Signed binary number 0011 1111 0111 1111 1111 1111 1111 1010 converted to an integer in base ten

How to convert a signed binary:
0011 1111 0111 1111 1111 1111 1111 1010(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.

0011 1111 0111 1111 1111 1111 1111 1010 is the binary representation of a positive integer, on 32 bits (4 Bytes).


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

0011 1111 0111 1111 1111 1111 1111 1010 = 011 1111 0111 1111 1111 1111 1111 1010

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

    • 230

      0

    • 229

      1

    • 228

      1

    • 227

      1

    • 226

      1

    • 225

      1

    • 224

      1

    • 223

      0

    • 222

      1

    • 221

      1

    • 220

      1

    • 219

      1

    • 218

      1

    • 217

      1

    • 216

      1

    • 215

      1

    • 214

      1

    • 213

      1

    • 212

      1

    • 211

      1

    • 210

      1

    • 29

      1

    • 28

      1

    • 27

      1

    • 26

      1

    • 25

      1

    • 24

      1

    • 23

      1

    • 22

      0

    • 21

      1

    • 20

      0

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

011 1111 0111 1111 1111 1111 1111 1010(2) =

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

(0 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 0 + 2 + 0)(10) =

(536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 2)(10) =

1 065 353 210(10)

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

0011 1111 0111 1111 1111 1111 1111 1010(2) = 1 065 353 210(10)

Conclusion:
Number 0011 1111 0111 1111 1111 1111 1111 1010(2) converted from signed binary to an integer in decimal system (in base 10):

0011 1111 0111 1111 1111 1111 1111 1010(2) = 1 065 353 210(10)

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

More operations of this kind:

0011 1111 0111 1111 1111 1111 1111 1001 = ?

0011 1111 0111 1111 1111 1111 1111 1011 = ?


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)

0011 1111 0111 1111 1111 1111 1111 1010 = 1,065,353,210 Jan 19 05:25 UTC (GMT)
0000 0000 0111 1111 1111 1011 1110 0001 = 8,387,553 Jan 19 05:23 UTC (GMT)
0000 0000 1011 0001 0001 1010 0111 0010 = 11,606,642 Jan 19 05:22 UTC (GMT)
0010 0001 = 33 Jan 19 05:22 UTC (GMT)
0010 0001 = 33 Jan 19 05:22 UTC (GMT)
0101 0010 1110 1111 1111 1111 1111 1010 = 1,391,460,346 Jan 19 05:22 UTC (GMT)
1100 0100 1111 1100 1100 0000 0000 0000 = -1,157,414,912 Jan 19 05:21 UTC (GMT)
0010 0001 = 33 Jan 19 05:21 UTC (GMT)
0000 0000 1010 0110 0010 0100 1100 1111 = 10,888,399 Jan 19 05:21 UTC (GMT)
0000 1111 1100 1110 = 4,046 Jan 19 05:21 UTC (GMT)
0101 0001 1001 1100 = 20,892 Jan 19 05:21 UTC (GMT)
0000 0000 1000 1111 = 143 Jan 19 05:20 UTC (GMT)
1011 1110 1110 0000 0000 0000 0000 0000 = -1,054,867,456 Jan 19 05:20 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