Convert 1 010 001 000 099 562 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 010 001 000 099 562(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 010 001 000 099 562 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;
  • 1 010 001 000 099 562 ÷ 2 = 505 000 500 049 781 + 0;
  • 505 000 500 049 781 ÷ 2 = 252 500 250 024 890 + 1;
  • 252 500 250 024 890 ÷ 2 = 126 250 125 012 445 + 0;
  • 126 250 125 012 445 ÷ 2 = 63 125 062 506 222 + 1;
  • 63 125 062 506 222 ÷ 2 = 31 562 531 253 111 + 0;
  • 31 562 531 253 111 ÷ 2 = 15 781 265 626 555 + 1;
  • 15 781 265 626 555 ÷ 2 = 7 890 632 813 277 + 1;
  • 7 890 632 813 277 ÷ 2 = 3 945 316 406 638 + 1;
  • 3 945 316 406 638 ÷ 2 = 1 972 658 203 319 + 0;
  • 1 972 658 203 319 ÷ 2 = 986 329 101 659 + 1;
  • 986 329 101 659 ÷ 2 = 493 164 550 829 + 1;
  • 493 164 550 829 ÷ 2 = 246 582 275 414 + 1;
  • 246 582 275 414 ÷ 2 = 123 291 137 707 + 0;
  • 123 291 137 707 ÷ 2 = 61 645 568 853 + 1;
  • 61 645 568 853 ÷ 2 = 30 822 784 426 + 1;
  • 30 822 784 426 ÷ 2 = 15 411 392 213 + 0;
  • 15 411 392 213 ÷ 2 = 7 705 696 106 + 1;
  • 7 705 696 106 ÷ 2 = 3 852 848 053 + 0;
  • 3 852 848 053 ÷ 2 = 1 926 424 026 + 1;
  • 1 926 424 026 ÷ 2 = 963 212 013 + 0;
  • 963 212 013 ÷ 2 = 481 606 006 + 1;
  • 481 606 006 ÷ 2 = 240 803 003 + 0;
  • 240 803 003 ÷ 2 = 120 401 501 + 1;
  • 120 401 501 ÷ 2 = 60 200 750 + 1;
  • 60 200 750 ÷ 2 = 30 100 375 + 0;
  • 30 100 375 ÷ 2 = 15 050 187 + 1;
  • 15 050 187 ÷ 2 = 7 525 093 + 1;
  • 7 525 093 ÷ 2 = 3 762 546 + 1;
  • 3 762 546 ÷ 2 = 1 881 273 + 0;
  • 1 881 273 ÷ 2 = 940 636 + 1;
  • 940 636 ÷ 2 = 470 318 + 0;
  • 470 318 ÷ 2 = 235 159 + 0;
  • 235 159 ÷ 2 = 117 579 + 1;
  • 117 579 ÷ 2 = 58 789 + 1;
  • 58 789 ÷ 2 = 29 394 + 1;
  • 29 394 ÷ 2 = 14 697 + 0;
  • 14 697 ÷ 2 = 7 348 + 1;
  • 7 348 ÷ 2 = 3 674 + 0;
  • 3 674 ÷ 2 = 1 837 + 0;
  • 1 837 ÷ 2 = 918 + 1;
  • 918 ÷ 2 = 459 + 0;
  • 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.

1 010 001 000 099 562(10) = 11 1001 0110 1001 0111 0010 1110 1101 0101 0110 1110 1110 1010(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 1 010 001 000 099 562(10) converted to signed binary in two's complement representation:

1 010 001 000 099 562(10) = 0000 0000 0000 0011 1001 0110 1001 0111 0010 1110 1101 0101 0110 1110 1110 1010

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