Signed: Binary ↘ Integer: 1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101 is the binary representation of a negative 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:

1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101 = 100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101


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

    0

  • 258

    1

  • 257

    0

  • 256

    1

  • 255

    0

  • 254

    1

  • 253

    0

  • 252

    1

  • 251

    0

  • 250

    0

  • 249

    0

  • 248

    1

  • 247

    0

  • 246

    1

  • 245

    0

  • 244

    1

  • 243

    0

  • 242

    0

  • 241

    0

  • 240

    1

  • 239

    1

  • 238

    1

  • 237

    0

  • 236

    0

  • 235

    1

  • 234

    0

  • 233

    1

  • 232

    0

  • 231

    1

  • 230

    0

  • 229

    1

  • 228

    0

  • 227

    1

  • 226

    0

  • 225

    1

  • 224

    1

  • 223

    1

  • 222

    1

  • 221

    1

  • 220

    0

  • 219

    0

  • 218

    0

  • 217

    0

  • 216

    1

  • 215

    1

  • 214

    1

  • 213

    0

  • 212

    1

  • 211

    0

  • 210

    1

  • 29

    0

  • 28

    1

  • 27

    0

  • 26

    1

  • 25

    1

  • 24

    0

  • 23

    1

  • 22

    1

  • 21

    0

  • 20

    1

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

100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101(2) =

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

(4 611 686 018 427 387 904 + 0 + 0 + 0 + 288 230 376 151 711 744 + 0 + 72 057 594 037 927 936 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 0 + 0 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 0 + 0 + 0 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 0 + 0 + 34 359 738 368 + 0 + 8 589 934 592 + 0 + 2 147 483 648 + 0 + 536 870 912 + 0 + 134 217 728 + 0 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 0 + 0 + 0 + 0 + 65 536 + 32 768 + 16 384 + 0 + 4 096 + 0 + 1 024 + 0 + 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 1)(10) =

(4 611 686 018 427 387 904 + 288 230 376 151 711 744 + 72 057 594 037 927 936 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 281 474 976 710 656 + 70 368 744 177 664 + 17 592 186 044 416 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 34 359 738 368 + 8 589 934 592 + 2 147 483 648 + 536 870 912 + 134 217 728 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 65 536 + 32 768 + 16 384 + 4 096 + 1 024 + 256 + 64 + 32 + 8 + 4 + 1)(10) =

4 994 863 392 639 538 541(10)

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

1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101(2) = -4 994 863 392 639 538 541(10)

The number 1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1101(2) = -4 994 863 392 639 538 541(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 1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1100 (in base two), converted and written as an integer in decimal system (in base ten) = ?

» The signed binary number 1100 0101 0101 0001 0101 0001 1100 1010 1010 1011 1110 0001 1101 0101 0110 1110 (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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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