Convert 10 011 101 010 011 260 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 011 101 010 011 260(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 011 101 010 011 260 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;
  • 10 011 101 010 011 260 ÷ 2 = 5 005 550 505 005 630 + 0;
  • 5 005 550 505 005 630 ÷ 2 = 2 502 775 252 502 815 + 0;
  • 2 502 775 252 502 815 ÷ 2 = 1 251 387 626 251 407 + 1;
  • 1 251 387 626 251 407 ÷ 2 = 625 693 813 125 703 + 1;
  • 625 693 813 125 703 ÷ 2 = 312 846 906 562 851 + 1;
  • 312 846 906 562 851 ÷ 2 = 156 423 453 281 425 + 1;
  • 156 423 453 281 425 ÷ 2 = 78 211 726 640 712 + 1;
  • 78 211 726 640 712 ÷ 2 = 39 105 863 320 356 + 0;
  • 39 105 863 320 356 ÷ 2 = 19 552 931 660 178 + 0;
  • 19 552 931 660 178 ÷ 2 = 9 776 465 830 089 + 0;
  • 9 776 465 830 089 ÷ 2 = 4 888 232 915 044 + 1;
  • 4 888 232 915 044 ÷ 2 = 2 444 116 457 522 + 0;
  • 2 444 116 457 522 ÷ 2 = 1 222 058 228 761 + 0;
  • 1 222 058 228 761 ÷ 2 = 611 029 114 380 + 1;
  • 611 029 114 380 ÷ 2 = 305 514 557 190 + 0;
  • 305 514 557 190 ÷ 2 = 152 757 278 595 + 0;
  • 152 757 278 595 ÷ 2 = 76 378 639 297 + 1;
  • 76 378 639 297 ÷ 2 = 38 189 319 648 + 1;
  • 38 189 319 648 ÷ 2 = 19 094 659 824 + 0;
  • 19 094 659 824 ÷ 2 = 9 547 329 912 + 0;
  • 9 547 329 912 ÷ 2 = 4 773 664 956 + 0;
  • 4 773 664 956 ÷ 2 = 2 386 832 478 + 0;
  • 2 386 832 478 ÷ 2 = 1 193 416 239 + 0;
  • 1 193 416 239 ÷ 2 = 596 708 119 + 1;
  • 596 708 119 ÷ 2 = 298 354 059 + 1;
  • 298 354 059 ÷ 2 = 149 177 029 + 1;
  • 149 177 029 ÷ 2 = 74 588 514 + 1;
  • 74 588 514 ÷ 2 = 37 294 257 + 0;
  • 37 294 257 ÷ 2 = 18 647 128 + 1;
  • 18 647 128 ÷ 2 = 9 323 564 + 0;
  • 9 323 564 ÷ 2 = 4 661 782 + 0;
  • 4 661 782 ÷ 2 = 2 330 891 + 0;
  • 2 330 891 ÷ 2 = 1 165 445 + 1;
  • 1 165 445 ÷ 2 = 582 722 + 1;
  • 582 722 ÷ 2 = 291 361 + 0;
  • 291 361 ÷ 2 = 145 680 + 1;
  • 145 680 ÷ 2 = 72 840 + 0;
  • 72 840 ÷ 2 = 36 420 + 0;
  • 36 420 ÷ 2 = 18 210 + 0;
  • 18 210 ÷ 2 = 9 105 + 0;
  • 9 105 ÷ 2 = 4 552 + 1;
  • 4 552 ÷ 2 = 2 276 + 0;
  • 2 276 ÷ 2 = 1 138 + 0;
  • 1 138 ÷ 2 = 569 + 0;
  • 569 ÷ 2 = 284 + 1;
  • 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.

10 011 101 010 011 260(10) = 10 0011 1001 0001 0000 1011 0001 0111 1000 0011 0010 0100 0111 1100(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 10 011 101 010 011 260(10) converted to signed binary in two's complement representation:

10 011 101 010 011 260(10) = 0000 0000 0010 0011 1001 0001 0000 1011 0001 0111 1000 0011 0010 0100 0111 1100

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