Signed Binary to Integer: Number 1111 1000 1110 1101 0001 1100 1010 0101 Converted and Written as a Base Ten Integer, in Decimal System

Signed binary number 1111 1000 1110 1101 0001 1100 1010 0101(2) written as a base ten integer, in decimal system

1. Is this a positive or a negative number?

1111 1000 1110 1101 0001 1100 1010 0101 is the binary representation of a negative integer, on 32 bits (4 Bytes).


In a signed binary, the first bit (the leftmost) is reserved for the sign,

1 = negative, 0 = positive. This bit does not count when calculating the absolute value.


2. Construct the unsigned binary number.

Exclude the first bit (the leftmost), that is reserved for the sign:

1111 1000 1110 1101 0001 1100 1010 0101 = 111 1000 1110 1101 0001 1100 1010 0101


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

    1

  • 221

    1

  • 220

    0

  • 219

    1

  • 218

    1

  • 217

    0

  • 216

    1

  • 215

    0

  • 214

    0

  • 213

    0

  • 212

    1

  • 211

    1

  • 210

    1

  • 29

    0

  • 28

    0

  • 27

    1

  • 26

    0

  • 25

    1

  • 24

    0

  • 23

    0

  • 22

    1

  • 21

    0

  • 20

    1

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

111 1000 1110 1101 0001 1100 1010 0101(2) =

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

(1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 0 + 0 + 0 + 8 388 608 + 4 194 304 + 2 097 152 + 0 + 524 288 + 262 144 + 0 + 65 536 + 0 + 0 + 0 + 4 096 + 2 048 + 1 024 + 0 + 0 + 128 + 0 + 32 + 0 + 0 + 4 + 0 + 1)(10) =

(1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 8 388 608 + 4 194 304 + 2 097 152 + 524 288 + 262 144 + 65 536 + 4 096 + 2 048 + 1 024 + 128 + 32 + 4 + 1)(10) =

2 028 805 285(10)

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

1111 1000 1110 1101 0001 1100 1010 0101(2) = -2 028 805 285(10)

The number 1111 1000 1110 1101 0001 1100 1010 0101(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
1111 1000 1110 1101 0001 1100 1010 0101(2) = -2 028 805 285(10)

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

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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