Signed: Binary ↘ Integer: 0000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000 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 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000 = 000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000


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

    1

  • 248

    1

  • 247

    1

  • 246

    1

  • 245

    1

  • 244

    1

  • 243

    1

  • 242

    1

  • 241

    1

  • 240

    1

  • 239

    1

  • 238

    1

  • 237

    1

  • 236

    1

  • 235

    1

  • 234

    1

  • 233

    1

  • 232

    1

  • 231

    1

  • 230

    1

  • 229

    1

  • 228

    1

  • 227

    1

  • 226

    1

  • 225

    1

  • 224

    1

  • 223

    1

  • 222

    1

  • 221

    1

  • 220

    1

  • 219

    1

  • 218

    1

  • 217

    1

  • 216

    1

  • 215

    1

  • 214

    1

  • 213

    1

  • 212

    1

  • 211

    1

  • 210

    1

  • 29

    1

  • 28

    1

  • 27

    1

  • 26

    1

  • 25

    1

  • 24

    1

  • 23

    1

  • 22

    0

  • 21

    0

  • 20

    0

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

000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000(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 + 1 × 249 + 1 × 248 + 1 × 247 + 1 × 246 + 1 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 1 × 241 + 1 × 240 + 1 × 239 + 1 × 238 + 1 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 1 × 231 + 1 × 230 + 1 × 229 + 1 × 228 + 1 × 227 + 1 × 226 + 1 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 1 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 1 × 217 + 1 × 216 + 1 × 215 + 1 × 214 + 1 × 213 + 1 × 212 + 1 × 211 + 1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =

(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 0 + 0 + 0)(10) =

(562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 67 108 864 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 4 096 + 2 048 + 1 024 + 512 + 256 + 128 + 64 + 32 + 16 + 8)(10) =

1 125 899 906 842 616(10)

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

0000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000(2) = 1 125 899 906 842 616(10)

The number 0000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000(2) = 1 125 899 906 842 616(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 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 (in base two), converted and written as an integer in decimal system (in base ten) = ?

» The signed binary number 0000 0000 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 (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 0100 0000 0001 0110 0110 0110 1010 1011, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0011 1111 0111 0011 0011 0011 0111 1000, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0011 1111 0010 0111 1111 1111 1111 0000, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0011 1110 0111 0000 0000 0000 0010 1000, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0011 1010 1101 1110 0110 1000 1001 0010, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0010 1111 0100 1111 1111 1111 1111 1111, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0011 1101 1100 1100 1100 1100 1100 1000, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0010 1000 0110 0001 1111 1111 1101 0000, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0010 1000 0010 0000, write it as a decimal system integer number (written in base ten) Apr 19 03:42 UTC (GMT)
Convert the signed binary number 0000 0000 1111 1011 0110 1110 1100 1001, write it as a decimal system integer number (written in base ten) Apr 19 03:42 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