Signed: Binary ↘ Integer: 0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101(2) to an integer (with sign) in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101 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:

0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101 = 101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101


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

  • 262

    1

  • 261

    0

  • 260

    1

  • 259

    1

  • 258

    0

  • 257

    1

  • 256

    0

  • 255

    0

  • 254

    1

  • 253

    0

  • 252

    1

  • 251

    0

  • 250

    1

  • 249

    1

  • 248

    1

  • 247

    0

  • 246

    1

  • 245

    1

  • 244

    0

  • 243

    0

  • 242

    1

  • 241

    1

  • 240

    0

  • 239

    1

  • 238

    1

  • 237

    1

  • 236

    0

  • 235

    1

  • 234

    0

  • 233

    1

  • 232

    0

  • 231

    1

  • 230

    1

  • 229

    1

  • 228

    1

  • 227

    0

  • 226

    1

  • 225

    1

  • 224

    1

  • 223

    0

  • 222

    1

  • 221

    1

  • 220

    1

  • 219

    0

  • 218

    1

  • 217

    0

  • 216

    1

  • 215

    1

  • 214

    1

  • 213

    0

  • 212

    0

  • 211

    1

  • 210

    1

  • 29

    1

  • 28

    1

  • 27

    1

  • 26

    0

  • 25

    0

  • 24

    0

  • 23

    0

  • 22

    1

  • 21

    0

  • 20

    1

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

101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101(2) =

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

(4 611 686 018 427 387 904 + 0 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 0 + 144 115 188 075 855 872 + 0 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 0 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 0 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 0 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 0 + 34 359 738 368 + 0 + 8 589 934 592 + 0 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 0 + 67 108 864 + 33 554 432 + 16 777 216 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 0 + 262 144 + 0 + 65 536 + 32 768 + 16 384 + 0 + 0 + 2 048 + 1 024 + 512 + 256 + 128 + 0 + 0 + 0 + 0 + 4 + 0 + 1)(10) =

(4 611 686 018 427 387 904 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 144 115 188 075 855 872 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 70 368 744 177 664 + 35 184 372 088 832 + 4 398 046 511 104 + 2 199 023 255 552 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 34 359 738 368 + 8 589 934 592 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 67 108 864 + 33 554 432 + 16 777 216 + 4 194 304 + 2 097 152 + 1 048 576 + 262 144 + 65 536 + 32 768 + 16 384 + 2 048 + 1 024 + 512 + 256 + 128 + 4 + 1)(10) =

6 509 784 945 747 414 917(10)

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

0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101(2) = 6 509 784 945 747 414 917(10)

The number 0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0101(2) = 6 509 784 945 747 414 917(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 0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0100 (in base two), converted and written as an integer in decimal system (in base ten) = ?

» The signed binary number 0101 1010 0101 0111 0110 0110 1110 1010 1111 0111 0111 0101 1100 1111 1000 0110 (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)

Convert the signed binary number 0000 0001 1010 1000 1100 0000 1111 1000, write it as a decimal system integer number (written in base ten) May 01 04:45 UTC (GMT)
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Convert the signed binary number 0000 0000 0011 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0100, write it as a decimal system integer number (written in base ten) May 01 04:44 UTC (GMT)
Convert the signed binary number 0000 0000 0001 0000 1000 0000 0111 0010, write it as a decimal system integer number (written in base ten) May 01 04:44 UTC (GMT)
Convert the signed binary number 1001 0111 0110 0101 0101 0010 1101 1110, write it as a decimal system integer number (written in base ten) May 01 04:44 UTC (GMT)
Convert the signed binary number 1000 0000 0000 0000 0101 0100 1011 0000, write it as a decimal system integer number (written in base ten) May 01 04:44 UTC (GMT)
Convert the signed binary number 1110 0000 0000 0100 0000 0001 0000 0000 0001 0010 0100 1111 1011 0110 0010 0110, write it as a decimal system integer number (written in base ten) May 01 04:44 UTC (GMT)
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