Convert 10 001 111 000 525 to a Signed Binary (Base 2)

How to convert 10 001 111 000 525(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 10 001 111 000 525 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;
  • 10 001 111 000 525 ÷ 2 = 5 000 555 500 262 + 1;
  • 5 000 555 500 262 ÷ 2 = 2 500 277 750 131 + 0;
  • 2 500 277 750 131 ÷ 2 = 1 250 138 875 065 + 1;
  • 1 250 138 875 065 ÷ 2 = 625 069 437 532 + 1;
  • 625 069 437 532 ÷ 2 = 312 534 718 766 + 0;
  • 312 534 718 766 ÷ 2 = 156 267 359 383 + 0;
  • 156 267 359 383 ÷ 2 = 78 133 679 691 + 1;
  • 78 133 679 691 ÷ 2 = 39 066 839 845 + 1;
  • 39 066 839 845 ÷ 2 = 19 533 419 922 + 1;
  • 19 533 419 922 ÷ 2 = 9 766 709 961 + 0;
  • 9 766 709 961 ÷ 2 = 4 883 354 980 + 1;
  • 4 883 354 980 ÷ 2 = 2 441 677 490 + 0;
  • 2 441 677 490 ÷ 2 = 1 220 838 745 + 0;
  • 1 220 838 745 ÷ 2 = 610 419 372 + 1;
  • 610 419 372 ÷ 2 = 305 209 686 + 0;
  • 305 209 686 ÷ 2 = 152 604 843 + 0;
  • 152 604 843 ÷ 2 = 76 302 421 + 1;
  • 76 302 421 ÷ 2 = 38 151 210 + 1;
  • 38 151 210 ÷ 2 = 19 075 605 + 0;
  • 19 075 605 ÷ 2 = 9 537 802 + 1;
  • 9 537 802 ÷ 2 = 4 768 901 + 0;
  • 4 768 901 ÷ 2 = 2 384 450 + 1;
  • 2 384 450 ÷ 2 = 1 192 225 + 0;
  • 1 192 225 ÷ 2 = 596 112 + 1;
  • 596 112 ÷ 2 = 298 056 + 0;
  • 298 056 ÷ 2 = 149 028 + 0;
  • 149 028 ÷ 2 = 74 514 + 0;
  • 74 514 ÷ 2 = 37 257 + 0;
  • 37 257 ÷ 2 = 18 628 + 1;
  • 18 628 ÷ 2 = 9 314 + 0;
  • 9 314 ÷ 2 = 4 657 + 0;
  • 4 657 ÷ 2 = 2 328 + 1;
  • 2 328 ÷ 2 = 1 164 + 0;
  • 1 164 ÷ 2 = 582 + 0;
  • 582 ÷ 2 = 291 + 0;
  • 291 ÷ 2 = 145 + 1;
  • 145 ÷ 2 = 72 + 1;
  • 72 ÷ 2 = 36 + 0;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 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.

10 001 111 000 525(10) = 1001 0001 1000 1001 0000 1010 1011 0010 0101 1100 1101(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 44.

  • 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, 44,

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:


10 001 111 000 525(10) Base 10 integer number converted and written as a signed binary code (in base 2):

10 001 111 000 525(10) = 0000 0000 0000 0000 0000 1001 0001 1000 1001 0000 1010 1011 0010 0101 1100 1101

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