Convert -2 400 000 123 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -2 400 000 123(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-2 400 000 123 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:

|-2 400 000 123| = 2 400 000 123

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;
  • 2 400 000 123 ÷ 2 = 1 200 000 061 + 1;
  • 1 200 000 061 ÷ 2 = 600 000 030 + 1;
  • 600 000 030 ÷ 2 = 300 000 015 + 0;
  • 300 000 015 ÷ 2 = 150 000 007 + 1;
  • 150 000 007 ÷ 2 = 75 000 003 + 1;
  • 75 000 003 ÷ 2 = 37 500 001 + 1;
  • 37 500 001 ÷ 2 = 18 750 000 + 1;
  • 18 750 000 ÷ 2 = 9 375 000 + 0;
  • 9 375 000 ÷ 2 = 4 687 500 + 0;
  • 4 687 500 ÷ 2 = 2 343 750 + 0;
  • 2 343 750 ÷ 2 = 1 171 875 + 0;
  • 1 171 875 ÷ 2 = 585 937 + 1;
  • 585 937 ÷ 2 = 292 968 + 1;
  • 292 968 ÷ 2 = 146 484 + 0;
  • 146 484 ÷ 2 = 73 242 + 0;
  • 73 242 ÷ 2 = 36 621 + 0;
  • 36 621 ÷ 2 = 18 310 + 1;
  • 18 310 ÷ 2 = 9 155 + 0;
  • 9 155 ÷ 2 = 4 577 + 1;
  • 4 577 ÷ 2 = 2 288 + 1;
  • 2 288 ÷ 2 = 1 144 + 0;
  • 1 144 ÷ 2 = 572 + 0;
  • 572 ÷ 2 = 286 + 0;
  • 286 ÷ 2 = 143 + 0;
  • 143 ÷ 2 = 71 + 1;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.

2 400 000 123(10) = 1000 1111 0000 1101 0001 1000 0111 1011(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 32.

  • 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, 32,

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.


2 400 000 123(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1000 1111 0000 1101 0001 1000 0111 1011

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 0000 0000 0000 0000 0000 1000 1111 0000 1101 0001 1000 0111 1011)

= 1111 1111 1111 1111 1111 1111 1111 1111 0111 0000 1111 0010 1110 0111 1000 0100


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 1111 1111 1111 1111 1111 0111 0000 1111 0010 1110 0111 1000 0100 (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):

-2 400 000 123 =

1111 1111 1111 1111 1111 1111 1111 1111 0111 0000 1111 0010 1110 0111 1000 0100 + 1


Decimal Number -2 400 000 123(10) converted to signed binary in two's complement representation:

-2 400 000 123(10) = 1111 1111 1111 1111 1111 1111 1111 1111 0111 0000 1111 0010 1110 0111 1000 0101

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