Signed binary number 1111 0111 1111 1111 converted to an integer in base ten

How to convert a signed binary:
1111 0111 1111 1111(2)
to an integer in decimal system (in base 10)

1. Is this a positive or a negative number?


In a signed binary, first bit (the leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

1111 0111 1111 1111 is the binary representation of a negative integer, on 16 bits (2 Bytes).


2. Construct the unsigned binary number, exclude the first bit (the leftmost), that is reserved for the sign:

1111 0111 1111 1111 = 111 0111 1111 1111

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

    • 214

      1

    • 213

      1

    • 212

      1

    • 211

      0

    • 210

      1

    • 29

      1

    • 28

      1

    • 27

      1

    • 26

      1

    • 25

      1

    • 24

      1

    • 23

      1

    • 22

      1

    • 21

      1

    • 20

      1

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

111 0111 1111 1111(2) =

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

(16 384 + 8 192 + 4 096 + 0 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1)(10) =

(16 384 + 8 192 + 4 096 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1)(10) =

30 719(10)

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

1111 0111 1111 1111(2) = -30 719(10)

Conclusion:

Number 1111 0111 1111 1111(2) converted from signed binary to an integer in decimal system (in base 10):

1111 0111 1111 1111(2) = -30 719(10)

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

Convert signed binary numbers to integers in decimal system (base 10)

First bit (the leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

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 to an integer in base ten:

1) Construct the unsigned binary number: exclude the first bit (the leftmost); this bit is reserved for the sign, 1 = negative, 0 = positive and does not count when calculating the absolute value (without sign).

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

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

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

Latest signed binary numbers converted to signed integers in decimal system (base ten)

1111 0111 1111 1111 = -30,719 Dec 12 00:13 UTC (GMT)
0000 1000 1101 1100 = 2,268 Dec 12 00:13 UTC (GMT)
0111 0011 1100 0100 = 29,636 Dec 12 00:13 UTC (GMT)
1111 1111 1101 0100 0011 1111 1111 1000 = -2,144,616,440 Dec 12 00:12 UTC (GMT)
0000 0000 0000 0000 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 = 140,737,488,355,326 Dec 12 00:12 UTC (GMT)
0000 0011 1110 0011 = 995 Dec 12 00:12 UTC (GMT)
0000 0101 0100 1100 = 1,356 Dec 12 00:12 UTC (GMT)
1100 0001 0011 0000 0000 0000 0000 0000 = -1,093,664,768 Dec 12 00:12 UTC (GMT)
0000 0001 0001 0100 = 276 Dec 12 00:12 UTC (GMT)
0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 4,611,686,018,427,387,903 Dec 12 00:11 UTC (GMT)
0000 0000 0000 1000 0010 1100 1010 0111 0100 1101 1011 1011 1000 0100 0111 0000 = 2,300,896,888,980,592 Dec 12 00:11 UTC (GMT)
0000 0000 0000 0000 0000 0000 1110 0111 0101 1011 1110 0000 0010 1000 1011 0111 = 993,678,862,519 Dec 12 00:10 UTC (GMT)
1110 0111 = -103 Dec 12 00:10 UTC (GMT)
All the converted signed binary numbers to integers in base ten

How to convert signed binary numbers from binary system to decimal (base ten)

To understand how to convert a signed binary number from binary system to decimal (base ten), the easiest way is to do it through an example - convert the binary number, 1001 1110, to base ten:

  • In a signed binary, the first bit (leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value (value without sign). The first bit is 1, so our number is negative.
  • Write bellow the binary number 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 and increasing each corresonding power of 2 by exactly one unit, but ignoring the very first bit (the leftmost, the one representing the sign):
  • powers of 2:   6 5 4 3 2 1 0
    digits: 1 0 0 1 1 1 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, but also taking care of the number sign:

    1001 1110 =

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

    - (0 + 0 + 16 + 8 + 4 + 2 + 0)(10) =

    - (16 + 8 + 4 + 2)(10) =

    -30(10)

  • Binary signed number, 1001 1110 = -30(10), signed negative integer in base 10