Two's Complement: Integer ↗ Binary: 4 607 182 418 800 017 466 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number 4 607 182 418 800 017 466(10) converted and written as a signed binary in two's complement representation (base 2) = ?

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;
  • 4 607 182 418 800 017 466 ÷ 2 = 2 303 591 209 400 008 733 + 0;
  • 2 303 591 209 400 008 733 ÷ 2 = 1 151 795 604 700 004 366 + 1;
  • 1 151 795 604 700 004 366 ÷ 2 = 575 897 802 350 002 183 + 0;
  • 575 897 802 350 002 183 ÷ 2 = 287 948 901 175 001 091 + 1;
  • 287 948 901 175 001 091 ÷ 2 = 143 974 450 587 500 545 + 1;
  • 143 974 450 587 500 545 ÷ 2 = 71 987 225 293 750 272 + 1;
  • 71 987 225 293 750 272 ÷ 2 = 35 993 612 646 875 136 + 0;
  • 35 993 612 646 875 136 ÷ 2 = 17 996 806 323 437 568 + 0;
  • 17 996 806 323 437 568 ÷ 2 = 8 998 403 161 718 784 + 0;
  • 8 998 403 161 718 784 ÷ 2 = 4 499 201 580 859 392 + 0;
  • 4 499 201 580 859 392 ÷ 2 = 2 249 600 790 429 696 + 0;
  • 2 249 600 790 429 696 ÷ 2 = 1 124 800 395 214 848 + 0;
  • 1 124 800 395 214 848 ÷ 2 = 562 400 197 607 424 + 0;
  • 562 400 197 607 424 ÷ 2 = 281 200 098 803 712 + 0;
  • 281 200 098 803 712 ÷ 2 = 140 600 049 401 856 + 0;
  • 140 600 049 401 856 ÷ 2 = 70 300 024 700 928 + 0;
  • 70 300 024 700 928 ÷ 2 = 35 150 012 350 464 + 0;
  • 35 150 012 350 464 ÷ 2 = 17 575 006 175 232 + 0;
  • 17 575 006 175 232 ÷ 2 = 8 787 503 087 616 + 0;
  • 8 787 503 087 616 ÷ 2 = 4 393 751 543 808 + 0;
  • 4 393 751 543 808 ÷ 2 = 2 196 875 771 904 + 0;
  • 2 196 875 771 904 ÷ 2 = 1 098 437 885 952 + 0;
  • 1 098 437 885 952 ÷ 2 = 549 218 942 976 + 0;
  • 549 218 942 976 ÷ 2 = 274 609 471 488 + 0;
  • 274 609 471 488 ÷ 2 = 137 304 735 744 + 0;
  • 137 304 735 744 ÷ 2 = 68 652 367 872 + 0;
  • 68 652 367 872 ÷ 2 = 34 326 183 936 + 0;
  • 34 326 183 936 ÷ 2 = 17 163 091 968 + 0;
  • 17 163 091 968 ÷ 2 = 8 581 545 984 + 0;
  • 8 581 545 984 ÷ 2 = 4 290 772 992 + 0;
  • 4 290 772 992 ÷ 2 = 2 145 386 496 + 0;
  • 2 145 386 496 ÷ 2 = 1 072 693 248 + 0;
  • 1 072 693 248 ÷ 2 = 536 346 624 + 0;
  • 536 346 624 ÷ 2 = 268 173 312 + 0;
  • 268 173 312 ÷ 2 = 134 086 656 + 0;
  • 134 086 656 ÷ 2 = 67 043 328 + 0;
  • 67 043 328 ÷ 2 = 33 521 664 + 0;
  • 33 521 664 ÷ 2 = 16 760 832 + 0;
  • 16 760 832 ÷ 2 = 8 380 416 + 0;
  • 8 380 416 ÷ 2 = 4 190 208 + 0;
  • 4 190 208 ÷ 2 = 2 095 104 + 0;
  • 2 095 104 ÷ 2 = 1 047 552 + 0;
  • 1 047 552 ÷ 2 = 523 776 + 0;
  • 523 776 ÷ 2 = 261 888 + 0;
  • 261 888 ÷ 2 = 130 944 + 0;
  • 130 944 ÷ 2 = 65 472 + 0;
  • 65 472 ÷ 2 = 32 736 + 0;
  • 32 736 ÷ 2 = 16 368 + 0;
  • 16 368 ÷ 2 = 8 184 + 0;
  • 8 184 ÷ 2 = 4 092 + 0;
  • 4 092 ÷ 2 = 2 046 + 0;
  • 2 046 ÷ 2 = 1 023 + 0;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 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.


4 607 182 418 800 017 466(10) = 11 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010(2)


3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 62.

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, 62,

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.


Number 4 607 182 418 800 017 466(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:

4 607 182 418 800 017 466(10) = 0011 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1010

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

More operations on integers written as signed binary numbers in two's complement representation:

» Signed integer number 4 607 182 418 800 017 465 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

» Signed integer number 4 607 182 418 800 017 467 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

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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