Convert 100 000 111 110 011 to signed binary, from a base 10 decimal system signed integer number

How to convert the signed integer in decimal system (in base 10):
100 000 111 110 011(10)
to a signed binary

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;
  • 100 000 111 110 011 ÷ 2 = 50 000 055 555 005 + 1;
  • 50 000 055 555 005 ÷ 2 = 25 000 027 777 502 + 1;
  • 25 000 027 777 502 ÷ 2 = 12 500 013 888 751 + 0;
  • 12 500 013 888 751 ÷ 2 = 6 250 006 944 375 + 1;
  • 6 250 006 944 375 ÷ 2 = 3 125 003 472 187 + 1;
  • 3 125 003 472 187 ÷ 2 = 1 562 501 736 093 + 1;
  • 1 562 501 736 093 ÷ 2 = 781 250 868 046 + 1;
  • 781 250 868 046 ÷ 2 = 390 625 434 023 + 0;
  • 390 625 434 023 ÷ 2 = 195 312 717 011 + 1;
  • 195 312 717 011 ÷ 2 = 97 656 358 505 + 1;
  • 97 656 358 505 ÷ 2 = 48 828 179 252 + 1;
  • 48 828 179 252 ÷ 2 = 24 414 089 626 + 0;
  • 24 414 089 626 ÷ 2 = 12 207 044 813 + 0;
  • 12 207 044 813 ÷ 2 = 6 103 522 406 + 1;
  • 6 103 522 406 ÷ 2 = 3 051 761 203 + 0;
  • 3 051 761 203 ÷ 2 = 1 525 880 601 + 1;
  • 1 525 880 601 ÷ 2 = 762 940 300 + 1;
  • 762 940 300 ÷ 2 = 381 470 150 + 0;
  • 381 470 150 ÷ 2 = 190 735 075 + 0;
  • 190 735 075 ÷ 2 = 95 367 537 + 1;
  • 95 367 537 ÷ 2 = 47 683 768 + 1;
  • 47 683 768 ÷ 2 = 23 841 884 + 0;
  • 23 841 884 ÷ 2 = 11 920 942 + 0;
  • 11 920 942 ÷ 2 = 5 960 471 + 0;
  • 5 960 471 ÷ 2 = 2 980 235 + 1;
  • 2 980 235 ÷ 2 = 1 490 117 + 1;
  • 1 490 117 ÷ 2 = 745 058 + 1;
  • 745 058 ÷ 2 = 372 529 + 0;
  • 372 529 ÷ 2 = 186 264 + 1;
  • 186 264 ÷ 2 = 93 132 + 0;
  • 93 132 ÷ 2 = 46 566 + 0;
  • 46 566 ÷ 2 = 23 283 + 0;
  • 23 283 ÷ 2 = 11 641 + 1;
  • 11 641 ÷ 2 = 5 820 + 1;
  • 5 820 ÷ 2 = 2 910 + 0;
  • 2 910 ÷ 2 = 1 455 + 0;
  • 1 455 ÷ 2 = 727 + 1;
  • 727 ÷ 2 = 363 + 1;
  • 363 ÷ 2 = 181 + 1;
  • 181 ÷ 2 = 90 + 1;
  • 90 ÷ 2 = 45 + 0;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.

100 000 111 110 011(10) = 101 1010 1111 0011 0001 0111 0001 1001 1010 0111 0111 1011(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 47.

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, 47,


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:

100 000 111 110 011(10) = 0000 0000 0000 0000 0101 1010 1111 0011 0001 0111 0001 1001 1010 0111 0111 1011


Conclusion:

Number 100 000 111 110 011, a signed integer, converted from decimal system (base 10) to signed binary:

100 000 111 110 011(10) = 0000 0000 0000 0000 0101 1010 1111 0011 0001 0111 0001 1001 1010 0111 0111 1011

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:

100 000 111 110 010 = ? | Signed integer 100 000 111 110 012 = ?


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