Signed binary two's complement number 1111 1010 1000 0010 converted to decimal system (base ten) signed integer

How to convert a signed binary two's complement:
1111 1010 1000 0010(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 1010 1000 0010 is the binary representation of a negative integer, on 16 bits (2 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 1010 1000 0010 - 1 = 1111 1010 1000 0001


2. 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 1010 1000 0001) = 0000 0101 0111 1110


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

    • 215

      0
    • 214

      0
    • 213

      0
    • 212

      0
    • 211

      0
    • 210

      1
    • 29

      0
    • 28

      1
    • 27

      0
    • 26

      1
    • 25

      1
    • 24

      1
    • 23

      1
    • 22

      1
    • 21

      1
    • 20

      0

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

0000 0101 0111 1110(2) =


(0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 1 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =


(0 + 0 + 0 + 0 + 0 + 1 024 + 0 + 256 + 0 + 64 + 32 + 16 + 8 + 4 + 2 + 0)(10) =


(1 024 + 256 + 64 + 32 + 16 + 8 + 4 + 2)(10) =


1 406(10)

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

1111 1010 1000 0010(2) = -1 406(10)

Conclusion:

Number 1111 1010 1000 0010(2) converted from signed binary two's complement representation to an integer in decimal system (in base 10):


1111 1010 1000 0010(2) = -1 406(10)

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

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 1010 1000 0010 = -1,406 Jun 18 09:37 UTC (GMT)
1000 0000 = -128 Jun 18 09:37 UTC (GMT)
1111 1101 = -3 Jun 18 09:37 UTC (GMT)
0000 0001 1111 1111 = 511 Jun 18 09:36 UTC (GMT)
0000 0000 0100 0101 = 69 Jun 18 09:36 UTC (GMT)
1100 1011 1101 0101 = -13,355 Jun 18 09:35 UTC (GMT)
1000 1111 = -113 Jun 18 09:35 UTC (GMT)
1000 1100 = -116 Jun 18 09:35 UTC (GMT)
0000 0000 1101 0101 = 213 Jun 18 09:31 UTC (GMT)
0000 0000 1000 1101 = 141 Jun 18 09:29 UTC (GMT)
1001 1000 0110 1000 = -26,520 Jun 18 09:28 UTC (GMT)
0111 1001 0111 0001 = 31,089 Jun 18 09:27 UTC (GMT)
1100 0110 0100 0110 = -14,778 Jun 18 09:26 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