Convert 1 017 101 610 150 838 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 017 101 610 150 838(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 017 101 610 150 838 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;
  • 1 017 101 610 150 838 ÷ 2 = 508 550 805 075 419 + 0;
  • 508 550 805 075 419 ÷ 2 = 254 275 402 537 709 + 1;
  • 254 275 402 537 709 ÷ 2 = 127 137 701 268 854 + 1;
  • 127 137 701 268 854 ÷ 2 = 63 568 850 634 427 + 0;
  • 63 568 850 634 427 ÷ 2 = 31 784 425 317 213 + 1;
  • 31 784 425 317 213 ÷ 2 = 15 892 212 658 606 + 1;
  • 15 892 212 658 606 ÷ 2 = 7 946 106 329 303 + 0;
  • 7 946 106 329 303 ÷ 2 = 3 973 053 164 651 + 1;
  • 3 973 053 164 651 ÷ 2 = 1 986 526 582 325 + 1;
  • 1 986 526 582 325 ÷ 2 = 993 263 291 162 + 1;
  • 993 263 291 162 ÷ 2 = 496 631 645 581 + 0;
  • 496 631 645 581 ÷ 2 = 248 315 822 790 + 1;
  • 248 315 822 790 ÷ 2 = 124 157 911 395 + 0;
  • 124 157 911 395 ÷ 2 = 62 078 955 697 + 1;
  • 62 078 955 697 ÷ 2 = 31 039 477 848 + 1;
  • 31 039 477 848 ÷ 2 = 15 519 738 924 + 0;
  • 15 519 738 924 ÷ 2 = 7 759 869 462 + 0;
  • 7 759 869 462 ÷ 2 = 3 879 934 731 + 0;
  • 3 879 934 731 ÷ 2 = 1 939 967 365 + 1;
  • 1 939 967 365 ÷ 2 = 969 983 682 + 1;
  • 969 983 682 ÷ 2 = 484 991 841 + 0;
  • 484 991 841 ÷ 2 = 242 495 920 + 1;
  • 242 495 920 ÷ 2 = 121 247 960 + 0;
  • 121 247 960 ÷ 2 = 60 623 980 + 0;
  • 60 623 980 ÷ 2 = 30 311 990 + 0;
  • 30 311 990 ÷ 2 = 15 155 995 + 0;
  • 15 155 995 ÷ 2 = 7 577 997 + 1;
  • 7 577 997 ÷ 2 = 3 788 998 + 1;
  • 3 788 998 ÷ 2 = 1 894 499 + 0;
  • 1 894 499 ÷ 2 = 947 249 + 1;
  • 947 249 ÷ 2 = 473 624 + 1;
  • 473 624 ÷ 2 = 236 812 + 0;
  • 236 812 ÷ 2 = 118 406 + 0;
  • 118 406 ÷ 2 = 59 203 + 0;
  • 59 203 ÷ 2 = 29 601 + 1;
  • 29 601 ÷ 2 = 14 800 + 1;
  • 14 800 ÷ 2 = 7 400 + 0;
  • 7 400 ÷ 2 = 3 700 + 0;
  • 3 700 ÷ 2 = 1 850 + 0;
  • 1 850 ÷ 2 = 925 + 0;
  • 925 ÷ 2 = 462 + 1;
  • 462 ÷ 2 = 231 + 0;
  • 231 ÷ 2 = 115 + 1;
  • 115 ÷ 2 = 57 + 1;
  • 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.

1 017 101 610 150 838(10) = 11 1001 1101 0000 1100 0110 1100 0010 1100 0110 1011 1011 0110(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 1 017 101 610 150 838(10) converted to signed binary in two's complement representation:

1 017 101 610 150 838(10) = 0000 0000 0000 0011 1001 1101 0000 1100 0110 1100 0010 1100 0110 1011 1011 0110

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