Convert 101 010 101 011 010 172 to a Signed Binary (Base 2)

How to convert 101 010 101 011 010 172(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 101 010 101 011 010 172 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;
  • 101 010 101 011 010 172 ÷ 2 = 50 505 050 505 505 086 + 0;
  • 50 505 050 505 505 086 ÷ 2 = 25 252 525 252 752 543 + 0;
  • 25 252 525 252 752 543 ÷ 2 = 12 626 262 626 376 271 + 1;
  • 12 626 262 626 376 271 ÷ 2 = 6 313 131 313 188 135 + 1;
  • 6 313 131 313 188 135 ÷ 2 = 3 156 565 656 594 067 + 1;
  • 3 156 565 656 594 067 ÷ 2 = 1 578 282 828 297 033 + 1;
  • 1 578 282 828 297 033 ÷ 2 = 789 141 414 148 516 + 1;
  • 789 141 414 148 516 ÷ 2 = 394 570 707 074 258 + 0;
  • 394 570 707 074 258 ÷ 2 = 197 285 353 537 129 + 0;
  • 197 285 353 537 129 ÷ 2 = 98 642 676 768 564 + 1;
  • 98 642 676 768 564 ÷ 2 = 49 321 338 384 282 + 0;
  • 49 321 338 384 282 ÷ 2 = 24 660 669 192 141 + 0;
  • 24 660 669 192 141 ÷ 2 = 12 330 334 596 070 + 1;
  • 12 330 334 596 070 ÷ 2 = 6 165 167 298 035 + 0;
  • 6 165 167 298 035 ÷ 2 = 3 082 583 649 017 + 1;
  • 3 082 583 649 017 ÷ 2 = 1 541 291 824 508 + 1;
  • 1 541 291 824 508 ÷ 2 = 770 645 912 254 + 0;
  • 770 645 912 254 ÷ 2 = 385 322 956 127 + 0;
  • 385 322 956 127 ÷ 2 = 192 661 478 063 + 1;
  • 192 661 478 063 ÷ 2 = 96 330 739 031 + 1;
  • 96 330 739 031 ÷ 2 = 48 165 369 515 + 1;
  • 48 165 369 515 ÷ 2 = 24 082 684 757 + 1;
  • 24 082 684 757 ÷ 2 = 12 041 342 378 + 1;
  • 12 041 342 378 ÷ 2 = 6 020 671 189 + 0;
  • 6 020 671 189 ÷ 2 = 3 010 335 594 + 1;
  • 3 010 335 594 ÷ 2 = 1 505 167 797 + 0;
  • 1 505 167 797 ÷ 2 = 752 583 898 + 1;
  • 752 583 898 ÷ 2 = 376 291 949 + 0;
  • 376 291 949 ÷ 2 = 188 145 974 + 1;
  • 188 145 974 ÷ 2 = 94 072 987 + 0;
  • 94 072 987 ÷ 2 = 47 036 493 + 1;
  • 47 036 493 ÷ 2 = 23 518 246 + 1;
  • 23 518 246 ÷ 2 = 11 759 123 + 0;
  • 11 759 123 ÷ 2 = 5 879 561 + 1;
  • 5 879 561 ÷ 2 = 2 939 780 + 1;
  • 2 939 780 ÷ 2 = 1 469 890 + 0;
  • 1 469 890 ÷ 2 = 734 945 + 0;
  • 734 945 ÷ 2 = 367 472 + 1;
  • 367 472 ÷ 2 = 183 736 + 0;
  • 183 736 ÷ 2 = 91 868 + 0;
  • 91 868 ÷ 2 = 45 934 + 0;
  • 45 934 ÷ 2 = 22 967 + 0;
  • 22 967 ÷ 2 = 11 483 + 1;
  • 11 483 ÷ 2 = 5 741 + 1;
  • 5 741 ÷ 2 = 2 870 + 1;
  • 2 870 ÷ 2 = 1 435 + 0;
  • 1 435 ÷ 2 = 717 + 1;
  • 717 ÷ 2 = 358 + 1;
  • 358 ÷ 2 = 179 + 0;
  • 179 ÷ 2 = 89 + 1;
  • 89 ÷ 2 = 44 + 1;
  • 44 ÷ 2 = 22 + 0;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.

101 010 101 011 010 172(10) = 1 0110 0110 1101 1100 0010 0110 1101 0101 0111 1100 1101 0010 0111 1100(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 57.

  • 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, 57,

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:


101 010 101 011 010 172(10) Base 10 integer number converted and written as a signed binary code (in base 2):

101 010 101 011 010 172(10) = 0000 0001 0110 0110 1101 1100 0010 0110 1101 0101 0111 1100 1101 0010 0111 1100

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