Signed: Binary ↘ Integer: 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 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 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101 = 000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 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

    1

  • 256

    1

  • 255

    1

  • 254

    1

  • 253

    1

  • 252

    1

  • 251

    1

  • 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

    1

  • 22

    1

  • 21

    0

  • 20

    1

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

000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2) =

(0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 1 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 1 × 253 + 1 × 252 + 1 × 251 + 1 × 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 + 1 × 22 + 0 × 21 + 1 × 20)(10) =

(0 + 0 + 0 + 0 + 0 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 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 + 4 + 0 + 1)(10) =

(144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 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 + 4 + 1)(10) =

288 230 376 151 711 741(10)

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

0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2) = 288 230 376 151 711 741(10)

The number 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1101(2) = 288 230 376 151 711 741(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 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1100 (in base two), converted and written as an integer in decimal system (in base ten) = ?

» The signed binary number 0000 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 (in base two), converted and written as an integer 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