Convert -1 000 000 000 000 122 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -1 000 000 000 000 122(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-1 000 000 000 000 122 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. Start with the positive version of the number:

|-1 000 000 000 000 122| = 1 000 000 000 000 122

2. 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;
  • 1 000 000 000 000 122 ÷ 2 = 500 000 000 000 061 + 0;
  • 500 000 000 000 061 ÷ 2 = 250 000 000 000 030 + 1;
  • 250 000 000 000 030 ÷ 2 = 125 000 000 000 015 + 0;
  • 125 000 000 000 015 ÷ 2 = 62 500 000 000 007 + 1;
  • 62 500 000 000 007 ÷ 2 = 31 250 000 000 003 + 1;
  • 31 250 000 000 003 ÷ 2 = 15 625 000 000 001 + 1;
  • 15 625 000 000 001 ÷ 2 = 7 812 500 000 000 + 1;
  • 7 812 500 000 000 ÷ 2 = 3 906 250 000 000 + 0;
  • 3 906 250 000 000 ÷ 2 = 1 953 125 000 000 + 0;
  • 1 953 125 000 000 ÷ 2 = 976 562 500 000 + 0;
  • 976 562 500 000 ÷ 2 = 488 281 250 000 + 0;
  • 488 281 250 000 ÷ 2 = 244 140 625 000 + 0;
  • 244 140 625 000 ÷ 2 = 122 070 312 500 + 0;
  • 122 070 312 500 ÷ 2 = 61 035 156 250 + 0;
  • 61 035 156 250 ÷ 2 = 30 517 578 125 + 0;
  • 30 517 578 125 ÷ 2 = 15 258 789 062 + 1;
  • 15 258 789 062 ÷ 2 = 7 629 394 531 + 0;
  • 7 629 394 531 ÷ 2 = 3 814 697 265 + 1;
  • 3 814 697 265 ÷ 2 = 1 907 348 632 + 1;
  • 1 907 348 632 ÷ 2 = 953 674 316 + 0;
  • 953 674 316 ÷ 2 = 476 837 158 + 0;
  • 476 837 158 ÷ 2 = 238 418 579 + 0;
  • 238 418 579 ÷ 2 = 119 209 289 + 1;
  • 119 209 289 ÷ 2 = 59 604 644 + 1;
  • 59 604 644 ÷ 2 = 29 802 322 + 0;
  • 29 802 322 ÷ 2 = 14 901 161 + 0;
  • 14 901 161 ÷ 2 = 7 450 580 + 1;
  • 7 450 580 ÷ 2 = 3 725 290 + 0;
  • 3 725 290 ÷ 2 = 1 862 645 + 0;
  • 1 862 645 ÷ 2 = 931 322 + 1;
  • 931 322 ÷ 2 = 465 661 + 0;
  • 465 661 ÷ 2 = 232 830 + 1;
  • 232 830 ÷ 2 = 116 415 + 0;
  • 116 415 ÷ 2 = 58 207 + 1;
  • 58 207 ÷ 2 = 29 103 + 1;
  • 29 103 ÷ 2 = 14 551 + 1;
  • 14 551 ÷ 2 = 7 275 + 1;
  • 7 275 ÷ 2 = 3 637 + 1;
  • 3 637 ÷ 2 = 1 818 + 1;
  • 1 818 ÷ 2 = 909 + 0;
  • 909 ÷ 2 = 454 + 1;
  • 454 ÷ 2 = 227 + 0;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

3. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

1 000 000 000 000 122(10) = 11 1000 1101 0111 1110 1010 0100 1100 0110 1000 0000 0111 1010(2)

4. 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.


5. 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.


1 000 000 000 000 122(10) = 0000 0000 0000 0011 1000 1101 0111 1110 1010 0100 1100 0110 1000 0000 0111 1010

6. Get the negative integer number representation. Part 1:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation, replace all the bits on 0 with 1s and all the bits set on 1 with 0s.

Reverse the digits, flip the digits:

Replace the bits set on 0 with 1s and the bits set on 1 with 0s.

!(0000 0000 0000 0011 1000 1101 0111 1110 1010 0100 1100 0110 1000 0000 0111 1010)

= 1111 1111 1111 1100 0111 0010 1000 0001 0101 1011 0011 1001 0111 1111 1000 0101


7. Get the negative integer number representation. Part 2:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in two's complement representation, add 1 to the number calculated above 1111 1111 1111 1100 0111 0010 1000 0001 0101 1011 0011 1001 0111 1111 1000 0101 (to the signed binary in one's complement representation).

Binary addition carries on a value of 2:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 10 = 11
  • 1 + 11 = 100

Add 1 to the number calculated above
(to the signed binary number in one's complement representation):

-1 000 000 000 000 122 =

1111 1111 1111 1100 0111 0010 1000 0001 0101 1011 0011 1001 0111 1111 1000 0101 + 1


Decimal Number -1 000 000 000 000 122(10) converted to signed binary in two's complement representation:

-1 000 000 000 000 122(10) = 1111 1111 1111 1100 0111 0010 1000 0001 0101 1011 0011 1001 0111 1111 1000 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