Convert 101 000 001 100 023 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):
101 000 001 100 023(10)
to a signed binary two's complement representation

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;
  • 101 000 001 100 023 ÷ 2 = 50 500 000 550 011 + 1;
  • 50 500 000 550 011 ÷ 2 = 25 250 000 275 005 + 1;
  • 25 250 000 275 005 ÷ 2 = 12 625 000 137 502 + 1;
  • 12 625 000 137 502 ÷ 2 = 6 312 500 068 751 + 0;
  • 6 312 500 068 751 ÷ 2 = 3 156 250 034 375 + 1;
  • 3 156 250 034 375 ÷ 2 = 1 578 125 017 187 + 1;
  • 1 578 125 017 187 ÷ 2 = 789 062 508 593 + 1;
  • 789 062 508 593 ÷ 2 = 394 531 254 296 + 1;
  • 394 531 254 296 ÷ 2 = 197 265 627 148 + 0;
  • 197 265 627 148 ÷ 2 = 98 632 813 574 + 0;
  • 98 632 813 574 ÷ 2 = 49 316 406 787 + 0;
  • 49 316 406 787 ÷ 2 = 24 658 203 393 + 1;
  • 24 658 203 393 ÷ 2 = 12 329 101 696 + 1;
  • 12 329 101 696 ÷ 2 = 6 164 550 848 + 0;
  • 6 164 550 848 ÷ 2 = 3 082 275 424 + 0;
  • 3 082 275 424 ÷ 2 = 1 541 137 712 + 0;
  • 1 541 137 712 ÷ 2 = 770 568 856 + 0;
  • 770 568 856 ÷ 2 = 385 284 428 + 0;
  • 385 284 428 ÷ 2 = 192 642 214 + 0;
  • 192 642 214 ÷ 2 = 96 321 107 + 0;
  • 96 321 107 ÷ 2 = 48 160 553 + 1;
  • 48 160 553 ÷ 2 = 24 080 276 + 1;
  • 24 080 276 ÷ 2 = 12 040 138 + 0;
  • 12 040 138 ÷ 2 = 6 020 069 + 0;
  • 6 020 069 ÷ 2 = 3 010 034 + 1;
  • 3 010 034 ÷ 2 = 1 505 017 + 0;
  • 1 505 017 ÷ 2 = 752 508 + 1;
  • 752 508 ÷ 2 = 376 254 + 0;
  • 376 254 ÷ 2 = 188 127 + 0;
  • 188 127 ÷ 2 = 94 063 + 1;
  • 94 063 ÷ 2 = 47 031 + 1;
  • 47 031 ÷ 2 = 23 515 + 1;
  • 23 515 ÷ 2 = 11 757 + 1;
  • 11 757 ÷ 2 = 5 878 + 1;
  • 5 878 ÷ 2 = 2 939 + 0;
  • 2 939 ÷ 2 = 1 469 + 1;
  • 1 469 ÷ 2 = 734 + 1;
  • 734 ÷ 2 = 367 + 0;
  • 367 ÷ 2 = 183 + 1;
  • 183 ÷ 2 = 91 + 1;
  • 91 ÷ 2 = 45 + 1;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.

101 000 001 100 023(10) = 101 1011 1101 1011 1110 0101 0011 0000 0001 1000 1111 0111(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, 47,


so that the first bit (leftmost) could be zero


(we deal with a positive number at this moment)


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, 64:

101 000 001 100 023(10) = 0000 0000 0000 0000 0101 1011 1101 1011 1110 0101 0011 0000 0001 1000 1111 0111


Conclusion:

Number 101 000 001 100 023, a signed integer, converted from decimal system (base 10) to a signed binary two's complement representation:

101 000 001 100 023(10) = 0000 0000 0000 0000 0101 1011 1101 1011 1110 0101 0011 0000 0001 1000 1111 0111

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


More operations of this kind:

101 000 001 100 022 = ? | 101 000 001 100 024 = ?


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

101,000,001,100,023 to signed binary two's complement = ? Jan 21 02:36 UTC (GMT)
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1,609 to signed binary two's complement = ? Jan 21 02:35 UTC (GMT)
12,824 to signed binary two's complement = ? Jan 21 02:35 UTC (GMT)
3,325 to signed binary two's complement = ? Jan 21 02:35 UTC (GMT)
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450 to signed binary two's complement = ? Jan 21 02:34 UTC (GMT)
-11,011,096 to signed binary two's complement = ? Jan 21 02:33 UTC (GMT)
-32,644 to signed binary two's complement = ? Jan 21 02:33 UTC (GMT)
-57,633 to signed binary two's complement = ? Jan 21 02:33 UTC (GMT)
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647,733 to signed binary two's complement = ? Jan 21 02:33 UTC (GMT)
All decimal integer numbers converted to signed 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