Convert 10 001 010 110 010 451 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 001 010 110 010 451(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 001 010 110 010 451 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;
  • 10 001 010 110 010 451 ÷ 2 = 5 000 505 055 005 225 + 1;
  • 5 000 505 055 005 225 ÷ 2 = 2 500 252 527 502 612 + 1;
  • 2 500 252 527 502 612 ÷ 2 = 1 250 126 263 751 306 + 0;
  • 1 250 126 263 751 306 ÷ 2 = 625 063 131 875 653 + 0;
  • 625 063 131 875 653 ÷ 2 = 312 531 565 937 826 + 1;
  • 312 531 565 937 826 ÷ 2 = 156 265 782 968 913 + 0;
  • 156 265 782 968 913 ÷ 2 = 78 132 891 484 456 + 1;
  • 78 132 891 484 456 ÷ 2 = 39 066 445 742 228 + 0;
  • 39 066 445 742 228 ÷ 2 = 19 533 222 871 114 + 0;
  • 19 533 222 871 114 ÷ 2 = 9 766 611 435 557 + 0;
  • 9 766 611 435 557 ÷ 2 = 4 883 305 717 778 + 1;
  • 4 883 305 717 778 ÷ 2 = 2 441 652 858 889 + 0;
  • 2 441 652 858 889 ÷ 2 = 1 220 826 429 444 + 1;
  • 1 220 826 429 444 ÷ 2 = 610 413 214 722 + 0;
  • 610 413 214 722 ÷ 2 = 305 206 607 361 + 0;
  • 305 206 607 361 ÷ 2 = 152 603 303 680 + 1;
  • 152 603 303 680 ÷ 2 = 76 301 651 840 + 0;
  • 76 301 651 840 ÷ 2 = 38 150 825 920 + 0;
  • 38 150 825 920 ÷ 2 = 19 075 412 960 + 0;
  • 19 075 412 960 ÷ 2 = 9 537 706 480 + 0;
  • 9 537 706 480 ÷ 2 = 4 768 853 240 + 0;
  • 4 768 853 240 ÷ 2 = 2 384 426 620 + 0;
  • 2 384 426 620 ÷ 2 = 1 192 213 310 + 0;
  • 1 192 213 310 ÷ 2 = 596 106 655 + 0;
  • 596 106 655 ÷ 2 = 298 053 327 + 1;
  • 298 053 327 ÷ 2 = 149 026 663 + 1;
  • 149 026 663 ÷ 2 = 74 513 331 + 1;
  • 74 513 331 ÷ 2 = 37 256 665 + 1;
  • 37 256 665 ÷ 2 = 18 628 332 + 1;
  • 18 628 332 ÷ 2 = 9 314 166 + 0;
  • 9 314 166 ÷ 2 = 4 657 083 + 0;
  • 4 657 083 ÷ 2 = 2 328 541 + 1;
  • 2 328 541 ÷ 2 = 1 164 270 + 1;
  • 1 164 270 ÷ 2 = 582 135 + 0;
  • 582 135 ÷ 2 = 291 067 + 1;
  • 291 067 ÷ 2 = 145 533 + 1;
  • 145 533 ÷ 2 = 72 766 + 1;
  • 72 766 ÷ 2 = 36 383 + 0;
  • 36 383 ÷ 2 = 18 191 + 1;
  • 18 191 ÷ 2 = 9 095 + 1;
  • 9 095 ÷ 2 = 4 547 + 1;
  • 4 547 ÷ 2 = 2 273 + 1;
  • 2 273 ÷ 2 = 1 136 + 1;
  • 1 136 ÷ 2 = 568 + 0;
  • 568 ÷ 2 = 284 + 0;
  • 284 ÷ 2 = 142 + 0;
  • 142 ÷ 2 = 71 + 0;
  • 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;

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

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

10 001 010 110 010 451(10) = 10 0011 1000 0111 1101 1101 1001 1111 0000 0000 1001 0100 0101 0011(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 54.

  • 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, 54,

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 10 001 010 110 010 451(10) converted to signed binary in two's complement representation:

10 001 010 110 010 451(10) = 0000 0000 0010 0011 1000 0111 1101 1101 1001 1111 0000 0000 1001 0100 0101 0011

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