Signed Binary: Integer Number 654 785 134 Converted and Written as a Signed Binary. Base Ten Decimal System Numbers Conversions

Integer number 654 785 134(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;
  • 654 785 134 ÷ 2 = 327 392 567 + 0;
  • 327 392 567 ÷ 2 = 163 696 283 + 1;
  • 163 696 283 ÷ 2 = 81 848 141 + 1;
  • 81 848 141 ÷ 2 = 40 924 070 + 1;
  • 40 924 070 ÷ 2 = 20 462 035 + 0;
  • 20 462 035 ÷ 2 = 10 231 017 + 1;
  • 10 231 017 ÷ 2 = 5 115 508 + 1;
  • 5 115 508 ÷ 2 = 2 557 754 + 0;
  • 2 557 754 ÷ 2 = 1 278 877 + 0;
  • 1 278 877 ÷ 2 = 639 438 + 1;
  • 639 438 ÷ 2 = 319 719 + 0;
  • 319 719 ÷ 2 = 159 859 + 1;
  • 159 859 ÷ 2 = 79 929 + 1;
  • 79 929 ÷ 2 = 39 964 + 1;
  • 39 964 ÷ 2 = 19 982 + 0;
  • 19 982 ÷ 2 = 9 991 + 0;
  • 9 991 ÷ 2 = 4 995 + 1;
  • 4 995 ÷ 2 = 2 497 + 1;
  • 2 497 ÷ 2 = 1 248 + 1;
  • 1 248 ÷ 2 = 624 + 0;
  • 624 ÷ 2 = 312 + 0;
  • 312 ÷ 2 = 156 + 0;
  • 156 ÷ 2 = 78 + 0;
  • 78 ÷ 2 = 39 + 0;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 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.


654 785 134(10) = 10 0111 0000 0111 0011 1010 0110 1110(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 30.

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, 30,

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

=== is: 32.


4. Get the 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:


Number 654 785 134(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):

654 785 134(10) = 0010 0111 0000 0111 0011 1010 0110 1110

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