Convert 1 101 001 111 009 457 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 101 001 111 009 457(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 101 001 111 009 457 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;
  • 1 101 001 111 009 457 ÷ 2 = 550 500 555 504 728 + 1;
  • 550 500 555 504 728 ÷ 2 = 275 250 277 752 364 + 0;
  • 275 250 277 752 364 ÷ 2 = 137 625 138 876 182 + 0;
  • 137 625 138 876 182 ÷ 2 = 68 812 569 438 091 + 0;
  • 68 812 569 438 091 ÷ 2 = 34 406 284 719 045 + 1;
  • 34 406 284 719 045 ÷ 2 = 17 203 142 359 522 + 1;
  • 17 203 142 359 522 ÷ 2 = 8 601 571 179 761 + 0;
  • 8 601 571 179 761 ÷ 2 = 4 300 785 589 880 + 1;
  • 4 300 785 589 880 ÷ 2 = 2 150 392 794 940 + 0;
  • 2 150 392 794 940 ÷ 2 = 1 075 196 397 470 + 0;
  • 1 075 196 397 470 ÷ 2 = 537 598 198 735 + 0;
  • 537 598 198 735 ÷ 2 = 268 799 099 367 + 1;
  • 268 799 099 367 ÷ 2 = 134 399 549 683 + 1;
  • 134 399 549 683 ÷ 2 = 67 199 774 841 + 1;
  • 67 199 774 841 ÷ 2 = 33 599 887 420 + 1;
  • 33 599 887 420 ÷ 2 = 16 799 943 710 + 0;
  • 16 799 943 710 ÷ 2 = 8 399 971 855 + 0;
  • 8 399 971 855 ÷ 2 = 4 199 985 927 + 1;
  • 4 199 985 927 ÷ 2 = 2 099 992 963 + 1;
  • 2 099 992 963 ÷ 2 = 1 049 996 481 + 1;
  • 1 049 996 481 ÷ 2 = 524 998 240 + 1;
  • 524 998 240 ÷ 2 = 262 499 120 + 0;
  • 262 499 120 ÷ 2 = 131 249 560 + 0;
  • 131 249 560 ÷ 2 = 65 624 780 + 0;
  • 65 624 780 ÷ 2 = 32 812 390 + 0;
  • 32 812 390 ÷ 2 = 16 406 195 + 0;
  • 16 406 195 ÷ 2 = 8 203 097 + 1;
  • 8 203 097 ÷ 2 = 4 101 548 + 1;
  • 4 101 548 ÷ 2 = 2 050 774 + 0;
  • 2 050 774 ÷ 2 = 1 025 387 + 0;
  • 1 025 387 ÷ 2 = 512 693 + 1;
  • 512 693 ÷ 2 = 256 346 + 1;
  • 256 346 ÷ 2 = 128 173 + 0;
  • 128 173 ÷ 2 = 64 086 + 1;
  • 64 086 ÷ 2 = 32 043 + 0;
  • 32 043 ÷ 2 = 16 021 + 1;
  • 16 021 ÷ 2 = 8 010 + 1;
  • 8 010 ÷ 2 = 4 005 + 0;
  • 4 005 ÷ 2 = 2 002 + 1;
  • 2 002 ÷ 2 = 1 001 + 0;
  • 1 001 ÷ 2 = 500 + 1;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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.

1 101 001 111 009 457(10) = 11 1110 1001 0101 1010 1100 1100 0001 1110 0111 1000 1011 0001(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 1 101 001 111 009 457(10) converted to signed binary in two's complement representation:

1 101 001 111 009 457(10) = 0000 0000 0000 0011 1110 1001 0101 1010 1100 1100 0001 1110 0111 1000 1011 0001

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