Convert 4 623 156 123 728 347 136 to a signed binary in two's complement representation, from a signed integer number in base 10 decimal system

How to convert a signed integer in decimal system (in base 10):
4 623 156 123 728 347 136(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;
  • 4 623 156 123 728 347 136 ÷ 2 = 2 311 578 061 864 173 568 + 0;
  • 2 311 578 061 864 173 568 ÷ 2 = 1 155 789 030 932 086 784 + 0;
  • 1 155 789 030 932 086 784 ÷ 2 = 577 894 515 466 043 392 + 0;
  • 577 894 515 466 043 392 ÷ 2 = 288 947 257 733 021 696 + 0;
  • 288 947 257 733 021 696 ÷ 2 = 144 473 628 866 510 848 + 0;
  • 144 473 628 866 510 848 ÷ 2 = 72 236 814 433 255 424 + 0;
  • 72 236 814 433 255 424 ÷ 2 = 36 118 407 216 627 712 + 0;
  • 36 118 407 216 627 712 ÷ 2 = 18 059 203 608 313 856 + 0;
  • 18 059 203 608 313 856 ÷ 2 = 9 029 601 804 156 928 + 0;
  • 9 029 601 804 156 928 ÷ 2 = 4 514 800 902 078 464 + 0;
  • 4 514 800 902 078 464 ÷ 2 = 2 257 400 451 039 232 + 0;
  • 2 257 400 451 039 232 ÷ 2 = 1 128 700 225 519 616 + 0;
  • 1 128 700 225 519 616 ÷ 2 = 564 350 112 759 808 + 0;
  • 564 350 112 759 808 ÷ 2 = 282 175 056 379 904 + 0;
  • 282 175 056 379 904 ÷ 2 = 141 087 528 189 952 + 0;
  • 141 087 528 189 952 ÷ 2 = 70 543 764 094 976 + 0;
  • 70 543 764 094 976 ÷ 2 = 35 271 882 047 488 + 0;
  • 35 271 882 047 488 ÷ 2 = 17 635 941 023 744 + 0;
  • 17 635 941 023 744 ÷ 2 = 8 817 970 511 872 + 0;
  • 8 817 970 511 872 ÷ 2 = 4 408 985 255 936 + 0;
  • 4 408 985 255 936 ÷ 2 = 2 204 492 627 968 + 0;
  • 2 204 492 627 968 ÷ 2 = 1 102 246 313 984 + 0;
  • 1 102 246 313 984 ÷ 2 = 551 123 156 992 + 0;
  • 551 123 156 992 ÷ 2 = 275 561 578 496 + 0;
  • 275 561 578 496 ÷ 2 = 137 780 789 248 + 0;
  • 137 780 789 248 ÷ 2 = 68 890 394 624 + 0;
  • 68 890 394 624 ÷ 2 = 34 445 197 312 + 0;
  • 34 445 197 312 ÷ 2 = 17 222 598 656 + 0;
  • 17 222 598 656 ÷ 2 = 8 611 299 328 + 0;
  • 8 611 299 328 ÷ 2 = 4 305 649 664 + 0;
  • 4 305 649 664 ÷ 2 = 2 152 824 832 + 0;
  • 2 152 824 832 ÷ 2 = 1 076 412 416 + 0;
  • 1 076 412 416 ÷ 2 = 538 206 208 + 0;
  • 538 206 208 ÷ 2 = 269 103 104 + 0;
  • 269 103 104 ÷ 2 = 134 551 552 + 0;
  • 134 551 552 ÷ 2 = 67 275 776 + 0;
  • 67 275 776 ÷ 2 = 33 637 888 + 0;
  • 33 637 888 ÷ 2 = 16 818 944 + 0;
  • 16 818 944 ÷ 2 = 8 409 472 + 0;
  • 8 409 472 ÷ 2 = 4 204 736 + 0;
  • 4 204 736 ÷ 2 = 2 102 368 + 0;
  • 2 102 368 ÷ 2 = 1 051 184 + 0;
  • 1 051 184 ÷ 2 = 525 592 + 0;
  • 525 592 ÷ 2 = 262 796 + 0;
  • 262 796 ÷ 2 = 131 398 + 0;
  • 131 398 ÷ 2 = 65 699 + 0;
  • 65 699 ÷ 2 = 32 849 + 1;
  • 32 849 ÷ 2 = 16 424 + 1;
  • 16 424 ÷ 2 = 8 212 + 0;
  • 8 212 ÷ 2 = 4 106 + 0;
  • 4 106 ÷ 2 = 2 053 + 0;
  • 2 053 ÷ 2 = 1 026 + 1;
  • 1 026 ÷ 2 = 513 + 0;
  • 513 ÷ 2 = 256 + 1;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 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, by taking all the remainders starting from the bottom of the list constructed above:

4 623 156 123 728 347 136(10) = 100 0000 0010 1000 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(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:

4 623 156 123 728 347 136(10) = 0100 0000 0010 1000 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

Conclusion:
Number 4 623 156 123 728 347 136, a signed integer, converted from decimal system (base 10) to a signed binary two's complement representation:
4 623 156 123 728 347 136(10) = 0100 0000 0010 1000 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

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


4 623 156 123 728 347 135 = ? | 4 623 156 123 728 347 137 = ?


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

How to convert a base 10 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 getting a quotient that is equal to 0.

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 0.

4) Only if the initial number is negative, switch all the bits from 0 to 1 and from 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