Convert 1 001 011 010 000 251 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 001 011 010 000 251(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 001 011 010 000 251 to a signed binary in two's (2's) complement representation?

  • 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.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 1 001 011 010 000 251 ÷ 2 = 500 505 505 000 125 + 1;
  • 500 505 505 000 125 ÷ 2 = 250 252 752 500 062 + 1;
  • 250 252 752 500 062 ÷ 2 = 125 126 376 250 031 + 0;
  • 125 126 376 250 031 ÷ 2 = 62 563 188 125 015 + 1;
  • 62 563 188 125 015 ÷ 2 = 31 281 594 062 507 + 1;
  • 31 281 594 062 507 ÷ 2 = 15 640 797 031 253 + 1;
  • 15 640 797 031 253 ÷ 2 = 7 820 398 515 626 + 1;
  • 7 820 398 515 626 ÷ 2 = 3 910 199 257 813 + 0;
  • 3 910 199 257 813 ÷ 2 = 1 955 099 628 906 + 1;
  • 1 955 099 628 906 ÷ 2 = 977 549 814 453 + 0;
  • 977 549 814 453 ÷ 2 = 488 774 907 226 + 1;
  • 488 774 907 226 ÷ 2 = 244 387 453 613 + 0;
  • 244 387 453 613 ÷ 2 = 122 193 726 806 + 1;
  • 122 193 726 806 ÷ 2 = 61 096 863 403 + 0;
  • 61 096 863 403 ÷ 2 = 30 548 431 701 + 1;
  • 30 548 431 701 ÷ 2 = 15 274 215 850 + 1;
  • 15 274 215 850 ÷ 2 = 7 637 107 925 + 0;
  • 7 637 107 925 ÷ 2 = 3 818 553 962 + 1;
  • 3 818 553 962 ÷ 2 = 1 909 276 981 + 0;
  • 1 909 276 981 ÷ 2 = 954 638 490 + 1;
  • 954 638 490 ÷ 2 = 477 319 245 + 0;
  • 477 319 245 ÷ 2 = 238 659 622 + 1;
  • 238 659 622 ÷ 2 = 119 329 811 + 0;
  • 119 329 811 ÷ 2 = 59 664 905 + 1;
  • 59 664 905 ÷ 2 = 29 832 452 + 1;
  • 29 832 452 ÷ 2 = 14 916 226 + 0;
  • 14 916 226 ÷ 2 = 7 458 113 + 0;
  • 7 458 113 ÷ 2 = 3 729 056 + 1;
  • 3 729 056 ÷ 2 = 1 864 528 + 0;
  • 1 864 528 ÷ 2 = 932 264 + 0;
  • 932 264 ÷ 2 = 466 132 + 0;
  • 466 132 ÷ 2 = 233 066 + 0;
  • 233 066 ÷ 2 = 116 533 + 0;
  • 116 533 ÷ 2 = 58 266 + 1;
  • 58 266 ÷ 2 = 29 133 + 0;
  • 29 133 ÷ 2 = 14 566 + 1;
  • 14 566 ÷ 2 = 7 283 + 0;
  • 7 283 ÷ 2 = 3 641 + 1;
  • 3 641 ÷ 2 = 1 820 + 1;
  • 1 820 ÷ 2 = 910 + 0;
  • 910 ÷ 2 = 455 + 0;
  • 455 ÷ 2 = 227 + 1;
  • 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.

1 001 011 010 000 251(10) = 11 1000 1110 0110 1010 0000 1001 1010 1010 1101 0101 0111 1011(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) indicates 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.


Decimal Number 1 001 011 010 000 251(10) converted to signed binary in two's complement representation:

1 001 011 010 000 251(10) = 0000 0000 0000 0011 1000 1110 0110 1010 0000 1001 1010 1010 1101 0101 0111 1011

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed integers from decimal system to signed binary in two's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in two's complement representation:

  • 1. If the number to be converted is negative, start with the positive version of the number.
  • 2. Divide repeatedly by 2 the positive representation of the integer number, keeping track of each remainder, until 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 must have 4, 8, 16, 32, 64, ... bit length (a power of 2) - if needed, add extra bits on 0 in front (to the left) of the base 2 number above, up to the required length, so that the first bit (the leftmost) will be 0, correctly representing a positive number.
  • 5. To get the negative integer number representation in signed binary one's complement, replace all 0 bits with 1s and all 1 bits with 0s (reversing the digits).
  • 6. To get the negative integer number, in signed binary two's complement representation, add 1 to the number above.

Example: convert the negative number -60 from the decimal system (base ten) to signed binary in two's complement:

  • 1. Start with the positive version of the number: |-60| = 60
  • 2. Divide repeatedly 60 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 60 ÷ 2 = 30 + 0
    • 30 ÷ 2 = 15 + 0
    • 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:
    60(10) = 11 1100(2)
  • 4. Bit length of base 2 representation number is 6, so the positive binary computer representation of a signed binary will take in this particular case 8 bits (the least power of 2 larger than 6) - add extra 0 digits in front of the base 2 number, up to the required length:
    60(10) = 0011 1100(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all the 0 bits with 1s and all 1 bits with 0s (reversing the digits):
    !(0011 1100) = 1100 0011
  • 6. To get the negative integer number, signed binary in two's complement representation, add 1 to the number above:
    -60(10) = 1100 0011 + 1 = 1100 0100
  • Number -60(10), signed integer, converted from decimal system (base 10) to signed binary two's complement representation = 1100 0100