Signed Binary to Integer: Number 1010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101 Converted and Written as a Base Ten Integer, in Decimal System

Signed binary number 1010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101(2) written as a base ten integer, in decimal system

1. Is this a positive or a negative number?

1010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101 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:

1010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101 = 010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101


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

  • 262

    0

  • 261

    1

  • 260

    0

  • 259

    1

  • 258

    0

  • 257

    0

  • 256

    1

  • 255

    0

  • 254

    1

  • 253

    0

  • 252

    1

  • 251

    0

  • 250

    1

  • 249

    0

  • 248

    1

  • 247

    0

  • 246

    1

  • 245

    0

  • 244

    1

  • 243

    0

  • 242

    1

  • 241

    1

  • 240

    1

  • 239

    1

  • 238

    0

  • 237

    1

  • 236

    0

  • 235

    0

  • 234

    1

  • 233

    0

  • 232

    1

  • 231

    0

  • 230

    1

  • 229

    0

  • 228

    0

  • 227

    1

  • 226

    0

  • 225

    0

  • 224

    0

  • 223

    1

  • 222

    0

  • 221

    1

  • 220

    0

  • 219

    1

  • 218

    0

  • 217

    1

  • 216

    0

  • 215

    1

  • 214

    0

  • 213

    1

  • 212

    0

  • 211

    1

  • 210

    0

  • 29

    1

  • 28

    1

  • 27

    1

  • 26

    1

  • 25

    1

  • 24

    1

  • 23

    0

  • 22

    1

  • 21

    0

  • 20

    1

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

010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101(2) =

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

(0 + 2 305 843 009 213 693 952 + 0 + 576 460 752 303 423 488 + 0 + 0 + 72 057 594 037 927 936 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 1 125 899 906 842 624 + 0 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 0 + 137 438 953 472 + 0 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 0 + 1 073 741 824 + 0 + 0 + 134 217 728 + 0 + 0 + 0 + 8 388 608 + 0 + 2 097 152 + 0 + 524 288 + 0 + 131 072 + 0 + 32 768 + 0 + 8 192 + 0 + 2 048 + 0 + 512 + 256 + 128 + 64 + 32 + 16 + 0 + 4 + 0 + 1)(10) =

(2 305 843 009 213 693 952 + 576 460 752 303 423 488 + 72 057 594 037 927 936 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 1 125 899 906 842 624 + 281 474 976 710 656 + 70 368 744 177 664 + 17 592 186 044 416 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 137 438 953 472 + 17 179 869 184 + 4 294 967 296 + 1 073 741 824 + 134 217 728 + 8 388 608 + 2 097 152 + 524 288 + 131 072 + 32 768 + 8 192 + 2 048 + 512 + 256 + 128 + 64 + 32 + 16 + 4 + 1)(10) =

2 978 383 095 975 816 181(10)

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

1010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101(2) = -2 978 383 095 975 816 181(10)

The number 1010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
1010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0101(2) = -2 978 383 095 975 816 181(10)

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

» More ops: The signed binary number 1010 1001 0101 0101 0101 0111 1010 0101 0100 1000 1010 1010 1010 1011 1111 0100 converted and written as an integer in decimal system (in base ten)

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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