Signed binary number 0101 1010 0001 0111 converted to an integer in base ten

Signed binary 0101 1010 0001 0111(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.

0101 1010 0001 0111 is the binary representation of a positive integer, on 16 bits (2 Bytes).


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

0101 1010 0001 0111 = 101 1010 0001 0111

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

    • 214

      1

    • 213

      0

    • 212

      1

    • 211

      1

    • 210

      0

    • 29

      1

    • 28

      0

    • 27

      0

    • 26

      0

    • 25

      0

    • 24

      1

    • 23

      0

    • 22

      1

    • 21

      1

    • 20

      1

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

101 1010 0001 0111(2) =

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

(16 384 + 0 + 4 096 + 2 048 + 0 + 512 + 0 + 0 + 0 + 0 + 16 + 0 + 4 + 2 + 1)(10) =

(16 384 + 4 096 + 2 048 + 512 + 16 + 4 + 2 + 1)(10) =

23 063(10)

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

0101 1010 0001 0111(2) = 23 063(10)

Number 0101 1010 0001 0111(2) converted from signed binary to an integer in decimal system (in base 10):
0101 1010 0001 0111(2) = 23 063(10)

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

More operations of this kind:

0101 1010 0001 0110 = ?

0101 1010 0001 1000 = ?


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)

0101 1010 0001 0111 = 23,063 Feb 24 18:09 UTC (GMT)
1100 1110 0010 1000 = -20,008 Feb 24 18:09 UTC (GMT)
0011 1011 1000 1110 = 15,246 Feb 24 18:09 UTC (GMT)
1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -9,223,372,036,854,775,807 Feb 24 18:09 UTC (GMT)
0100 0110 = 70 Feb 24 18:09 UTC (GMT)
1101 0100 0101 1100 = -21,596 Feb 24 18:09 UTC (GMT)
0000 0000 1000 0000 1100 1100 1100 1010 = 8,441,034 Feb 24 18:09 UTC (GMT)
0001 0000 0000 0001 0000 0000 0110 1001 = 268,501,097 Feb 24 18:08 UTC (GMT)
1111 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 = -8,935,141,660,703,064,058 Feb 24 18:08 UTC (GMT)
1101 = -5 Feb 24 18:08 UTC (GMT)
1011 1111 1100 1111 1111 1111 1110 1001 = -1,070,596,073 Feb 24 18:08 UTC (GMT)
1111 1101 1010 1010 = -32,170 Feb 24 18:08 UTC (GMT)
0000 0111 0011 1110 = 1,854 Feb 24 18:08 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