Convert 100 010 001 000 005 to a Signed Binary (Base 2)

How to convert 100 010 001 000 005(10), a signed base 10 integer number? How to write it as a signed binary code in base 2

What are the required steps to convert base 10 integer
number 100 010 001 000 005 to signed binary code (in base 2)?

  • A signed integer, written in base ten, or a decimal system number, is a number written using the digits 0 through 9 and the sign, which can be positive (+) or negative (-). If positive, the sign is usually not written. A number written in base two, or binary, is a number written using only the digits 0 and 1.

1. Divide the number repeatedly by 2:

Keep track of each remainder.

Stop when you get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 100 010 001 000 005 ÷ 2 = 50 005 000 500 002 + 1;
  • 50 005 000 500 002 ÷ 2 = 25 002 500 250 001 + 0;
  • 25 002 500 250 001 ÷ 2 = 12 501 250 125 000 + 1;
  • 12 501 250 125 000 ÷ 2 = 6 250 625 062 500 + 0;
  • 6 250 625 062 500 ÷ 2 = 3 125 312 531 250 + 0;
  • 3 125 312 531 250 ÷ 2 = 1 562 656 265 625 + 0;
  • 1 562 656 265 625 ÷ 2 = 781 328 132 812 + 1;
  • 781 328 132 812 ÷ 2 = 390 664 066 406 + 0;
  • 390 664 066 406 ÷ 2 = 195 332 033 203 + 0;
  • 195 332 033 203 ÷ 2 = 97 666 016 601 + 1;
  • 97 666 016 601 ÷ 2 = 48 833 008 300 + 1;
  • 48 833 008 300 ÷ 2 = 24 416 504 150 + 0;
  • 24 416 504 150 ÷ 2 = 12 208 252 075 + 0;
  • 12 208 252 075 ÷ 2 = 6 104 126 037 + 1;
  • 6 104 126 037 ÷ 2 = 3 052 063 018 + 1;
  • 3 052 063 018 ÷ 2 = 1 526 031 509 + 0;
  • 1 526 031 509 ÷ 2 = 763 015 754 + 1;
  • 763 015 754 ÷ 2 = 381 507 877 + 0;
  • 381 507 877 ÷ 2 = 190 753 938 + 1;
  • 190 753 938 ÷ 2 = 95 376 969 + 0;
  • 95 376 969 ÷ 2 = 47 688 484 + 1;
  • 47 688 484 ÷ 2 = 23 844 242 + 0;
  • 23 844 242 ÷ 2 = 11 922 121 + 0;
  • 11 922 121 ÷ 2 = 5 961 060 + 1;
  • 5 961 060 ÷ 2 = 2 980 530 + 0;
  • 2 980 530 ÷ 2 = 1 490 265 + 0;
  • 1 490 265 ÷ 2 = 745 132 + 1;
  • 745 132 ÷ 2 = 372 566 + 0;
  • 372 566 ÷ 2 = 186 283 + 0;
  • 186 283 ÷ 2 = 93 141 + 1;
  • 93 141 ÷ 2 = 46 570 + 1;
  • 46 570 ÷ 2 = 23 285 + 0;
  • 23 285 ÷ 2 = 11 642 + 1;
  • 11 642 ÷ 2 = 5 821 + 0;
  • 5 821 ÷ 2 = 2 910 + 1;
  • 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.

100 010 001 000 005(10) = 101 1010 1111 0101 0110 0100 1001 0101 0110 0110 0100 0101(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; ...
  • The first bit (the leftmost) is reserved for the sign:
  • 0 = positive integer number, 1 = negative integer number

The least number that is:

1) a power of 2

2) and is larger than the actual length, 47,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


4. Get the 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:


100 010 001 000 005(10) Base 10 integer number converted and written as a signed binary code (in base 2):

100 010 001 000 005(10) = 0000 0000 0000 0000 0101 1010 1111 0101 0110 0100 1001 0101 0110 0110 0100 0101

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

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How to convert signed base 10 integers in decimal 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