Convert 1 000 100 010 009 934 to a Signed Binary (Base 2)

How to convert 1 000 100 010 009 934(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 1 000 100 010 009 934 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;
  • 1 000 100 010 009 934 ÷ 2 = 500 050 005 004 967 + 0;
  • 500 050 005 004 967 ÷ 2 = 250 025 002 502 483 + 1;
  • 250 025 002 502 483 ÷ 2 = 125 012 501 251 241 + 1;
  • 125 012 501 251 241 ÷ 2 = 62 506 250 625 620 + 1;
  • 62 506 250 625 620 ÷ 2 = 31 253 125 312 810 + 0;
  • 31 253 125 312 810 ÷ 2 = 15 626 562 656 405 + 0;
  • 15 626 562 656 405 ÷ 2 = 7 813 281 328 202 + 1;
  • 7 813 281 328 202 ÷ 2 = 3 906 640 664 101 + 0;
  • 3 906 640 664 101 ÷ 2 = 1 953 320 332 050 + 1;
  • 1 953 320 332 050 ÷ 2 = 976 660 166 025 + 0;
  • 976 660 166 025 ÷ 2 = 488 330 083 012 + 1;
  • 488 330 083 012 ÷ 2 = 244 165 041 506 + 0;
  • 244 165 041 506 ÷ 2 = 122 082 520 753 + 0;
  • 122 082 520 753 ÷ 2 = 61 041 260 376 + 1;
  • 61 041 260 376 ÷ 2 = 30 520 630 188 + 0;
  • 30 520 630 188 ÷ 2 = 15 260 315 094 + 0;
  • 15 260 315 094 ÷ 2 = 7 630 157 547 + 0;
  • 7 630 157 547 ÷ 2 = 3 815 078 773 + 1;
  • 3 815 078 773 ÷ 2 = 1 907 539 386 + 1;
  • 1 907 539 386 ÷ 2 = 953 769 693 + 0;
  • 953 769 693 ÷ 2 = 476 884 846 + 1;
  • 476 884 846 ÷ 2 = 238 442 423 + 0;
  • 238 442 423 ÷ 2 = 119 221 211 + 1;
  • 119 221 211 ÷ 2 = 59 610 605 + 1;
  • 59 610 605 ÷ 2 = 29 805 302 + 1;
  • 29 805 302 ÷ 2 = 14 902 651 + 0;
  • 14 902 651 ÷ 2 = 7 451 325 + 1;
  • 7 451 325 ÷ 2 = 3 725 662 + 1;
  • 3 725 662 ÷ 2 = 1 862 831 + 0;
  • 1 862 831 ÷ 2 = 931 415 + 1;
  • 931 415 ÷ 2 = 465 707 + 1;
  • 465 707 ÷ 2 = 232 853 + 1;
  • 232 853 ÷ 2 = 116 426 + 1;
  • 116 426 ÷ 2 = 58 213 + 0;
  • 58 213 ÷ 2 = 29 106 + 1;
  • 29 106 ÷ 2 = 14 553 + 0;
  • 14 553 ÷ 2 = 7 276 + 1;
  • 7 276 ÷ 2 = 3 638 + 0;
  • 3 638 ÷ 2 = 1 819 + 0;
  • 1 819 ÷ 2 = 909 + 1;
  • 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.

1 000 100 010 009 934(10) = 11 1000 1101 1001 0101 1110 1101 1101 0110 0010 0101 0100 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:


1 000 100 010 009 934(10) Base 10 integer number converted and written as a signed binary code (in base 2):

1 000 100 010 009 934(10) = 0000 0000 0000 0011 1000 1101 1001 0101 1110 1101 1101 0110 0010 0101 0100 1110

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