Signed: Integer ↗ Binary: 1 000 000 000 101 102 Convert the Integer Number to a Signed Binary. Converting and Writing the Base Ten Decimal System Signed Integer as Binary Code (Written in Base Two)

Signed integer number 1 000 000 000 101 102(10)
converted and written as a signed binary (base 2) = ?

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;
  • 1 000 000 000 101 102 ÷ 2 = 500 000 000 050 551 + 0;
  • 500 000 000 050 551 ÷ 2 = 250 000 000 025 275 + 1;
  • 250 000 000 025 275 ÷ 2 = 125 000 000 012 637 + 1;
  • 125 000 000 012 637 ÷ 2 = 62 500 000 006 318 + 1;
  • 62 500 000 006 318 ÷ 2 = 31 250 000 003 159 + 0;
  • 31 250 000 003 159 ÷ 2 = 15 625 000 001 579 + 1;
  • 15 625 000 001 579 ÷ 2 = 7 812 500 000 789 + 1;
  • 7 812 500 000 789 ÷ 2 = 3 906 250 000 394 + 1;
  • 3 906 250 000 394 ÷ 2 = 1 953 125 000 197 + 0;
  • 1 953 125 000 197 ÷ 2 = 976 562 500 098 + 1;
  • 976 562 500 098 ÷ 2 = 488 281 250 049 + 0;
  • 488 281 250 049 ÷ 2 = 244 140 625 024 + 1;
  • 244 140 625 024 ÷ 2 = 122 070 312 512 + 0;
  • 122 070 312 512 ÷ 2 = 61 035 156 256 + 0;
  • 61 035 156 256 ÷ 2 = 30 517 578 128 + 0;
  • 30 517 578 128 ÷ 2 = 15 258 789 064 + 0;
  • 15 258 789 064 ÷ 2 = 7 629 394 532 + 0;
  • 7 629 394 532 ÷ 2 = 3 814 697 266 + 0;
  • 3 814 697 266 ÷ 2 = 1 907 348 633 + 0;
  • 1 907 348 633 ÷ 2 = 953 674 316 + 1;
  • 953 674 316 ÷ 2 = 476 837 158 + 0;
  • 476 837 158 ÷ 2 = 238 418 579 + 0;
  • 238 418 579 ÷ 2 = 119 209 289 + 1;
  • 119 209 289 ÷ 2 = 59 604 644 + 1;
  • 59 604 644 ÷ 2 = 29 802 322 + 0;
  • 29 802 322 ÷ 2 = 14 901 161 + 0;
  • 14 901 161 ÷ 2 = 7 450 580 + 1;
  • 7 450 580 ÷ 2 = 3 725 290 + 0;
  • 3 725 290 ÷ 2 = 1 862 645 + 0;
  • 1 862 645 ÷ 2 = 931 322 + 1;
  • 931 322 ÷ 2 = 465 661 + 0;
  • 465 661 ÷ 2 = 232 830 + 1;
  • 232 830 ÷ 2 = 116 415 + 0;
  • 116 415 ÷ 2 = 58 207 + 1;
  • 58 207 ÷ 2 = 29 103 + 1;
  • 29 103 ÷ 2 = 14 551 + 1;
  • 14 551 ÷ 2 = 7 275 + 1;
  • 7 275 ÷ 2 = 3 637 + 1;
  • 3 637 ÷ 2 = 1 818 + 1;
  • 1 818 ÷ 2 = 909 + 0;
  • 909 ÷ 2 = 454 + 1;
  • 454 ÷ 2 = 227 + 0;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 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.


1 000 000 000 101 102(10) = 11 1000 1101 0111 1110 1010 0100 1100 1000 0000 1010 1110 1110(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 50.

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, 50,

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 1 000 000 000 101 102(10), a signed integer number (with sign),
converted from decimal system (from base 10)
and written as a signed binary (in base 2):

1 000 000 000 101 102(10) = 0000 0000 0000 0011 1000 1101 0111 1110 1010 0100 1100 1000 0000 1010 1110 1110

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

More operations on signed integer numbers:

» Signed integer number 1 000 000 000 101 101 converted from decimal system (from base 10) and written as a signed binary (in base 2) = ?

» Signed integer number 1 000 000 000 101 103 converted from decimal system (from base 10) and written as a signed binary (in base 2) = ?

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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