = 83 237 409 215 + 1;
  • 83 237 409 215 ÷ 2 = 41 618 704 607 + 1;
  • 41 618 704 607 ÷ 2 = 20 809 352 303 + 1;
  • 20 809 352 303 ÷ 2 = 10 404 676 151 + 1;
  • 10 404 676 151 ÷ 2 = 5 202 338 075 + 1;
  • 5 202 338 075 ÷ 2 = 2 601 169 037 + 1;
  • 2 601 169 037 ÷ 2 = 1 300 584 518 + 1;
  • 1 300 584 518 ÷ 2 = 650 292 259 + 0;
  • 650 292 259 ÷ 2 = 325 146 129 + 1;
  • 325 146 129 ÷ 2 = 162 573 064 + 1;
  • 162 573 064 ÷ 2 = 81 286 532 + 0;
  • 81 286 532 ÷ 2 = 40 643 266 + 0;
  • 40 643 266 ÷ 2 = 20 321 633 + 0;
  • 20 321 633 ÷ 2 = 10 160 816 + 1;
  • 10 160 816 ÷ 2 = 5 080 408 + 0;
  • 5 080 408 ÷ 2 = 2 540 204 + 0;
  • 2 540 204 ÷ 2 = 1 270 102 + 0;
  • 1 270 102 ÷ 2 = 635 051 + 0;
  • 635 051 ÷ 2 = 317 525 + 1;
  • 317 525 ÷ 2 = 158 762 + 1;
  • 158 762 ÷ 2 = 79 381 + 0;
  • 79 381 ÷ 2 = 39 690 + 1;
  • 39 690 ÷ 2 = 19 845 + 0;
  • 19 845 ÷ 2 = 9 922 + 1;
  • 9 922 ÷ 2 = 4 961 + 0;
  • 4 961 ÷ 2 = 2 480 + 1;
  • 2 480 ÷ 2 = 1 240 + 0;
  • 1 240 ÷ 2 = 620 + 0;
  • 620 ÷ 2 = 310 + 0;
  • 310 ÷ 2 = 155 + 0;
  • 155 ÷ 2 = 77 + 1;
  • 77 ÷ 2 = 38 + 1;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 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.

10 654 388 379 646(10) = 1001 1011 0000 1010 1011 0000 1000 1101 1111 1111 1110(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; ...

First bit (the leftmost) is reserved for the sign:
0 = positive integer number, 1 = negative integer number

The least number that is:


a power of 2


and is larger than the actual length, 44,


so that the first bit (leftmost) could be zero


(we deal with a positive number at this moment)


is: 64.


5. 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:

10 654 388 379 646(10) = 0000 0000 0000 0000 0000 1001 1011 0000 1010 1011 0000 1000 1101 1111 1111 1110


6. Get the negative integer number representation:

To get the negative integer number representation on 64 bits (8 Bytes),


change the first bit (the leftmost), from 0 to 1:


-10 654 388 379 646(10) =


1000 0000 0000 0000 0000 1001 1011 0000 1010 1011 0000 1000 1101 1111 1111 1110


Conclusion:

Number -10 654 388 379 646, a signed integer, converted from decimal system (base 10) to signed binary:

-10 654 388 379 646(10) = 1000 0000 0000 0000 0000 1001 1011 0000 1010 1011 0000 1000 1101 1111 1111 1110

First bit (the leftmost) is reserved for the sign:
0 = positive integer number, 1 = negative integer number

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


More operations of this kind:

-10 654 388 379 647 = ? | Signed integer -10 654 388 379 645 = ?


Convert signed integer numbers from the decimal system (base ten) to signed binary

How to convert a base 10 signed integer number to signed binary:

1) Divide the positive version of number repeatedly by 2, keeping track of each remainder, till getting a quotient that is 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, change the first bit (the leftmost), from 0 to 1. The leftmost bit is reserved for the sign, 1 = negative, 0 = positive.

Latest signed integer numbers in decimal (base ten) converted to signed binary

How to convert signed integers from decimal system to binary code system

Follow the steps below to convert a signed base ten integer number to signed binary:

  • 1. In a signed binary, first bit (the leftmost) is reserved for sign: 0 = positive integer number, 1 = positive integer number. If the number to be converted is negative, start with its positive version.
  • 2. Divide repeatedly by 2 the positive integer number keeping track of each remainder. STOP when 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 have a length of 4, 8, 16, 32, 64, ... bits (power of 2) - if needed, fill in extra '0' bits in front of the base 2 number (to the left), up to the right length; this way the first bit (the leftmost one) is always '0', as for a positive representation.
  • 5. To get the negative reprezentation of the number, simply switch the first bit (the leftmost one), from '0' to '1'.

Example: convert the negative number -63 from decimal system (base ten) to signed binary code system:

  • 1. Start with the positive version of the number: |-63| = 63;
  • 2. Divide repeatedly 63 by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder
    • 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, by taking all the remainders starting from the bottom of the list constructed above:
    63(10) = 11 1111(2)
  • 4. The actual length of base 2 representation number is 6, so the positive binary computer representation length of the signed binary will take in this case 8 bits (the least power of 2 higher than 6) - add extra '0's in front (to the left), up to the required length; this way the first bit (the leftmost one) is to be '0', as for a positive number:
    63(10) = 0011 1111(2)
  • 5. To get the negative integer number representation simply change the first bit (the leftmost), from '0' to '1':
    -63(10) = 1011 1111
  • Number -63(10), signed integer, converted from decimal system (base 10) to signed binary = 1011 1111