Convert 1 000 001 001 101 031 to a Signed Binary (Base 2)

How to convert 1 000 001 001 101 031(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 1 000 001 001 101 031 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;
  • 1 000 001 001 101 031 ÷ 2 = 500 000 500 550 515 + 1;
  • 500 000 500 550 515 ÷ 2 = 250 000 250 275 257 + 1;
  • 250 000 250 275 257 ÷ 2 = 125 000 125 137 628 + 1;
  • 125 000 125 137 628 ÷ 2 = 62 500 062 568 814 + 0;
  • 62 500 062 568 814 ÷ 2 = 31 250 031 284 407 + 0;
  • 31 250 031 284 407 ÷ 2 = 15 625 015 642 203 + 1;
  • 15 625 015 642 203 ÷ 2 = 7 812 507 821 101 + 1;
  • 7 812 507 821 101 ÷ 2 = 3 906 253 910 550 + 1;
  • 3 906 253 910 550 ÷ 2 = 1 953 126 955 275 + 0;
  • 1 953 126 955 275 ÷ 2 = 976 563 477 637 + 1;
  • 976 563 477 637 ÷ 2 = 488 281 738 818 + 1;
  • 488 281 738 818 ÷ 2 = 244 140 869 409 + 0;
  • 244 140 869 409 ÷ 2 = 122 070 434 704 + 1;
  • 122 070 434 704 ÷ 2 = 61 035 217 352 + 0;
  • 61 035 217 352 ÷ 2 = 30 517 608 676 + 0;
  • 30 517 608 676 ÷ 2 = 15 258 804 338 + 0;
  • 15 258 804 338 ÷ 2 = 7 629 402 169 + 0;
  • 7 629 402 169 ÷ 2 = 3 814 701 084 + 1;
  • 3 814 701 084 ÷ 2 = 1 907 350 542 + 0;
  • 1 907 350 542 ÷ 2 = 953 675 271 + 0;
  • 953 675 271 ÷ 2 = 476 837 635 + 1;
  • 476 837 635 ÷ 2 = 238 418 817 + 1;
  • 238 418 817 ÷ 2 = 119 209 408 + 1;
  • 119 209 408 ÷ 2 = 59 604 704 + 0;
  • 59 604 704 ÷ 2 = 29 802 352 + 0;
  • 29 802 352 ÷ 2 = 14 901 176 + 0;
  • 14 901 176 ÷ 2 = 7 450 588 + 0;
  • 7 450 588 ÷ 2 = 3 725 294 + 0;
  • 3 725 294 ÷ 2 = 1 862 647 + 0;
  • 1 862 647 ÷ 2 = 931 323 + 1;
  • 931 323 ÷ 2 = 465 661 + 1;
  • 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.

1 000 001 001 101 031(10) = 11 1000 1101 0111 1110 1110 0000 0111 0010 0001 0110 1110 0111(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:


1 000 001 001 101 031(10) Base 10 integer number converted and written as a signed binary code (in base 2):

1 000 001 001 101 031(10) = 0000 0000 0000 0011 1000 1101 0111 1110 1110 0000 0111 0010 0001 0110 1110 0111

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