Convert 1 011 011 101 101 048 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 011 011 101 101 048(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 011 011 101 101 048 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;
  • 1 011 011 101 101 048 ÷ 2 = 505 505 550 550 524 + 0;
  • 505 505 550 550 524 ÷ 2 = 252 752 775 275 262 + 0;
  • 252 752 775 275 262 ÷ 2 = 126 376 387 637 631 + 0;
  • 126 376 387 637 631 ÷ 2 = 63 188 193 818 815 + 1;
  • 63 188 193 818 815 ÷ 2 = 31 594 096 909 407 + 1;
  • 31 594 096 909 407 ÷ 2 = 15 797 048 454 703 + 1;
  • 15 797 048 454 703 ÷ 2 = 7 898 524 227 351 + 1;
  • 7 898 524 227 351 ÷ 2 = 3 949 262 113 675 + 1;
  • 3 949 262 113 675 ÷ 2 = 1 974 631 056 837 + 1;
  • 1 974 631 056 837 ÷ 2 = 987 315 528 418 + 1;
  • 987 315 528 418 ÷ 2 = 493 657 764 209 + 0;
  • 493 657 764 209 ÷ 2 = 246 828 882 104 + 1;
  • 246 828 882 104 ÷ 2 = 123 414 441 052 + 0;
  • 123 414 441 052 ÷ 2 = 61 707 220 526 + 0;
  • 61 707 220 526 ÷ 2 = 30 853 610 263 + 0;
  • 30 853 610 263 ÷ 2 = 15 426 805 131 + 1;
  • 15 426 805 131 ÷ 2 = 7 713 402 565 + 1;
  • 7 713 402 565 ÷ 2 = 3 856 701 282 + 1;
  • 3 856 701 282 ÷ 2 = 1 928 350 641 + 0;
  • 1 928 350 641 ÷ 2 = 964 175 320 + 1;
  • 964 175 320 ÷ 2 = 482 087 660 + 0;
  • 482 087 660 ÷ 2 = 241 043 830 + 0;
  • 241 043 830 ÷ 2 = 120 521 915 + 0;
  • 120 521 915 ÷ 2 = 60 260 957 + 1;
  • 60 260 957 ÷ 2 = 30 130 478 + 1;
  • 30 130 478 ÷ 2 = 15 065 239 + 0;
  • 15 065 239 ÷ 2 = 7 532 619 + 1;
  • 7 532 619 ÷ 2 = 3 766 309 + 1;
  • 3 766 309 ÷ 2 = 1 883 154 + 1;
  • 1 883 154 ÷ 2 = 941 577 + 0;
  • 941 577 ÷ 2 = 470 788 + 1;
  • 470 788 ÷ 2 = 235 394 + 0;
  • 235 394 ÷ 2 = 117 697 + 0;
  • 117 697 ÷ 2 = 58 848 + 1;
  • 58 848 ÷ 2 = 29 424 + 0;
  • 29 424 ÷ 2 = 14 712 + 0;
  • 14 712 ÷ 2 = 7 356 + 0;
  • 7 356 ÷ 2 = 3 678 + 0;
  • 3 678 ÷ 2 = 1 839 + 0;
  • 1 839 ÷ 2 = 919 + 1;
  • 919 ÷ 2 = 459 + 1;
  • 459 ÷ 2 = 229 + 1;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 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.

1 011 011 101 101 048(10) = 11 1001 0111 1000 0010 0101 1101 1000 1011 1000 1011 1111 1000(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 1 011 011 101 101 048(10) converted to signed binary in two's complement representation:

1 011 011 101 101 048(10) = 0000 0000 0000 0011 1001 0111 1000 0010 0101 1101 1000 1011 1000 1011 1111 1000

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