Signed Binary to Integer: Number 0001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 1011 Converted and Written as a Base Ten Integer, in Decimal System

Signed binary number 0001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 1011(2) written as a base ten integer, in decimal system

1. Is this a positive or a negative number?

0001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 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:

0001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 1011 = 001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 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

    1

  • 259

    0

  • 258

    1

  • 257

    0

  • 256

    0

  • 255

    0

  • 254

    0

  • 253

    0

  • 252

    0

  • 251

    1

  • 250

    1

  • 249

    1

  • 248

    1

  • 247

    1

  • 246

    1

  • 245

    1

  • 244

    1

  • 243

    1

  • 242

    1

  • 241

    0

  • 240

    1

  • 239

    1

  • 238

    1

  • 237

    1

  • 236

    0

  • 235

    0

  • 234

    1

  • 233

    0

  • 232

    1

  • 231

    0

  • 230

    0

  • 229

    0

  • 228

    1

  • 227

    1

  • 226

    0

  • 225

    0

  • 224

    1

  • 223

    1

  • 222

    1

  • 221

    0

  • 220

    0

  • 219

    0

  • 218

    1

  • 217

    0

  • 216

    1

  • 215

    0

  • 214

    0

  • 213

    0

  • 212

    1

  • 211

    1

  • 210

    1

  • 29

    1

  • 28

    0

  • 27

    1

  • 26

    0

  • 25

    1

  • 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.

001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 1011(2) =

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

(0 + 0 + 1 152 921 504 606 846 976 + 0 + 288 230 376 151 711 744 + 0 + 0 + 0 + 0 + 0 + 0 + 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 + 0 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 0 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 0 + 0 + 0 + 268 435 456 + 134 217 728 + 0 + 0 + 16 777 216 + 8 388 608 + 4 194 304 + 0 + 0 + 0 + 262 144 + 0 + 65 536 + 0 + 0 + 0 + 4 096 + 2 048 + 1 024 + 512 + 0 + 128 + 0 + 32 + 0 + 8 + 0 + 2 + 1)(10) =

(1 152 921 504 606 846 976 + 288 230 376 151 711 744 + 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 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 17 179 869 184 + 4 294 967 296 + 268 435 456 + 134 217 728 + 16 777 216 + 8 388 608 + 4 194 304 + 262 144 + 65 536 + 4 096 + 2 048 + 1 024 + 512 + 128 + 32 + 8 + 2 + 1)(10) =

1 445 653 165 830 905 515(10)

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

0001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 1011(2) = 1 445 653 165 830 905 515(10)

The number 0001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 1011(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0001 0100 0000 1111 1111 1101 1110 0101 0001 1001 1100 0101 0001 1110 1010 1011(2) = 1 445 653 165 830 905 515(10)

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

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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