Convert 1 011 101 010 111 127 to a Signed Binary (Base 2)

How to convert 1 011 101 010 111 127(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 1 011 101 010 111 127 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;
  • 1 011 101 010 111 127 ÷ 2 = 505 550 505 055 563 + 1;
  • 505 550 505 055 563 ÷ 2 = 252 775 252 527 781 + 1;
  • 252 775 252 527 781 ÷ 2 = 126 387 626 263 890 + 1;
  • 126 387 626 263 890 ÷ 2 = 63 193 813 131 945 + 0;
  • 63 193 813 131 945 ÷ 2 = 31 596 906 565 972 + 1;
  • 31 596 906 565 972 ÷ 2 = 15 798 453 282 986 + 0;
  • 15 798 453 282 986 ÷ 2 = 7 899 226 641 493 + 0;
  • 7 899 226 641 493 ÷ 2 = 3 949 613 320 746 + 1;
  • 3 949 613 320 746 ÷ 2 = 1 974 806 660 373 + 0;
  • 1 974 806 660 373 ÷ 2 = 987 403 330 186 + 1;
  • 987 403 330 186 ÷ 2 = 493 701 665 093 + 0;
  • 493 701 665 093 ÷ 2 = 246 850 832 546 + 1;
  • 246 850 832 546 ÷ 2 = 123 425 416 273 + 0;
  • 123 425 416 273 ÷ 2 = 61 712 708 136 + 1;
  • 61 712 708 136 ÷ 2 = 30 856 354 068 + 0;
  • 30 856 354 068 ÷ 2 = 15 428 177 034 + 0;
  • 15 428 177 034 ÷ 2 = 7 714 088 517 + 0;
  • 7 714 088 517 ÷ 2 = 3 857 044 258 + 1;
  • 3 857 044 258 ÷ 2 = 1 928 522 129 + 0;
  • 1 928 522 129 ÷ 2 = 964 261 064 + 1;
  • 964 261 064 ÷ 2 = 482 130 532 + 0;
  • 482 130 532 ÷ 2 = 241 065 266 + 0;
  • 241 065 266 ÷ 2 = 120 532 633 + 0;
  • 120 532 633 ÷ 2 = 60 266 316 + 1;
  • 60 266 316 ÷ 2 = 30 133 158 + 0;
  • 30 133 158 ÷ 2 = 15 066 579 + 0;
  • 15 066 579 ÷ 2 = 7 533 289 + 1;
  • 7 533 289 ÷ 2 = 3 766 644 + 1;
  • 3 766 644 ÷ 2 = 1 883 322 + 0;
  • 1 883 322 ÷ 2 = 941 661 + 0;
  • 941 661 ÷ 2 = 470 830 + 1;
  • 470 830 ÷ 2 = 235 415 + 0;
  • 235 415 ÷ 2 = 117 707 + 1;
  • 117 707 ÷ 2 = 58 853 + 1;
  • 58 853 ÷ 2 = 29 426 + 1;
  • 29 426 ÷ 2 = 14 713 + 0;
  • 14 713 ÷ 2 = 7 356 + 1;
  • 7 356 ÷ 2 = 3 678 + 0;
  • 3 678 ÷ 2 = 1 839 + 0;
  • 1 839 ÷ 2 = 919 + 1;
  • 919 ÷ 2 = 459 + 1;
  • 459 ÷ 2 = 229 + 1;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 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.

1 011 101 010 111 127(10) = 11 1001 0111 1001 0111 0100 1100 1000 1010 0010 1010 1001 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:


1 011 101 010 111 127(10) Base 10 integer number converted and written as a signed binary code (in base 2):

1 011 101 010 111 127(10) = 0000 0000 0000 0011 1001 0111 1001 0111 0100 1100 1000 1010 0010 1010 1001 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