Signed: Binary ↘ Integer: 0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001 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:

0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001 = 101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001


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

  • 262

    1

  • 261

    0

  • 260

    1

  • 259

    0

  • 258

    0

  • 257

    0

  • 256

    1

  • 255

    0

  • 254

    0

  • 253

    1

  • 252

    0

  • 251

    1

  • 250

    0

  • 249

    0

  • 248

    1

  • 247

    1

  • 246

    0

  • 245

    1

  • 244

    1

  • 243

    0

  • 242

    0

  • 241

    1

  • 240

    0

  • 239

    0

  • 238

    0

  • 237

    0

  • 236

    0

  • 235

    0

  • 234

    1

  • 233

    0

  • 232

    0

  • 231

    0

  • 230

    1

  • 229

    0

  • 228

    0

  • 227

    0

  • 226

    1

  • 225

    0

  • 224

    0

  • 223

    1

  • 222

    1

  • 221

    1

  • 220

    1

  • 219

    0

  • 218

    1

  • 217

    0

  • 216

    0

  • 215

    1

  • 214

    0

  • 213

    0

  • 212

    1

  • 211

    0

  • 210

    1

  • 29

    1

  • 28

    0

  • 27

    0

  • 26

    0

  • 25

    1

  • 24

    0

  • 23

    1

  • 22

    0

  • 21

    0

  • 20

    1

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

101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001(2) =

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

(4 611 686 018 427 387 904 + 0 + 1 152 921 504 606 846 976 + 0 + 0 + 0 + 72 057 594 037 927 936 + 0 + 0 + 9 007 199 254 740 992 + 0 + 2 251 799 813 685 248 + 0 + 0 + 281 474 976 710 656 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 0 + 0 + 2 199 023 255 552 + 0 + 0 + 0 + 0 + 0 + 0 + 17 179 869 184 + 0 + 0 + 0 + 1 073 741 824 + 0 + 0 + 0 + 67 108 864 + 0 + 0 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 0 + 262 144 + 0 + 0 + 32 768 + 0 + 0 + 4 096 + 0 + 1 024 + 512 + 0 + 0 + 0 + 32 + 0 + 8 + 0 + 0 + 1)(10) =

(4 611 686 018 427 387 904 + 1 152 921 504 606 846 976 + 72 057 594 037 927 936 + 9 007 199 254 740 992 + 2 251 799 813 685 248 + 281 474 976 710 656 + 140 737 488 355 328 + 35 184 372 088 832 + 17 592 186 044 416 + 2 199 023 255 552 + 17 179 869 184 + 1 073 741 824 + 67 108 864 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 262 144 + 32 768 + 4 096 + 1 024 + 512 + 32 + 8 + 1)(10) =

5 848 401 322 523 792 937(10)

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

0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001(2) = 5 848 401 322 523 792 937(10)

The number 0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1001(2) = 5 848 401 322 523 792 937(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 0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1000 (in base two), converted and written as an integer in decimal system (in base ten) = ?

» The signed binary number 0101 0001 0010 1001 1011 0010 0000 0100 0100 0100 1111 0100 1001 0110 0010 1010 (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)

Convert the signed binary number 0000 0010 0010 1110 0001 1000 1000 1110, write it as a decimal system integer number (written in base ten) Apr 26 22:58 UTC (GMT)
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Convert the signed binary number 0000 0000 0000 0000 0011 1010 1011 0100, write it as a decimal system integer number (written in base ten) Apr 26 22:57 UTC (GMT)
Convert the signed binary number 0000 0000 0000 0000 0000 0000 0000 0001 1100 1010 0000 0000 0111 0000 1111 1101, write it as a decimal system integer number (written in base ten) Apr 26 22:57 UTC (GMT)
Convert the signed binary number 1011 1110 1110 1111, write it as a decimal system integer number (written in base ten) Apr 26 22:57 UTC (GMT)
Convert the signed binary number 1111 0101 1111 0111 1111 1111 1001 1101, write it as a decimal system integer number (written in base ten) Apr 26 22:57 UTC (GMT)
Convert the signed binary number 0000 0000 0000 0000 1001 1010 1100 0110, write it as a decimal system integer number (written in base ten) Apr 26 22:57 UTC (GMT)
Convert the signed binary number 1101 0111, write it as a decimal system integer number (written in base ten) Apr 26 22:57 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