Signed: Binary ↘ Integer: 0000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101 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 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101 = 000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101


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

    1

  • 253

    1

  • 252

    1

  • 251

    0

  • 250

    0

  • 249

    1

  • 248

    1

  • 247

    0

  • 246

    1

  • 245

    1

  • 244

    0

  • 243

    0

  • 242

    0

  • 241

    0

  • 240

    1

  • 239

    0

  • 238

    1

  • 237

    1

  • 236

    0

  • 235

    0

  • 234

    1

  • 233

    0

  • 232

    1

  • 231

    0

  • 230

    1

  • 229

    1

  • 228

    0

  • 227

    0

  • 226

    0

  • 225

    0

  • 224

    1

  • 223

    0

  • 222

    1

  • 221

    0

  • 220

    1

  • 219

    0

  • 218

    1

  • 217

    1

  • 216

    0

  • 215

    0

  • 214

    0

  • 213

    1

  • 212

    1

  • 211

    0

  • 210

    1

  • 29

    0

  • 28

    0

  • 27

    1

  • 26

    0

  • 25

    0

  • 24

    0

  • 23

    1

  • 22

    1

  • 21

    0

  • 20

    1

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

000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101(2) =

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

(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 0 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 0 + 0 + 0 + 0 + 1 099 511 627 776 + 0 + 274 877 906 944 + 137 438 953 472 + 0 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 0 + 1 073 741 824 + 536 870 912 + 0 + 0 + 0 + 0 + 16 777 216 + 0 + 4 194 304 + 0 + 1 048 576 + 0 + 262 144 + 131 072 + 0 + 0 + 0 + 8 192 + 4 096 + 0 + 1 024 + 0 + 0 + 128 + 0 + 0 + 0 + 8 + 4 + 0 + 1)(10) =

(18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 562 949 953 421 312 + 281 474 976 710 656 + 70 368 744 177 664 + 35 184 372 088 832 + 1 099 511 627 776 + 274 877 906 944 + 137 438 953 472 + 17 179 869 184 + 4 294 967 296 + 1 073 741 824 + 536 870 912 + 16 777 216 + 4 194 304 + 1 048 576 + 262 144 + 131 072 + 8 192 + 4 096 + 1 024 + 128 + 8 + 4 + 1)(10) =

32 476 710 374 356 109(10)

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

0000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101(2) = 32 476 710 374 356 109(10)

The number 0000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1101(2) = 32 476 710 374 356 109(10)

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

More operations with signed binary numbers:

» The signed binary number 0000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1100 (in base two), converted and written as an integer in decimal system (in base ten) = ?

» The signed binary number 0000 0000 0111 0011 0110 0001 0110 0101 0110 0001 0101 0110 0011 0100 1000 1110 (in base two), converted and written as an integer in decimal system (in base ten) = ?

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

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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