Integer to Signed Binary: Number 106 048 781 380 917 Converted and Written as a Signed Binary. Base Ten Decimal System Conversion

Integer number 106 048 781 380 917(10) written as a signed binary number

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;
  • 106 048 781 380 917 ÷ 2 = 53 024 390 690 458 + 1;
  • 53 024 390 690 458 ÷ 2 = 26 512 195 345 229 + 0;
  • 26 512 195 345 229 ÷ 2 = 13 256 097 672 614 + 1;
  • 13 256 097 672 614 ÷ 2 = 6 628 048 836 307 + 0;
  • 6 628 048 836 307 ÷ 2 = 3 314 024 418 153 + 1;
  • 3 314 024 418 153 ÷ 2 = 1 657 012 209 076 + 1;
  • 1 657 012 209 076 ÷ 2 = 828 506 104 538 + 0;
  • 828 506 104 538 ÷ 2 = 414 253 052 269 + 0;
  • 414 253 052 269 ÷ 2 = 207 126 526 134 + 1;
  • 207 126 526 134 ÷ 2 = 103 563 263 067 + 0;
  • 103 563 263 067 ÷ 2 = 51 781 631 533 + 1;
  • 51 781 631 533 ÷ 2 = 25 890 815 766 + 1;
  • 25 890 815 766 ÷ 2 = 12 945 407 883 + 0;
  • 12 945 407 883 ÷ 2 = 6 472 703 941 + 1;
  • 6 472 703 941 ÷ 2 = 3 236 351 970 + 1;
  • 3 236 351 970 ÷ 2 = 1 618 175 985 + 0;
  • 1 618 175 985 ÷ 2 = 809 087 992 + 1;
  • 809 087 992 ÷ 2 = 404 543 996 + 0;
  • 404 543 996 ÷ 2 = 202 271 998 + 0;
  • 202 271 998 ÷ 2 = 101 135 999 + 0;
  • 101 135 999 ÷ 2 = 50 567 999 + 1;
  • 50 567 999 ÷ 2 = 25 283 999 + 1;
  • 25 283 999 ÷ 2 = 12 641 999 + 1;
  • 12 641 999 ÷ 2 = 6 320 999 + 1;
  • 6 320 999 ÷ 2 = 3 160 499 + 1;
  • 3 160 499 ÷ 2 = 1 580 249 + 1;
  • 1 580 249 ÷ 2 = 790 124 + 1;
  • 790 124 ÷ 2 = 395 062 + 0;
  • 395 062 ÷ 2 = 197 531 + 0;
  • 197 531 ÷ 2 = 98 765 + 1;
  • 98 765 ÷ 2 = 49 382 + 1;
  • 49 382 ÷ 2 = 24 691 + 0;
  • 24 691 ÷ 2 = 12 345 + 1;
  • 12 345 ÷ 2 = 6 172 + 1;
  • 6 172 ÷ 2 = 3 086 + 0;
  • 3 086 ÷ 2 = 1 543 + 0;
  • 1 543 ÷ 2 = 771 + 1;
  • 771 ÷ 2 = 385 + 1;
  • 385 ÷ 2 = 192 + 1;
  • 192 ÷ 2 = 96 + 0;
  • 96 ÷ 2 = 48 + 0;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

106 048 781 380 917(10) = 110 0000 0111 0011 0110 0111 1111 0001 0110 1101 0011 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:


Number 106 048 781 380 917(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):

106 048 781 380 917(10) = 0000 0000 0000 0000 0110 0000 0111 0011 0110 0111 1111 0001 0110 1101 0011 0101

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

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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