Signed: Binary ↘ Integer: 0100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 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 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000


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

    0

  • 257

    0

  • 256

    0

  • 255

    0

  • 254

    1

  • 253

    0

  • 252

    1

  • 251

    0

  • 250

    1

  • 249

    0

  • 248

    0

  • 247

    1

  • 246

    1

  • 245

    0

  • 244

    0

  • 243

    0

  • 242

    0

  • 241

    0

  • 240

    0

  • 239

    0

  • 238

    0

  • 237

    0

  • 236

    0

  • 235

    0

  • 234

    0

  • 233

    0

  • 232

    0

  • 231

    0

  • 230

    0

  • 229

    0

  • 228

    0

  • 227

    0

  • 226

    0

  • 225

    0

  • 224

    0

  • 223

    0

  • 222

    0

  • 221

    0

  • 220

    0

  • 219

    0

  • 218

    0

  • 217

    0

  • 216

    0

  • 215

    0

  • 214

    0

  • 213

    0

  • 212

    0

  • 211

    0

  • 210

    0

  • 29

    0

  • 28

    0

  • 27

    0

  • 26

    0

  • 25

    0

  • 24

    0

  • 23

    0

  • 22

    0

  • 21

    0

  • 20

    0

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

100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =

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

(4 611 686 018 427 387 904 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 1 125 899 906 842 624 + 0 + 0 + 140 737 488 355 328 + 70 368 744 177 664 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0)(10) =

(4 611 686 018 427 387 904 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 1 125 899 906 842 624 + 140 737 488 355 328 + 70 368 744 177 664)(10) =

4 635 541 022 703 616 000(10)

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

0100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) = 4 635 541 022 703 616 000(10)

The number 0100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) = 4 635 541 022 703 616 000(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 0000 0101 0100 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 (in base two), converted and written as an integer in decimal system (in base ten) = ?

» The signed binary number 0100 0000 0101 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 (in base two), 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