Convert 10 111 011 101 115 to a Signed Binary (Base 2)

How to convert 10 111 011 101 115(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 10 111 011 101 115 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;
  • 10 111 011 101 115 ÷ 2 = 5 055 505 550 557 + 1;
  • 5 055 505 550 557 ÷ 2 = 2 527 752 775 278 + 1;
  • 2 527 752 775 278 ÷ 2 = 1 263 876 387 639 + 0;
  • 1 263 876 387 639 ÷ 2 = 631 938 193 819 + 1;
  • 631 938 193 819 ÷ 2 = 315 969 096 909 + 1;
  • 315 969 096 909 ÷ 2 = 157 984 548 454 + 1;
  • 157 984 548 454 ÷ 2 = 78 992 274 227 + 0;
  • 78 992 274 227 ÷ 2 = 39 496 137 113 + 1;
  • 39 496 137 113 ÷ 2 = 19 748 068 556 + 1;
  • 19 748 068 556 ÷ 2 = 9 874 034 278 + 0;
  • 9 874 034 278 ÷ 2 = 4 937 017 139 + 0;
  • 4 937 017 139 ÷ 2 = 2 468 508 569 + 1;
  • 2 468 508 569 ÷ 2 = 1 234 254 284 + 1;
  • 1 234 254 284 ÷ 2 = 617 127 142 + 0;
  • 617 127 142 ÷ 2 = 308 563 571 + 0;
  • 308 563 571 ÷ 2 = 154 281 785 + 1;
  • 154 281 785 ÷ 2 = 77 140 892 + 1;
  • 77 140 892 ÷ 2 = 38 570 446 + 0;
  • 38 570 446 ÷ 2 = 19 285 223 + 0;
  • 19 285 223 ÷ 2 = 9 642 611 + 1;
  • 9 642 611 ÷ 2 = 4 821 305 + 1;
  • 4 821 305 ÷ 2 = 2 410 652 + 1;
  • 2 410 652 ÷ 2 = 1 205 326 + 0;
  • 1 205 326 ÷ 2 = 602 663 + 0;
  • 602 663 ÷ 2 = 301 331 + 1;
  • 301 331 ÷ 2 = 150 665 + 1;
  • 150 665 ÷ 2 = 75 332 + 1;
  • 75 332 ÷ 2 = 37 666 + 0;
  • 37 666 ÷ 2 = 18 833 + 0;
  • 18 833 ÷ 2 = 9 416 + 1;
  • 9 416 ÷ 2 = 4 708 + 0;
  • 4 708 ÷ 2 = 2 354 + 0;
  • 2 354 ÷ 2 = 1 177 + 0;
  • 1 177 ÷ 2 = 588 + 1;
  • 588 ÷ 2 = 294 + 0;
  • 294 ÷ 2 = 147 + 0;
  • 147 ÷ 2 = 73 + 1;
  • 73 ÷ 2 = 36 + 1;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 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.

10 111 011 101 115(10) = 1001 0011 0010 0010 0111 0011 1001 1001 1001 1011 1011(2)


3. 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; ...
  • 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, 44,

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:


10 111 011 101 115(10) Base 10 integer number converted and written as a signed binary code (in base 2):

10 111 011 101 115(10) = 0000 0000 0000 0000 0000 1001 0011 0010 0010 0111 0011 1001 1001 1001 1011 1011

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