Base ten decimal system signed integer number 7 177 689 270 052 918 071 converted to signed binary in two's complement representation

How to convert a signed integer in decimal system (in base 10):
7 177 689 270 052 918 071(10)
to a signed binary two's complement representation

1. Divide the number repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:

  • division = quotient + remainder;
  • 7 177 689 270 052 918 071 ÷ 2 = 3 588 844 635 026 459 035 + 1;
  • 3 588 844 635 026 459 035 ÷ 2 = 1 794 422 317 513 229 517 + 1;
  • 1 794 422 317 513 229 517 ÷ 2 = 897 211 158 756 614 758 + 1;
  • 897 211 158 756 614 758 ÷ 2 = 448 605 579 378 307 379 + 0;
  • 448 605 579 378 307 379 ÷ 2 = 224 302 789 689 153 689 + 1;
  • 224 302 789 689 153 689 ÷ 2 = 112 151 394 844 576 844 + 1;
  • 112 151 394 844 576 844 ÷ 2 = 56 075 697 422 288 422 + 0;
  • 56 075 697 422 288 422 ÷ 2 = 28 037 848 711 144 211 + 0;
  • 28 037 848 711 144 211 ÷ 2 = 14 018 924 355 572 105 + 1;
  • 14 018 924 355 572 105 ÷ 2 = 7 009 462 177 786 052 + 1;
  • 7 009 462 177 786 052 ÷ 2 = 3 504 731 088 893 026 + 0;
  • 3 504 731 088 893 026 ÷ 2 = 1 752 365 544 446 513 + 0;
  • 1 752 365 544 446 513 ÷ 2 = 876 182 772 223 256 + 1;
  • 876 182 772 223 256 ÷ 2 = 438 091 386 111 628 + 0;
  • 438 091 386 111 628 ÷ 2 = 219 045 693 055 814 + 0;
  • 219 045 693 055 814 ÷ 2 = 109 522 846 527 907 + 0;
  • 109 522 846 527 907 ÷ 2 = 54 761 423 263 953 + 1;
  • 54 761 423 263 953 ÷ 2 = 27 380 711 631 976 + 1;
  • 27 380 711 631 976 ÷ 2 = 13 690 355 815 988 + 0;
  • 13 690 355 815 988 ÷ 2 = 6 845 177 907 994 + 0;
  • 6 845 177 907 994 ÷ 2 = 3 422 588 953 997 + 0;
  • 3 422 588 953 997 ÷ 2 = 1 711 294 476 998 + 1;
  • 1 711 294 476 998 ÷ 2 = 855 647 238 499 + 0;
  • 855 647 238 499 ÷ 2 = 427 823 619 249 + 1;
  • 427 823 619 249 ÷ 2 = 213 911 809 624 + 1;
  • 213 911 809 624 ÷ 2 = 106 955 904 812 + 0;
  • 106 955 904 812 ÷ 2 = 53 477 952 406 + 0;
  • 53 477 952 406 ÷ 2 = 26 738 976 203 + 0;
  • 26 738 976 203 ÷ 2 = 13 369 488 101 + 1;
  • 13 369 488 101 ÷ 2 = 6 684 744 050 + 1;
  • 6 684 744 050 ÷ 2 = 3 342 372 025 + 0;
  • 3 342 372 025 ÷ 2 = 1 671 186 012 + 1;
  • 1 671 186 012 ÷ 2 = 835 593 006 + 0;
  • 835 593 006 ÷ 2 = 417 796 503 + 0;
  • 417 796 503 ÷ 2 = 208 898 251 + 1;
  • 208 898 251 ÷ 2 = 104 449 125 + 1;
  • 104 449 125 ÷ 2 = 52 224 562 + 1;
  • 52 224 562 ÷ 2 = 26 112 281 + 0;
  • 26 112 281 ÷ 2 = 13 056 140 + 1;
  • 13 056 140 ÷ 2 = 6 528 070 + 0;
  • 6 528 070 ÷ 2 = 3 264 035 + 0;
  • 3 264 035 ÷ 2 = 1 632 017 + 1;
  • 1 632 017 ÷ 2 = 816 008 + 1;
  • 816 008 ÷ 2 = 408 004 + 0;
  • 408 004 ÷ 2 = 204 002 + 0;
  • 204 002 ÷ 2 = 102 001 + 0;
  • 102 001 ÷ 2 = 51 000 + 1;
  • 51 000 ÷ 2 = 25 500 + 0;
  • 25 500 ÷ 2 = 12 750 + 0;
  • 12 750 ÷ 2 = 6 375 + 0;
  • 6 375 ÷ 2 = 3 187 + 1;
  • 3 187 ÷ 2 = 1 593 + 1;
  • 1 593 ÷ 2 = 796 + 1;
  • 796 ÷ 2 = 398 + 0;
  • 398 ÷ 2 = 199 + 0;
  • 199 ÷ 2 = 99 + 1;
  • 99 ÷ 2 = 49 + 1;
  • 49 ÷ 2 = 24 + 1;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:

7 177 689 270 052 918 071(10) = 110 0011 1001 1100 0100 0110 0101 1100 1011 0001 1010 0011 0001 0011 0011 0111(2)

3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 63.

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; ...

First bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.

The least number that is a power of 2 and is larger than the actual length so that the first bit (leftmost) could be zero is: 64.

4. 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:

7 177 689 270 052 918 071(10) = 0110 0011 1001 1100 0100 0110 0101 1100 1011 0001 1010 0011 0001 0011 0011 0111

Conclusion:

Number 7 177 689 270 052 918 071, a signed integer, converted from decimal system (base 10) to a signed binary two's complement representation:
7 177 689 270 052 918 071(10) = 0110 0011 1001 1100 0100 0110 0101 1100 1011 0001 1010 0011 0001 0011 0011 0111

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

Convert signed integer numbers from the decimal system (base ten) to signed binary two's complement representation

How to convert a base ten signed integer number to signed binary in two's complement representation:

1) Divide the positive version of number repeatedly by 2, keeping track of each remainder, till we get a quotient that is zero.

2) Construct the base 2 representation by taking the previously calculated remainders starting from the last remainder up to the first one, in that order.

3) Construct the positive binary computer representation so that the first bit is zero.

4) Only if the initial number is negative, switch all the bits on 0 to 1 and all the bits on 1 to 0 (reversing the digits).

5) Only if the initial number is negative, add 1 to the number at the previous point.

Latest signed integers converted from decimal system to binary two's complement representation

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