Convert 10 111 101 010 100 986 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 111 101 010 100 986(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 111 101 010 100 986 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;
  • 10 111 101 010 100 986 ÷ 2 = 5 055 550 505 050 493 + 0;
  • 5 055 550 505 050 493 ÷ 2 = 2 527 775 252 525 246 + 1;
  • 2 527 775 252 525 246 ÷ 2 = 1 263 887 626 262 623 + 0;
  • 1 263 887 626 262 623 ÷ 2 = 631 943 813 131 311 + 1;
  • 631 943 813 131 311 ÷ 2 = 315 971 906 565 655 + 1;
  • 315 971 906 565 655 ÷ 2 = 157 985 953 282 827 + 1;
  • 157 985 953 282 827 ÷ 2 = 78 992 976 641 413 + 1;
  • 78 992 976 641 413 ÷ 2 = 39 496 488 320 706 + 1;
  • 39 496 488 320 706 ÷ 2 = 19 748 244 160 353 + 0;
  • 19 748 244 160 353 ÷ 2 = 9 874 122 080 176 + 1;
  • 9 874 122 080 176 ÷ 2 = 4 937 061 040 088 + 0;
  • 4 937 061 040 088 ÷ 2 = 2 468 530 520 044 + 0;
  • 2 468 530 520 044 ÷ 2 = 1 234 265 260 022 + 0;
  • 1 234 265 260 022 ÷ 2 = 617 132 630 011 + 0;
  • 617 132 630 011 ÷ 2 = 308 566 315 005 + 1;
  • 308 566 315 005 ÷ 2 = 154 283 157 502 + 1;
  • 154 283 157 502 ÷ 2 = 77 141 578 751 + 0;
  • 77 141 578 751 ÷ 2 = 38 570 789 375 + 1;
  • 38 570 789 375 ÷ 2 = 19 285 394 687 + 1;
  • 19 285 394 687 ÷ 2 = 9 642 697 343 + 1;
  • 9 642 697 343 ÷ 2 = 4 821 348 671 + 1;
  • 4 821 348 671 ÷ 2 = 2 410 674 335 + 1;
  • 2 410 674 335 ÷ 2 = 1 205 337 167 + 1;
  • 1 205 337 167 ÷ 2 = 602 668 583 + 1;
  • 602 668 583 ÷ 2 = 301 334 291 + 1;
  • 301 334 291 ÷ 2 = 150 667 145 + 1;
  • 150 667 145 ÷ 2 = 75 333 572 + 1;
  • 75 333 572 ÷ 2 = 37 666 786 + 0;
  • 37 666 786 ÷ 2 = 18 833 393 + 0;
  • 18 833 393 ÷ 2 = 9 416 696 + 1;
  • 9 416 696 ÷ 2 = 4 708 348 + 0;
  • 4 708 348 ÷ 2 = 2 354 174 + 0;
  • 2 354 174 ÷ 2 = 1 177 087 + 0;
  • 1 177 087 ÷ 2 = 588 543 + 1;
  • 588 543 ÷ 2 = 294 271 + 1;
  • 294 271 ÷ 2 = 147 135 + 1;
  • 147 135 ÷ 2 = 73 567 + 1;
  • 73 567 ÷ 2 = 36 783 + 1;
  • 36 783 ÷ 2 = 18 391 + 1;
  • 18 391 ÷ 2 = 9 195 + 1;
  • 9 195 ÷ 2 = 4 597 + 1;
  • 4 597 ÷ 2 = 2 298 + 1;
  • 2 298 ÷ 2 = 1 149 + 0;
  • 1 149 ÷ 2 = 574 + 1;
  • 574 ÷ 2 = 287 + 0;
  • 287 ÷ 2 = 143 + 1;
  • 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;

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

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

10 111 101 010 100 986(10) = 10 0011 1110 1011 1111 1110 0010 0111 1111 1110 1100 0010 1111 1010(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 10 111 101 010 100 986(10) converted to signed binary in two's complement representation:

10 111 101 010 100 986(10) = 0000 0000 0010 0011 1110 1011 1111 1110 0010 0111 1111 1110 1100 0010 1111 1010

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