Convert 1 001 101 101 111 103 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 001 101 101 111 103(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 001 101 101 111 103 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;
  • 1 001 101 101 111 103 ÷ 2 = 500 550 550 555 551 + 1;
  • 500 550 550 555 551 ÷ 2 = 250 275 275 277 775 + 1;
  • 250 275 275 277 775 ÷ 2 = 125 137 637 638 887 + 1;
  • 125 137 637 638 887 ÷ 2 = 62 568 818 819 443 + 1;
  • 62 568 818 819 443 ÷ 2 = 31 284 409 409 721 + 1;
  • 31 284 409 409 721 ÷ 2 = 15 642 204 704 860 + 1;
  • 15 642 204 704 860 ÷ 2 = 7 821 102 352 430 + 0;
  • 7 821 102 352 430 ÷ 2 = 3 910 551 176 215 + 0;
  • 3 910 551 176 215 ÷ 2 = 1 955 275 588 107 + 1;
  • 1 955 275 588 107 ÷ 2 = 977 637 794 053 + 1;
  • 977 637 794 053 ÷ 2 = 488 818 897 026 + 1;
  • 488 818 897 026 ÷ 2 = 244 409 448 513 + 0;
  • 244 409 448 513 ÷ 2 = 122 204 724 256 + 1;
  • 122 204 724 256 ÷ 2 = 61 102 362 128 + 0;
  • 61 102 362 128 ÷ 2 = 30 551 181 064 + 0;
  • 30 551 181 064 ÷ 2 = 15 275 590 532 + 0;
  • 15 275 590 532 ÷ 2 = 7 637 795 266 + 0;
  • 7 637 795 266 ÷ 2 = 3 818 897 633 + 0;
  • 3 818 897 633 ÷ 2 = 1 909 448 816 + 1;
  • 1 909 448 816 ÷ 2 = 954 724 408 + 0;
  • 954 724 408 ÷ 2 = 477 362 204 + 0;
  • 477 362 204 ÷ 2 = 238 681 102 + 0;
  • 238 681 102 ÷ 2 = 119 340 551 + 0;
  • 119 340 551 ÷ 2 = 59 670 275 + 1;
  • 59 670 275 ÷ 2 = 29 835 137 + 1;
  • 29 835 137 ÷ 2 = 14 917 568 + 1;
  • 14 917 568 ÷ 2 = 7 458 784 + 0;
  • 7 458 784 ÷ 2 = 3 729 392 + 0;
  • 3 729 392 ÷ 2 = 1 864 696 + 0;
  • 1 864 696 ÷ 2 = 932 348 + 0;
  • 932 348 ÷ 2 = 466 174 + 0;
  • 466 174 ÷ 2 = 233 087 + 0;
  • 233 087 ÷ 2 = 116 543 + 1;
  • 116 543 ÷ 2 = 58 271 + 1;
  • 58 271 ÷ 2 = 29 135 + 1;
  • 29 135 ÷ 2 = 14 567 + 1;
  • 14 567 ÷ 2 = 7 283 + 1;
  • 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.

1 001 101 101 111 103(10) = 11 1000 1110 0111 1111 0000 0011 1000 0100 0001 0111 0011 1111(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 1 001 101 101 111 103(10) converted to signed binary in two's complement representation:

1 001 101 101 111 103(10) = 0000 0000 0000 0011 1000 1110 0111 1111 0000 0011 1000 0100 0001 0111 0011 1111

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