Convert 1 163 670 569 177 170 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 163 670 569 177 170(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 163 670 569 177 170 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;
  • 1 163 670 569 177 170 ÷ 2 = 581 835 284 588 585 + 0;
  • 581 835 284 588 585 ÷ 2 = 290 917 642 294 292 + 1;
  • 290 917 642 294 292 ÷ 2 = 145 458 821 147 146 + 0;
  • 145 458 821 147 146 ÷ 2 = 72 729 410 573 573 + 0;
  • 72 729 410 573 573 ÷ 2 = 36 364 705 286 786 + 1;
  • 36 364 705 286 786 ÷ 2 = 18 182 352 643 393 + 0;
  • 18 182 352 643 393 ÷ 2 = 9 091 176 321 696 + 1;
  • 9 091 176 321 696 ÷ 2 = 4 545 588 160 848 + 0;
  • 4 545 588 160 848 ÷ 2 = 2 272 794 080 424 + 0;
  • 2 272 794 080 424 ÷ 2 = 1 136 397 040 212 + 0;
  • 1 136 397 040 212 ÷ 2 = 568 198 520 106 + 0;
  • 568 198 520 106 ÷ 2 = 284 099 260 053 + 0;
  • 284 099 260 053 ÷ 2 = 142 049 630 026 + 1;
  • 142 049 630 026 ÷ 2 = 71 024 815 013 + 0;
  • 71 024 815 013 ÷ 2 = 35 512 407 506 + 1;
  • 35 512 407 506 ÷ 2 = 17 756 203 753 + 0;
  • 17 756 203 753 ÷ 2 = 8 878 101 876 + 1;
  • 8 878 101 876 ÷ 2 = 4 439 050 938 + 0;
  • 4 439 050 938 ÷ 2 = 2 219 525 469 + 0;
  • 2 219 525 469 ÷ 2 = 1 109 762 734 + 1;
  • 1 109 762 734 ÷ 2 = 554 881 367 + 0;
  • 554 881 367 ÷ 2 = 277 440 683 + 1;
  • 277 440 683 ÷ 2 = 138 720 341 + 1;
  • 138 720 341 ÷ 2 = 69 360 170 + 1;
  • 69 360 170 ÷ 2 = 34 680 085 + 0;
  • 34 680 085 ÷ 2 = 17 340 042 + 1;
  • 17 340 042 ÷ 2 = 8 670 021 + 0;
  • 8 670 021 ÷ 2 = 4 335 010 + 1;
  • 4 335 010 ÷ 2 = 2 167 505 + 0;
  • 2 167 505 ÷ 2 = 1 083 752 + 1;
  • 1 083 752 ÷ 2 = 541 876 + 0;
  • 541 876 ÷ 2 = 270 938 + 0;
  • 270 938 ÷ 2 = 135 469 + 0;
  • 135 469 ÷ 2 = 67 734 + 1;
  • 67 734 ÷ 2 = 33 867 + 0;
  • 33 867 ÷ 2 = 16 933 + 1;
  • 16 933 ÷ 2 = 8 466 + 1;
  • 8 466 ÷ 2 = 4 233 + 0;
  • 4 233 ÷ 2 = 2 116 + 1;
  • 2 116 ÷ 2 = 1 058 + 0;
  • 1 058 ÷ 2 = 529 + 0;
  • 529 ÷ 2 = 264 + 1;
  • 264 ÷ 2 = 132 + 0;
  • 132 ÷ 2 = 66 + 0;
  • 66 ÷ 2 = 33 + 0;
  • 33 ÷ 2 = 16 + 1;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 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.

1 163 670 569 177 170(10) = 100 0010 0010 0101 1010 0010 1010 1110 1001 0101 0000 0101 0010(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 51.

  • 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, 51,

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 1 163 670 569 177 170(10) converted to signed binary in two's complement representation:

1 163 670 569 177 170(10) = 0000 0000 0000 0100 0010 0010 0101 1010 0010 1010 1110 1001 0101 0000 0101 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