Base ten decimal system signed integer number 123 456 789 012 345 converted to signed binary in one's complement representation

How to convert a signed integer in decimal system (in base 10):
123 456 789 012 345(10)
to a signed binary one'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;
  • 123 456 789 012 345 ÷ 2 = 61 728 394 506 172 + 1;
  • 61 728 394 506 172 ÷ 2 = 30 864 197 253 086 + 0;
  • 30 864 197 253 086 ÷ 2 = 15 432 098 626 543 + 0;
  • 15 432 098 626 543 ÷ 2 = 7 716 049 313 271 + 1;
  • 7 716 049 313 271 ÷ 2 = 3 858 024 656 635 + 1;
  • 3 858 024 656 635 ÷ 2 = 1 929 012 328 317 + 1;
  • 1 929 012 328 317 ÷ 2 = 964 506 164 158 + 1;
  • 964 506 164 158 ÷ 2 = 482 253 082 079 + 0;
  • 482 253 082 079 ÷ 2 = 241 126 541 039 + 1;
  • 241 126 541 039 ÷ 2 = 120 563 270 519 + 1;
  • 120 563 270 519 ÷ 2 = 60 281 635 259 + 1;
  • 60 281 635 259 ÷ 2 = 30 140 817 629 + 1;
  • 30 140 817 629 ÷ 2 = 15 070 408 814 + 1;
  • 15 070 408 814 ÷ 2 = 7 535 204 407 + 0;
  • 7 535 204 407 ÷ 2 = 3 767 602 203 + 1;
  • 3 767 602 203 ÷ 2 = 1 883 801 101 + 1;
  • 1 883 801 101 ÷ 2 = 941 900 550 + 1;
  • 941 900 550 ÷ 2 = 470 950 275 + 0;
  • 470 950 275 ÷ 2 = 235 475 137 + 1;
  • 235 475 137 ÷ 2 = 117 737 568 + 1;
  • 117 737 568 ÷ 2 = 58 868 784 + 0;
  • 58 868 784 ÷ 2 = 29 434 392 + 0;
  • 29 434 392 ÷ 2 = 14 717 196 + 0;
  • 14 717 196 ÷ 2 = 7 358 598 + 0;
  • 7 358 598 ÷ 2 = 3 679 299 + 0;
  • 3 679 299 ÷ 2 = 1 839 649 + 1;
  • 1 839 649 ÷ 2 = 919 824 + 1;
  • 919 824 ÷ 2 = 459 912 + 0;
  • 459 912 ÷ 2 = 229 956 + 0;
  • 229 956 ÷ 2 = 114 978 + 0;
  • 114 978 ÷ 2 = 57 489 + 0;
  • 57 489 ÷ 2 = 28 744 + 1;
  • 28 744 ÷ 2 = 14 372 + 0;
  • 14 372 ÷ 2 = 7 186 + 0;
  • 7 186 ÷ 2 = 3 593 + 0;
  • 3 593 ÷ 2 = 1 796 + 1;
  • 1 796 ÷ 2 = 898 + 0;
  • 898 ÷ 2 = 449 + 0;
  • 449 ÷ 2 = 224 + 1;
  • 224 ÷ 2 = 112 + 0;
  • 112 ÷ 2 = 56 + 0;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 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:

123 456 789 012 345(10) = 111 0000 0100 1000 1000 0110 0000 1101 1101 1111 0111 1001(2)

3. Determine the signed binary number bit length:

The base 2 number's actual length, in bits: 47.

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:

123 456 789 012 345(10) = 0000 0000 0000 0000 0111 0000 0100 1000 1000 0110 0000 1101 1101 1111 0111 1001

Conclusion:

Number 123 456 789 012 345, a signed integer, converted from decimal system (base 10) to a signed binary one's complement representation:
123 456 789 012 345(10) = 0000 0000 0000 0000 0111 0000 0100 1000 1000 0110 0000 1101 1101 1111 0111 1001

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 one's complement representation

How to convert a base ten signed integer number to signed binary in one'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 from 0 to 1 and all the bits from 1 to 0 (reversing the digits).

Latest signed integers numbers converted from decimal system to signed binary in one's complement representation

How to convert signed integers from the decimal system to signed binary in one's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in one'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 that is to be converted to binary, keeping track of each remainder, until we get a quotient that is equal to 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, fill in '0' bits in front (to the left) of the base 2 number calculated above, up to the right length; this way the first bit (leftmost) will always 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 '1's and all '1' bits with '0's.

Example: convert the negative number -49 from the decimal system (base ten) to signed binary one's complement:

  • 1. Start with the positive version of the number: |-49| = 49
  • 2. Divide repeatedly 49 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 49 ÷ 2 = 24 + 1
    • 24 ÷ 2 = 12 + 0
    • 12 ÷ 2 = 6 + 0
    • 6 ÷ 2 = 3 + 0
    • 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:
    49(10) = 11 0001(2)
  • 4. The actual bit length of base 2 representation is 6, so the positive binary computer representation of a signed binary will take in this case 8 bits (the least power of 2 that is larger than 6) - add '0's in front of the base 2 number, up to the required length:
    49(10) = 0011 0001(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's:
    -49(10) = 1100 1110
  • Number -49(10), signed integer, converted from the decimal system (base 10) to signed binary in one's complement representation = 1100 1110