Convert 1 101 100 101 110 120 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 101 100 101 110 120(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 101 100 101 110 120 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;
  • 1 101 100 101 110 120 ÷ 2 = 550 550 050 555 060 + 0;
  • 550 550 050 555 060 ÷ 2 = 275 275 025 277 530 + 0;
  • 275 275 025 277 530 ÷ 2 = 137 637 512 638 765 + 0;
  • 137 637 512 638 765 ÷ 2 = 68 818 756 319 382 + 1;
  • 68 818 756 319 382 ÷ 2 = 34 409 378 159 691 + 0;
  • 34 409 378 159 691 ÷ 2 = 17 204 689 079 845 + 1;
  • 17 204 689 079 845 ÷ 2 = 8 602 344 539 922 + 1;
  • 8 602 344 539 922 ÷ 2 = 4 301 172 269 961 + 0;
  • 4 301 172 269 961 ÷ 2 = 2 150 586 134 980 + 1;
  • 2 150 586 134 980 ÷ 2 = 1 075 293 067 490 + 0;
  • 1 075 293 067 490 ÷ 2 = 537 646 533 745 + 0;
  • 537 646 533 745 ÷ 2 = 268 823 266 872 + 1;
  • 268 823 266 872 ÷ 2 = 134 411 633 436 + 0;
  • 134 411 633 436 ÷ 2 = 67 205 816 718 + 0;
  • 67 205 816 718 ÷ 2 = 33 602 908 359 + 0;
  • 33 602 908 359 ÷ 2 = 16 801 454 179 + 1;
  • 16 801 454 179 ÷ 2 = 8 400 727 089 + 1;
  • 8 400 727 089 ÷ 2 = 4 200 363 544 + 1;
  • 4 200 363 544 ÷ 2 = 2 100 181 772 + 0;
  • 2 100 181 772 ÷ 2 = 1 050 090 886 + 0;
  • 1 050 090 886 ÷ 2 = 525 045 443 + 0;
  • 525 045 443 ÷ 2 = 262 522 721 + 1;
  • 262 522 721 ÷ 2 = 131 261 360 + 1;
  • 131 261 360 ÷ 2 = 65 630 680 + 0;
  • 65 630 680 ÷ 2 = 32 815 340 + 0;
  • 32 815 340 ÷ 2 = 16 407 670 + 0;
  • 16 407 670 ÷ 2 = 8 203 835 + 0;
  • 8 203 835 ÷ 2 = 4 101 917 + 1;
  • 4 101 917 ÷ 2 = 2 050 958 + 1;
  • 2 050 958 ÷ 2 = 1 025 479 + 0;
  • 1 025 479 ÷ 2 = 512 739 + 1;
  • 512 739 ÷ 2 = 256 369 + 1;
  • 256 369 ÷ 2 = 128 184 + 1;
  • 128 184 ÷ 2 = 64 092 + 0;
  • 64 092 ÷ 2 = 32 046 + 0;
  • 32 046 ÷ 2 = 16 023 + 0;
  • 16 023 ÷ 2 = 8 011 + 1;
  • 8 011 ÷ 2 = 4 005 + 1;
  • 4 005 ÷ 2 = 2 002 + 1;
  • 2 002 ÷ 2 = 1 001 + 0;
  • 1 001 ÷ 2 = 500 + 1;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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.

1 101 100 101 110 120(10) = 11 1110 1001 0111 0001 1101 1000 0110 0011 1000 1001 0110 1000(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 1 101 100 101 110 120(10) converted to signed binary in two's complement representation:

1 101 100 101 110 120(10) = 0000 0000 0000 0011 1110 1001 0111 0001 1101 1000 0110 0011 1000 1001 0110 1000

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