Signed binary number 1100 0010 1001 1010 1101 0110 0100 0000 converted to an integer in base ten

How to convert a signed binary:
1100 0010 1001 1010 1101 0110 0100 0000(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.

1100 0010 1001 1010 1101 0110 0100 0000 is the binary representation of a negative integer, on 32 bits (4 Bytes).


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

1100 0010 1001 1010 1101 0110 0100 0000 = 100 0010 1001 1010 1101 0110 0100 0000

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

    • 230

      1
    • 229

      0
    • 228

      0
    • 227

      0
    • 226

      0
    • 225

      1
    • 224

      0
    • 223

      1
    • 222

      0
    • 221

      0
    • 220

      1
    • 219

      1
    • 218

      0
    • 217

      1
    • 216

      0
    • 215

      1
    • 214

      1
    • 213

      0
    • 212

      1
    • 211

      0
    • 210

      1
    • 29

      1
    • 28

      0
    • 27

      0
    • 26

      1
    • 25

      0
    • 24

      0
    • 23

      0
    • 22

      0
    • 21

      0
    • 20

      0

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

100 0010 1001 1010 1101 0110 0100 0000(2) =


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


(1 073 741 824 + 0 + 0 + 0 + 0 + 33 554 432 + 0 + 8 388 608 + 0 + 0 + 1 048 576 + 524 288 + 0 + 131 072 + 0 + 32 768 + 16 384 + 0 + 4 096 + 0 + 1 024 + 512 + 0 + 0 + 64 + 0 + 0 + 0 + 0 + 0 + 0)(10) =


(1 073 741 824 + 33 554 432 + 8 388 608 + 1 048 576 + 524 288 + 131 072 + 32 768 + 16 384 + 4 096 + 1 024 + 512 + 64)(10) =


1 117 443 648(10)

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

1100 0010 1001 1010 1101 0110 0100 0000(2) = -1 117 443 648(10)

Conclusion:
Number 1100 0010 1001 1010 1101 0110 0100 0000(2) converted from signed binary to an integer in decimal system (in base 10):


1100 0010 1001 1010 1101 0110 0100 0000(2) = -1 117 443 648(10)

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


More operations of this kind:

1100 0010 1001 1010 1101 0110 0011 1111 = ?

1100 0010 1001 1010 1101 0110 0100 0001 = ?


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)

1100 0010 1001 1010 1101 0110 0100 0000 = -1,117,443,648 Jan 26 13:10 UTC (GMT)
0011 0000 1110 0011 = 12,515 Jan 26 13:09 UTC (GMT)
0000 0000 0000 0000 0001 0000 0100 0101 = 4,165 Jan 26 13:08 UTC (GMT)
0111 1010 0000 0000 = 31,232 Jan 26 13:07 UTC (GMT)
1111 1111 1111 1111 1101 1110 1010 1101 = -2,147,475,117 Jan 26 13:07 UTC (GMT)
0000 0000 0000 1010 1101 1001 1110 1001 0001 1001 1011 1001 0000 1001 0011 1001 = 3,054,344,949,270,841 Jan 26 13:07 UTC (GMT)
1000 0101 0100 0101 = -1,349 Jan 26 13:07 UTC (GMT)
0010 0000 0111 0011 1111 1111 1111 1100 = 544,473,084 Jan 26 13:05 UTC (GMT)
1000 0010 0000 1000 0000 0000 1000 0000 = -34,078,848 Jan 26 13:05 UTC (GMT)
0001 0111 0001 0010 = 5,906 Jan 26 13:05 UTC (GMT)
0001 1000 1010 1111 = 6,319 Jan 26 13:05 UTC (GMT)
0000 0000 0000 0001 0011 1100 0110 1011 = 81,003 Jan 26 13:05 UTC (GMT)
1111 1110 1010 1010 = -32,426 Jan 26 13:04 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