Signed binary number 0111 1000 1011 1000 1001 1111 0000 1010 converted to an integer in base ten

How to convert a signed binary:
0111 1000 1011 1000 1001 1111 0000 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.

0111 1000 1011 1000 1001 1111 0000 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:

0111 1000 1011 1000 1001 1111 0000 1010 = 111 1000 1011 1000 1001 1111 0000 1010

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

    • 230

      1

    • 229

      1

    • 228

      1

    • 227

      1

    • 226

      0

    • 225

      0

    • 224

      0

    • 223

      1

    • 222

      0

    • 221

      1

    • 220

      1

    • 219

      1

    • 218

      0

    • 217

      0

    • 216

      0

    • 215

      1

    • 214

      0

    • 213

      0

    • 212

      1

    • 211

      1

    • 210

      1

    • 29

      1

    • 28

      1

    • 27

      0

    • 26

      0

    • 25

      0

    • 24

      0

    • 23

      1

    • 22

      0

    • 21

      1

    • 20

      0

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

111 1000 1011 1000 1001 1111 0000 1010(2) =

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

(1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 0 + 0 + 0 + 8 388 608 + 0 + 2 097 152 + 1 048 576 + 524 288 + 0 + 0 + 0 + 32 768 + 0 + 0 + 4 096 + 2 048 + 1 024 + 512 + 256 + 0 + 0 + 0 + 0 + 8 + 0 + 2 + 0)(10) =

(1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 8 388 608 + 2 097 152 + 1 048 576 + 524 288 + 32 768 + 4 096 + 2 048 + 1 024 + 512 + 256 + 8 + 2)(10) =

2 025 365 258(10)

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

0111 1000 1011 1000 1001 1111 0000 1010(2) = 2 025 365 258(10)

Conclusion:
Number 0111 1000 1011 1000 1001 1111 0000 1010(2) converted from signed binary to an integer in decimal system (in base 10):

0111 1000 1011 1000 1001 1111 0000 1010(2) = 2 025 365 258(10)

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

More operations of this kind:

0111 1000 1011 1000 1001 1111 0000 1001 = ?

0111 1000 1011 1000 1001 1111 0000 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)

0111 1000 1011 1000 1001 1111 0000 1010 = 2,025,365,258 Jan 26 12:17 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 0000 1011 1111 1111 0000 0000 0000 0000 0001 = 3,220,176,897 Jan 26 12:17 UTC (GMT)
1111 1101 1110 1101 = -32,237 Jan 26 12:17 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 0000 1010 1110 0010 1010 0000 0000 0010 1001 = 2,921,988,137 Jan 26 12:16 UTC (GMT)
0111 0010 0011 0100 1101 1111 0011 0010 = 1,916,067,634 Jan 26 12:16 UTC (GMT)
0000 0000 1111 1111 1111 1111 1011 1011 = 16,777,147 Jan 26 12:16 UTC (GMT)
1101 1111 1001 1001 = -24,473 Jan 26 12:15 UTC (GMT)
0111 1111 = 127 Jan 26 12:15 UTC (GMT)
1111 1110 1111 1010 = -32,506 Jan 26 12:15 UTC (GMT)
0011 1101 = 61 Jan 26 12:15 UTC (GMT)
0000 0000 0000 1100 0000 0000 0000 0001 = 786,433 Jan 26 12:14 UTC (GMT)
0111 0000 0110 0101 = 28,773 Jan 26 12:14 UTC (GMT)
0000 1001 1001 0000 = 2,448 Jan 26 12:14 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