Convert 9 000 000 000 000 000 000 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 9 000 000 000 000 000 000(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
9 000 000 000 000 000 000 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;
  • 9 000 000 000 000 000 000 ÷ 2 = 4 500 000 000 000 000 000 + 0;
  • 4 500 000 000 000 000 000 ÷ 2 = 2 250 000 000 000 000 000 + 0;
  • 2 250 000 000 000 000 000 ÷ 2 = 1 125 000 000 000 000 000 + 0;
  • 1 125 000 000 000 000 000 ÷ 2 = 562 500 000 000 000 000 + 0;
  • 562 500 000 000 000 000 ÷ 2 = 281 250 000 000 000 000 + 0;
  • 281 250 000 000 000 000 ÷ 2 = 140 625 000 000 000 000 + 0;
  • 140 625 000 000 000 000 ÷ 2 = 70 312 500 000 000 000 + 0;
  • 70 312 500 000 000 000 ÷ 2 = 35 156 250 000 000 000 + 0;
  • 35 156 250 000 000 000 ÷ 2 = 17 578 125 000 000 000 + 0;
  • 17 578 125 000 000 000 ÷ 2 = 8 789 062 500 000 000 + 0;
  • 8 789 062 500 000 000 ÷ 2 = 4 394 531 250 000 000 + 0;
  • 4 394 531 250 000 000 ÷ 2 = 2 197 265 625 000 000 + 0;
  • 2 197 265 625 000 000 ÷ 2 = 1 098 632 812 500 000 + 0;
  • 1 098 632 812 500 000 ÷ 2 = 549 316 406 250 000 + 0;
  • 549 316 406 250 000 ÷ 2 = 274 658 203 125 000 + 0;
  • 274 658 203 125 000 ÷ 2 = 137 329 101 562 500 + 0;
  • 137 329 101 562 500 ÷ 2 = 68 664 550 781 250 + 0;
  • 68 664 550 781 250 ÷ 2 = 34 332 275 390 625 + 0;
  • 34 332 275 390 625 ÷ 2 = 17 166 137 695 312 + 1;
  • 17 166 137 695 312 ÷ 2 = 8 583 068 847 656 + 0;
  • 8 583 068 847 656 ÷ 2 = 4 291 534 423 828 + 0;
  • 4 291 534 423 828 ÷ 2 = 2 145 767 211 914 + 0;
  • 2 145 767 211 914 ÷ 2 = 1 072 883 605 957 + 0;
  • 1 072 883 605 957 ÷ 2 = 536 441 802 978 + 1;
  • 536 441 802 978 ÷ 2 = 268 220 901 489 + 0;
  • 268 220 901 489 ÷ 2 = 134 110 450 744 + 1;
  • 134 110 450 744 ÷ 2 = 67 055 225 372 + 0;
  • 67 055 225 372 ÷ 2 = 33 527 612 686 + 0;
  • 33 527 612 686 ÷ 2 = 16 763 806 343 + 0;
  • 16 763 806 343 ÷ 2 = 8 381 903 171 + 1;
  • 8 381 903 171 ÷ 2 = 4 190 951 585 + 1;
  • 4 190 951 585 ÷ 2 = 2 095 475 792 + 1;
  • 2 095 475 792 ÷ 2 = 1 047 737 896 + 0;
  • 1 047 737 896 ÷ 2 = 523 868 948 + 0;
  • 523 868 948 ÷ 2 = 261 934 474 + 0;
  • 261 934 474 ÷ 2 = 130 967 237 + 0;
  • 130 967 237 ÷ 2 = 65 483 618 + 1;
  • 65 483 618 ÷ 2 = 32 741 809 + 0;
  • 32 741 809 ÷ 2 = 16 370 904 + 1;
  • 16 370 904 ÷ 2 = 8 185 452 + 0;
  • 8 185 452 ÷ 2 = 4 092 726 + 0;
  • 4 092 726 ÷ 2 = 2 046 363 + 0;
  • 2 046 363 ÷ 2 = 1 023 181 + 1;
  • 1 023 181 ÷ 2 = 511 590 + 1;
  • 511 590 ÷ 2 = 255 795 + 0;
  • 255 795 ÷ 2 = 127 897 + 1;
  • 127 897 ÷ 2 = 63 948 + 1;
  • 63 948 ÷ 2 = 31 974 + 0;
  • 31 974 ÷ 2 = 15 987 + 0;
  • 15 987 ÷ 2 = 7 993 + 1;
  • 7 993 ÷ 2 = 3 996 + 1;
  • 3 996 ÷ 2 = 1 998 + 0;
  • 1 998 ÷ 2 = 999 + 0;
  • 999 ÷ 2 = 499 + 1;
  • 499 ÷ 2 = 249 + 1;
  • 249 ÷ 2 = 124 + 1;
  • 124 ÷ 2 = 62 + 0;
  • 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.

9 000 000 000 000 000 000(10) = 111 1100 1110 0110 0110 1100 0101 0000 1110 0010 1000 0100 0000 0000 0000 0000(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 63.

  • 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, 63,

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 9 000 000 000 000 000 000(10) converted to signed binary in two's complement representation:

9 000 000 000 000 000 000(10) = 0111 1100 1110 0110 0110 1100 0101 0000 1110 0010 1000 0100 0000 0000 0000 0000

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