Convert 999 999 999 999 538 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 999 999 999 999 538(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
999 999 999 999 538 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;
  • 999 999 999 999 538 ÷ 2 = 499 999 999 999 769 + 0;
  • 499 999 999 999 769 ÷ 2 = 249 999 999 999 884 + 1;
  • 249 999 999 999 884 ÷ 2 = 124 999 999 999 942 + 0;
  • 124 999 999 999 942 ÷ 2 = 62 499 999 999 971 + 0;
  • 62 499 999 999 971 ÷ 2 = 31 249 999 999 985 + 1;
  • 31 249 999 999 985 ÷ 2 = 15 624 999 999 992 + 1;
  • 15 624 999 999 992 ÷ 2 = 7 812 499 999 996 + 0;
  • 7 812 499 999 996 ÷ 2 = 3 906 249 999 998 + 0;
  • 3 906 249 999 998 ÷ 2 = 1 953 124 999 999 + 0;
  • 1 953 124 999 999 ÷ 2 = 976 562 499 999 + 1;
  • 976 562 499 999 ÷ 2 = 488 281 249 999 + 1;
  • 488 281 249 999 ÷ 2 = 244 140 624 999 + 1;
  • 244 140 624 999 ÷ 2 = 122 070 312 499 + 1;
  • 122 070 312 499 ÷ 2 = 61 035 156 249 + 1;
  • 61 035 156 249 ÷ 2 = 30 517 578 124 + 1;
  • 30 517 578 124 ÷ 2 = 15 258 789 062 + 0;
  • 15 258 789 062 ÷ 2 = 7 629 394 531 + 0;
  • 7 629 394 531 ÷ 2 = 3 814 697 265 + 1;
  • 3 814 697 265 ÷ 2 = 1 907 348 632 + 1;
  • 1 907 348 632 ÷ 2 = 953 674 316 + 0;
  • 953 674 316 ÷ 2 = 476 837 158 + 0;
  • 476 837 158 ÷ 2 = 238 418 579 + 0;
  • 238 418 579 ÷ 2 = 119 209 289 + 1;
  • 119 209 289 ÷ 2 = 59 604 644 + 1;
  • 59 604 644 ÷ 2 = 29 802 322 + 0;
  • 29 802 322 ÷ 2 = 14 901 161 + 0;
  • 14 901 161 ÷ 2 = 7 450 580 + 1;
  • 7 450 580 ÷ 2 = 3 725 290 + 0;
  • 3 725 290 ÷ 2 = 1 862 645 + 0;
  • 1 862 645 ÷ 2 = 931 322 + 1;
  • 931 322 ÷ 2 = 465 661 + 0;
  • 465 661 ÷ 2 = 232 830 + 1;
  • 232 830 ÷ 2 = 116 415 + 0;
  • 116 415 ÷ 2 = 58 207 + 1;
  • 58 207 ÷ 2 = 29 103 + 1;
  • 29 103 ÷ 2 = 14 551 + 1;
  • 14 551 ÷ 2 = 7 275 + 1;
  • 7 275 ÷ 2 = 3 637 + 1;
  • 3 637 ÷ 2 = 1 818 + 1;
  • 1 818 ÷ 2 = 909 + 0;
  • 909 ÷ 2 = 454 + 1;
  • 454 ÷ 2 = 227 + 0;
  • 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.

999 999 999 999 538(10) = 11 1000 1101 0111 1110 1010 0100 1100 0110 0111 1110 0011 0010(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 999 999 999 999 538(10) converted to signed binary in two's complement representation:

999 999 999 999 538(10) = 0000 0000 0000 0011 1000 1101 0111 1110 1010 0100 1100 0110 0111 1110 0011 0010

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