Signed: Binary -> Integer: 0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011 is the binary representation of a positive integer, on 64 bits (8 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:


0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011 = 000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011


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

  • 262

    0
  • 261

    0
  • 260

    0
  • 259

    0
  • 258

    0
  • 257

    0
  • 256

    0
  • 255

    0
  • 254

    0
  • 253

    0
  • 252

    0
  • 251

    0
  • 250

    0
  • 249

    0
  • 248

    0
  • 247

    0
  • 246

    0
  • 245

    0
  • 244

    0
  • 243

    0
  • 242

    0
  • 241

    0
  • 240

    0
  • 239

    0
  • 238

    0
  • 237

    1
  • 236

    1
  • 235

    1
  • 234

    1
  • 233

    1
  • 232

    0
  • 231

    1
  • 230

    0
  • 229

    1
  • 228

    0
  • 227

    0
  • 226

    0
  • 225

    1
  • 224

    1
  • 223

    1
  • 222

    0
  • 221

    0
  • 220

    0
  • 219

    0
  • 218

    0
  • 217

    0
  • 216

    0
  • 215

    0
  • 214

    0
  • 213

    0
  • 212

    0
  • 211

    0
  • 210

    0
  • 29

    0
  • 28

    0
  • 27

    0
  • 26

    0
  • 25

    0
  • 24

    0
  • 23

    1
  • 22

    0
  • 21

    1
  • 20

    1

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

000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011(2) =


(0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 0 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 0 × 249 + 0 × 248 + 0 × 247 + 0 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 0 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 1 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 0 × 232 + 1 × 231 + 0 × 230 + 1 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 1 × 225 + 1 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20)(10) =


(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 0 + 2 147 483 648 + 0 + 536 870 912 + 0 + 0 + 0 + 33 554 432 + 16 777 216 + 8 388 608 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 + 0 + 2 + 1)(10) =


(137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 2 147 483 648 + 536 870 912 + 33 554 432 + 16 777 216 + 8 388 608 + 8 + 2 + 1)(10) =


269 031 047 179(10)

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

0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011(2) = 269 031 047 179(10)

The number 0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011(2) = 269 031 047 179(10)

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

The latest signed binary numbers converted and written as signed integers in decimal system (in base ten)

Convert the signed binary number 0000 0000 0000 0000 0000 0000 0011 1110 1010 0011 1000 0000 0000 0000 0000 1011, write it as a decimal system integer number (written in base ten) Feb 27 03:35 UTC (GMT)
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Convert the signed binary number 0000 0000 0000 0000 0000 0000 1110 1100 0110 1011 0110 0110 0001 0011 0100 1101, write it as a decimal system integer number (written in base ten) Feb 27 03:35 UTC (GMT)
Convert the signed binary number 1001, write it as a decimal system integer number (written in base ten) Feb 27 03:35 UTC (GMT)
Convert the signed binary number 0000 0000 1111 1111 0011 1111 1111 1111, write it as a decimal system integer number (written in base ten) Feb 27 03:35 UTC (GMT)
Convert the signed binary number 0000 0000 0000 0010 1010 1011 1011 1111, write it as a decimal system integer number (written in base ten) Feb 27 03:34 UTC (GMT)
Convert the signed binary number 1011 0010 1000 0100, write it as a decimal system integer number (written in base ten) Feb 27 03:34 UTC (GMT)
Convert the signed binary number 0111 1010, write it as a decimal system integer number (written in base ten) Feb 27 03:34 UTC (GMT)
All the signed binary numbers converted to integers in decimal system (written 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