Convert 1 921 681 003 to signed binary, from a base 10 decimal system signed integer number

How to convert the signed integer in decimal system (in base 10):
1 921 681 003(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;
  • 1 921 681 003 ÷ 2 = 960 840 501 + 1;
  • 960 840 501 ÷ 2 = 480 420 250 + 1;
  • 480 420 250 ÷ 2 = 240 210 125 + 0;
  • 240 210 125 ÷ 2 = 120 105 062 + 1;
  • 120 105 062 ÷ 2 = 60 052 531 + 0;
  • 60 052 531 ÷ 2 = 30 026 265 + 1;
  • 30 026 265 ÷ 2 = 15 013 132 + 1;
  • 15 013 132 ÷ 2 = 7 506 566 + 0;
  • 7 506 566 ÷ 2 = 3 753 283 + 0;
  • 3 753 283 ÷ 2 = 1 876 641 + 1;
  • 1 876 641 ÷ 2 = 938 320 + 1;
  • 938 320 ÷ 2 = 469 160 + 0;
  • 469 160 ÷ 2 = 234 580 + 0;
  • 234 580 ÷ 2 = 117 290 + 0;
  • 117 290 ÷ 2 = 58 645 + 0;
  • 58 645 ÷ 2 = 29 322 + 1;
  • 29 322 ÷ 2 = 14 661 + 0;
  • 14 661 ÷ 2 = 7 330 + 1;
  • 7 330 ÷ 2 = 3 665 + 0;
  • 3 665 ÷ 2 = 1 832 + 1;
  • 1 832 ÷ 2 = 916 + 0;
  • 916 ÷ 2 = 458 + 0;
  • 458 ÷ 2 = 229 + 0;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 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.

1 921 681 003(10) = 111 0010 1000 1010 1000 0110 0110 1011(2)


3. 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) 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, 31,


so that the first bit (leftmost) could be zero


(we deal with a positive number at this moment)


is: 32.


4. 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:

1 921 681 003(10) = 0111 0010 1000 1010 1000 0110 0110 1011


Conclusion:

Number 1 921 681 003, a signed integer, converted from decimal system (base 10) to signed binary:

1 921 681 003(10) = 0111 0010 1000 1010 1000 0110 0110 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:

1 921 681 002 = ? | Signed integer 1 921 681 004 = ?


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