Signed binary two's complement number 1111 1101 1110 0111 0110 1100 1000 0110 converted to decimal system (base ten) signed integer

How to convert a signed binary two's complement:
1111 1101 1110 0111 0110 1100 1000 0110(2)
to an integer in decimal system (in base 10)

1. Is this a positive or a negative number?


In a signed binary two's complement, first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.

1111 1101 1110 0111 0110 1100 1000 0110 is the binary representation of a negative integer, on 32 bits (4 Bytes).


2. Get the binary representation in one's complement:


* Run this step only if the number is negative *

Subtract 1 from the binary initial number:

1111 1101 1110 0111 0110 1100 1000 0110 - 1 = 1111 1101 1110 0111 0110 1100 1000 0101


3. Get the binary representation of the positive (unsigned) number:


* Run this step only if the number is negative *

Flip all the bits in the signed binary one's complement representation (reverse the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:

!(1111 1101 1110 0111 0110 1100 1000 0101) = 0000 0010 0001 1000 1001 0011 0111 1010


4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:

    • 231

      0

    • 230

      0

    • 229

      0

    • 228

      0

    • 227

      0

    • 226

      0

    • 225

      1

    • 224

      0

    • 223

      0

    • 222

      0

    • 221

      0

    • 220

      1

    • 219

      1

    • 218

      0

    • 217

      0

    • 216

      0

    • 215

      1

    • 214

      0

    • 213

      0

    • 212

      1

    • 211

      0

    • 210

      0

    • 29

      1

    • 28

      1

    • 27

      0

    • 26

      1

    • 25

      1

    • 24

      1

    • 23

      1

    • 22

      0

    • 21

      1

    • 20

      0

5. Multiply each bit by its corresponding power of 2 and add all the terms up:

0000 0010 0001 1000 1001 0011 0111 1010(2) =

(0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 1 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 1 × 220 + 1 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 1 × 215 + 0 × 214 + 0 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =

(0 + 0 + 0 + 0 + 0 + 0 + 33 554 432 + 0 + 0 + 0 + 0 + 1 048 576 + 524 288 + 0 + 0 + 0 + 32 768 + 0 + 0 + 4 096 + 0 + 0 + 512 + 256 + 0 + 64 + 32 + 16 + 8 + 0 + 2 + 0)(10) =

(33 554 432 + 1 048 576 + 524 288 + 32 768 + 4 096 + 512 + 256 + 64 + 32 + 16 + 8 + 2)(10) =

35 165 050(10)

6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:

1111 1101 1110 0111 0110 1100 1000 0110(2) = -35 165 050(10)

Conclusion:
Number 1111 1101 1110 0111 0110 1100 1000 0110(2) converted from signed binary two's complement representation to an integer in decimal system (in base 10):

1111 1101 1110 0111 0110 1100 1000 0110(2) = -35 165 050(10)

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

More operations of this kind:

1111 1101 1110 0111 0110 1100 1000 0101 = ?

1111 1101 1110 0111 0110 1100 1000 0111 = ?


Convert signed binary two's complement numbers to decimal system (base ten) integers

Entered binary number length must be: 2, 4, 8, 16, 32, or 64 - otherwise extra bits on 0 will be added in front (to the left).

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

1) In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive.

2) Get the signed binary representation in one's complement, subtract 1 from the initial number.

3) Construct the unsigned binary number: flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s.

4) Multiply each bit of the binary number by its corresponding power of 2 that its place value represents.

5) Add all the terms up to get the positive integer number in base ten.

6) Adjust the sign of the integer number by the first bit of the initial binary number.

Latest binary numbers in two's complement representation converted to signed integers in decimal system (base ten)

1111 1101 1110 0111 0110 1100 1000 0110 = -35,165,050 Jan 21 02:33 UTC (GMT)
1100 1011 = -53 Jan 21 02:32 UTC (GMT)
1011 1010 1010 0010 1001 0110 1000 0100 = -1,163,749,756 Jan 21 02:32 UTC (GMT)
0000 1001 0111 0011 = 2,419 Jan 21 02:32 UTC (GMT)
1110 1010 0110 1100 = -5,524 Jan 21 02:32 UTC (GMT)
0000 1001 0000 1001 = 2,313 Jan 21 02:31 UTC (GMT)
0000 0000 1010 0110 1110 1000 1101 1110 1110 0000 0100 0000 1100 0100 1101 0001 = 46,980,890,076,693,713 Jan 21 02:31 UTC (GMT)
1111 1100 1110 1001 = -791 Jan 21 02:31 UTC (GMT)
0001 0111 0010 1110 = 5,934 Jan 21 02:31 UTC (GMT)
1100 0000 0000 0000 0000 0000 0110 1001 = -1,073,741,719 Jan 21 02:30 UTC (GMT)
1011 0100 1010 1010 1011 1111 1010 1010 1111 0101 0101 0101 0101 0000 0000 1111 = -5,428,315,659,860,357,105 Jan 21 02:29 UTC (GMT)
1101 0110 0110 1101 = -10,643 Jan 21 02:29 UTC (GMT)
0000 0000 0000 0000 0000 1111 0000 0010 = 3,842 Jan 21 02:29 UTC (GMT)
All the converted signed binary two's complement numbers

How to convert signed binary numbers in two's complement representation from binary system to decimal

To understand how to convert a signed binary number in two's complement representation from the binary system to decimal (base ten), the easiest way is to do it by an example - convert binary, 1101 1110, to base ten:

  • In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
  • Get the signed binary representation in one's complement, subtract 1 from the initial number:
    1101 1110 - 1 = 1101 1101
  • Get the binary representation of the positive number, flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:
    !(1101 1101) = 0010 0010
  • Write bellow the positive binary number representation in base two, and above each bit that makes up the binary number write the corresponding power of 2 (numeral base) that its place value represents, starting with zero, from the right of the number (rightmost bit), walking to the left of the number, increasing each corresonding power of 2 by exactly one unit:
  • powers of 2: 7 6 5 4 3 2 1 0
    digits: 0 0 1 0 0 0 1 0
  • Build the representation of the positive number in base 10, by taking each digit of the binary number, multiplying it by the corresponding power of 2 and then adding all the terms up:

    0010 0010(2) =

    (0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =

    (0 + 0 + 32 + 0 + 0 + 0 + 2 + 0)(10) =

    (32 + 2)(10) =

    34(10)

  • Signed binary number in two's complement representation, 1101 1110 = -34(10), a signed negative integer in base 10