Convert 338 051 712 969 015 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 338 051 712 969 015(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
338 051 712 969 015 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;
  • 338 051 712 969 015 ÷ 2 = 169 025 856 484 507 + 1;
  • 169 025 856 484 507 ÷ 2 = 84 512 928 242 253 + 1;
  • 84 512 928 242 253 ÷ 2 = 42 256 464 121 126 + 1;
  • 42 256 464 121 126 ÷ 2 = 21 128 232 060 563 + 0;
  • 21 128 232 060 563 ÷ 2 = 10 564 116 030 281 + 1;
  • 10 564 116 030 281 ÷ 2 = 5 282 058 015 140 + 1;
  • 5 282 058 015 140 ÷ 2 = 2 641 029 007 570 + 0;
  • 2 641 029 007 570 ÷ 2 = 1 320 514 503 785 + 0;
  • 1 320 514 503 785 ÷ 2 = 660 257 251 892 + 1;
  • 660 257 251 892 ÷ 2 = 330 128 625 946 + 0;
  • 330 128 625 946 ÷ 2 = 165 064 312 973 + 0;
  • 165 064 312 973 ÷ 2 = 82 532 156 486 + 1;
  • 82 532 156 486 ÷ 2 = 41 266 078 243 + 0;
  • 41 266 078 243 ÷ 2 = 20 633 039 121 + 1;
  • 20 633 039 121 ÷ 2 = 10 316 519 560 + 1;
  • 10 316 519 560 ÷ 2 = 5 158 259 780 + 0;
  • 5 158 259 780 ÷ 2 = 2 579 129 890 + 0;
  • 2 579 129 890 ÷ 2 = 1 289 564 945 + 0;
  • 1 289 564 945 ÷ 2 = 644 782 472 + 1;
  • 644 782 472 ÷ 2 = 322 391 236 + 0;
  • 322 391 236 ÷ 2 = 161 195 618 + 0;
  • 161 195 618 ÷ 2 = 80 597 809 + 0;
  • 80 597 809 ÷ 2 = 40 298 904 + 1;
  • 40 298 904 ÷ 2 = 20 149 452 + 0;
  • 20 149 452 ÷ 2 = 10 074 726 + 0;
  • 10 074 726 ÷ 2 = 5 037 363 + 0;
  • 5 037 363 ÷ 2 = 2 518 681 + 1;
  • 2 518 681 ÷ 2 = 1 259 340 + 1;
  • 1 259 340 ÷ 2 = 629 670 + 0;
  • 629 670 ÷ 2 = 314 835 + 0;
  • 314 835 ÷ 2 = 157 417 + 1;
  • 157 417 ÷ 2 = 78 708 + 1;
  • 78 708 ÷ 2 = 39 354 + 0;
  • 39 354 ÷ 2 = 19 677 + 0;
  • 19 677 ÷ 2 = 9 838 + 1;
  • 9 838 ÷ 2 = 4 919 + 0;
  • 4 919 ÷ 2 = 2 459 + 1;
  • 2 459 ÷ 2 = 1 229 + 1;
  • 1 229 ÷ 2 = 614 + 1;
  • 614 ÷ 2 = 307 + 0;
  • 307 ÷ 2 = 153 + 1;
  • 153 ÷ 2 = 76 + 1;
  • 76 ÷ 2 = 38 + 0;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 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.

338 051 712 969 015(10) = 1 0011 0011 0111 0100 1100 1100 0100 0100 0110 1001 0011 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 49.

  • 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, 49,

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 338 051 712 969 015(10) converted to signed binary in two's complement representation:

338 051 712 969 015(10) = 0000 0000 0000 0001 0011 0011 0111 0100 1100 1100 0100 0100 0110 1001 0011 0111

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