Signed: Binary ↘ Integer: 0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111 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:

0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111 = 100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111


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

  • 262

    1

  • 261

    0

  • 260

    0

  • 259

    1

  • 258

    1

  • 257

    0

  • 256

    0

  • 255

    0

  • 254

    0

  • 253

    0

  • 252

    1

  • 251

    1

  • 250

    0

  • 249

    0

  • 248

    0

  • 247

    1

  • 246

    0

  • 245

    0

  • 244

    1

  • 243

    1

  • 242

    1

  • 241

    0

  • 240

    1

  • 239

    0

  • 238

    1

  • 237

    0

  • 236

    0

  • 235

    0

  • 234

    0

  • 233

    0

  • 232

    1

  • 231

    0

  • 230

    1

  • 229

    1

  • 228

    1

  • 227

    0

  • 226

    1

  • 225

    0

  • 224

    1

  • 223

    0

  • 222

    1

  • 221

    0

  • 220

    1

  • 219

    1

  • 218

    1

  • 217

    0

  • 216

    1

  • 215

    0

  • 214

    1

  • 213

    0

  • 212

    0

  • 211

    1

  • 210

    0

  • 29

    1

  • 28

    0

  • 27

    1

  • 26

    1

  • 25

    1

  • 24

    1

  • 23

    1

  • 22

    1

  • 21

    1

  • 20

    1

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

100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111(2) =

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

(4 611 686 018 427 387 904 + 0 + 0 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 0 + 0 + 0 + 0 + 0 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 0 + 0 + 0 + 140 737 488 355 328 + 0 + 0 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 0 + 274 877 906 944 + 0 + 0 + 0 + 0 + 0 + 4 294 967 296 + 0 + 1 073 741 824 + 536 870 912 + 268 435 456 + 0 + 67 108 864 + 0 + 16 777 216 + 0 + 4 194 304 + 0 + 1 048 576 + 524 288 + 262 144 + 0 + 65 536 + 0 + 16 384 + 0 + 0 + 2 048 + 0 + 512 + 0 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1)(10) =

(4 611 686 018 427 387 904 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 140 737 488 355 328 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 1 099 511 627 776 + 274 877 906 944 + 4 294 967 296 + 1 073 741 824 + 536 870 912 + 268 435 456 + 67 108 864 + 16 777 216 + 4 194 304 + 1 048 576 + 524 288 + 262 144 + 65 536 + 16 384 + 2 048 + 512 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1)(10) =

5 483 305 450 791 062 271(10)

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

0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111(2) = 5 483 305 450 791 062 271(10)

The number 0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1111(2) = 5 483 305 450 791 062 271(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 0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1010 1111 1110 (in base two), converted and written as an integer in decimal system (in base ten) = ?

» The signed binary number 0100 1100 0001 1000 1001 1101 0100 0001 0111 0101 0101 1101 0100 1011 0000 0000 (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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Convert the signed binary number 0000 0000 0010 0010 1010 0100 1010 1010, write it as a decimal system integer number (written in base ten) May 02 19:25 UTC (GMT)
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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