Convert 1 010 101 101 010 010 975 to a Signed Binary (Base 2)

How to convert 1 010 101 101 010 010 975(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 1 010 101 101 010 010 975 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;
  • 1 010 101 101 010 010 975 ÷ 2 = 505 050 550 505 005 487 + 1;
  • 505 050 550 505 005 487 ÷ 2 = 252 525 275 252 502 743 + 1;
  • 252 525 275 252 502 743 ÷ 2 = 126 262 637 626 251 371 + 1;
  • 126 262 637 626 251 371 ÷ 2 = 63 131 318 813 125 685 + 1;
  • 63 131 318 813 125 685 ÷ 2 = 31 565 659 406 562 842 + 1;
  • 31 565 659 406 562 842 ÷ 2 = 15 782 829 703 281 421 + 0;
  • 15 782 829 703 281 421 ÷ 2 = 7 891 414 851 640 710 + 1;
  • 7 891 414 851 640 710 ÷ 2 = 3 945 707 425 820 355 + 0;
  • 3 945 707 425 820 355 ÷ 2 = 1 972 853 712 910 177 + 1;
  • 1 972 853 712 910 177 ÷ 2 = 986 426 856 455 088 + 1;
  • 986 426 856 455 088 ÷ 2 = 493 213 428 227 544 + 0;
  • 493 213 428 227 544 ÷ 2 = 246 606 714 113 772 + 0;
  • 246 606 714 113 772 ÷ 2 = 123 303 357 056 886 + 0;
  • 123 303 357 056 886 ÷ 2 = 61 651 678 528 443 + 0;
  • 61 651 678 528 443 ÷ 2 = 30 825 839 264 221 + 1;
  • 30 825 839 264 221 ÷ 2 = 15 412 919 632 110 + 1;
  • 15 412 919 632 110 ÷ 2 = 7 706 459 816 055 + 0;
  • 7 706 459 816 055 ÷ 2 = 3 853 229 908 027 + 1;
  • 3 853 229 908 027 ÷ 2 = 1 926 614 954 013 + 1;
  • 1 926 614 954 013 ÷ 2 = 963 307 477 006 + 1;
  • 963 307 477 006 ÷ 2 = 481 653 738 503 + 0;
  • 481 653 738 503 ÷ 2 = 240 826 869 251 + 1;
  • 240 826 869 251 ÷ 2 = 120 413 434 625 + 1;
  • 120 413 434 625 ÷ 2 = 60 206 717 312 + 1;
  • 60 206 717 312 ÷ 2 = 30 103 358 656 + 0;
  • 30 103 358 656 ÷ 2 = 15 051 679 328 + 0;
  • 15 051 679 328 ÷ 2 = 7 525 839 664 + 0;
  • 7 525 839 664 ÷ 2 = 3 762 919 832 + 0;
  • 3 762 919 832 ÷ 2 = 1 881 459 916 + 0;
  • 1 881 459 916 ÷ 2 = 940 729 958 + 0;
  • 940 729 958 ÷ 2 = 470 364 979 + 0;
  • 470 364 979 ÷ 2 = 235 182 489 + 1;
  • 235 182 489 ÷ 2 = 117 591 244 + 1;
  • 117 591 244 ÷ 2 = 58 795 622 + 0;
  • 58 795 622 ÷ 2 = 29 397 811 + 0;
  • 29 397 811 ÷ 2 = 14 698 905 + 1;
  • 14 698 905 ÷ 2 = 7 349 452 + 1;
  • 7 349 452 ÷ 2 = 3 674 726 + 0;
  • 3 674 726 ÷ 2 = 1 837 363 + 0;
  • 1 837 363 ÷ 2 = 918 681 + 1;
  • 918 681 ÷ 2 = 459 340 + 1;
  • 459 340 ÷ 2 = 229 670 + 0;
  • 229 670 ÷ 2 = 114 835 + 0;
  • 114 835 ÷ 2 = 57 417 + 1;
  • 57 417 ÷ 2 = 28 708 + 1;
  • 28 708 ÷ 2 = 14 354 + 0;
  • 14 354 ÷ 2 = 7 177 + 0;
  • 7 177 ÷ 2 = 3 588 + 1;
  • 3 588 ÷ 2 = 1 794 + 0;
  • 1 794 ÷ 2 = 897 + 0;
  • 897 ÷ 2 = 448 + 1;
  • 448 ÷ 2 = 224 + 0;
  • 224 ÷ 2 = 112 + 0;
  • 112 ÷ 2 = 56 + 0;
  • 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.

1 010 101 101 010 010 975(10) = 1110 0000 0100 1001 1001 1001 1001 1000 0000 1110 1110 1100 0011 0101 1111(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 60.

  • 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, 60,

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:


1 010 101 101 010 010 975(10) Base 10 integer number converted and written as a signed binary code (in base 2):

1 010 101 101 010 010 975(10) = 0000 1110 0000 0100 1001 1001 1001 1001 1000 0000 1110 1110 1100 0011 0101 1111

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