Convert 500 000 498 970 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 500 000 498 970(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
500 000 498 970 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;
  • 500 000 498 970 ÷ 2 = 250 000 249 485 + 0;
  • 250 000 249 485 ÷ 2 = 125 000 124 742 + 1;
  • 125 000 124 742 ÷ 2 = 62 500 062 371 + 0;
  • 62 500 062 371 ÷ 2 = 31 250 031 185 + 1;
  • 31 250 031 185 ÷ 2 = 15 625 015 592 + 1;
  • 15 625 015 592 ÷ 2 = 7 812 507 796 + 0;
  • 7 812 507 796 ÷ 2 = 3 906 253 898 + 0;
  • 3 906 253 898 ÷ 2 = 1 953 126 949 + 0;
  • 1 953 126 949 ÷ 2 = 976 563 474 + 1;
  • 976 563 474 ÷ 2 = 488 281 737 + 0;
  • 488 281 737 ÷ 2 = 244 140 868 + 1;
  • 244 140 868 ÷ 2 = 122 070 434 + 0;
  • 122 070 434 ÷ 2 = 61 035 217 + 0;
  • 61 035 217 ÷ 2 = 30 517 608 + 1;
  • 30 517 608 ÷ 2 = 15 258 804 + 0;
  • 15 258 804 ÷ 2 = 7 629 402 + 0;
  • 7 629 402 ÷ 2 = 3 814 701 + 0;
  • 3 814 701 ÷ 2 = 1 907 350 + 1;
  • 1 907 350 ÷ 2 = 953 675 + 0;
  • 953 675 ÷ 2 = 476 837 + 1;
  • 476 837 ÷ 2 = 238 418 + 1;
  • 238 418 ÷ 2 = 119 209 + 0;
  • 119 209 ÷ 2 = 59 604 + 1;
  • 59 604 ÷ 2 = 29 802 + 0;
  • 29 802 ÷ 2 = 14 901 + 0;
  • 14 901 ÷ 2 = 7 450 + 1;
  • 7 450 ÷ 2 = 3 725 + 0;
  • 3 725 ÷ 2 = 1 862 + 1;
  • 1 862 ÷ 2 = 931 + 0;
  • 931 ÷ 2 = 465 + 1;
  • 465 ÷ 2 = 232 + 1;
  • 232 ÷ 2 = 116 + 0;
  • 116 ÷ 2 = 58 + 0;
  • 58 ÷ 2 = 29 + 0;
  • 29 ÷ 2 = 14 + 1;
  • 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.

500 000 498 970(10) = 111 0100 0110 1010 0101 1010 0010 0101 0001 1010(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 39.

  • 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, 39,

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 500 000 498 970(10) converted to signed binary in two's complement representation:

500 000 498 970(10) = 0000 0000 0000 0000 0000 0000 0111 0100 0110 1010 0101 1010 0010 0101 0001 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