Signed binary number 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101 converted to an integer in base ten

How to convert a signed binary:
1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101(2)
to an integer in decimal system (in base 10)

1. Is this a positive or a negative number?


In a signed binary, first bit (the leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101 is the binary representation of a negative integer, on 64 bits (8 Bytes).


2. Construct the unsigned binary number, exclude the first bit (the leftmost), that is reserved for the sign:

1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101 = 111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101

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

    • 262

      1
    • 261

      1
    • 260

      1
    • 259

      1
    • 258

      1
    • 257

      1
    • 256

      1
    • 255

      1
    • 254

      1
    • 253

      1
    • 252

      1
    • 251

      1
    • 250

      1
    • 249

      1
    • 248

      1
    • 247

      1
    • 246

      1
    • 245

      1
    • 244

      1
    • 243

      1
    • 242

      1
    • 241

      1
    • 240

      1
    • 239

      1
    • 238

      1
    • 237

      1
    • 236

      1
    • 235

      1
    • 234

      1
    • 233

      1
    • 232

      1
    • 231

      1
    • 230

      1
    • 229

      1
    • 228

      1
    • 227

      1
    • 226

      0
    • 225

      1
    • 224

      0
    • 223

      0
    • 222

      1
    • 221

      1
    • 220

      1
    • 219

      1
    • 218

      0
    • 217

      1
    • 216

      0
    • 215

      0
    • 214

      0
    • 213

      0
    • 212

      0
    • 211

      1
    • 210

      0
    • 29

      0
    • 28

      0
    • 27

      1
    • 26

      0
    • 25

      0
    • 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:

111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101(2) =


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


(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 0 + 33 554 432 + 0 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 0 + 131 072 + 0 + 0 + 0 + 0 + 0 + 2 048 + 0 + 0 + 0 + 128 + 0 + 0 + 16 + 0 + 4 + 0 + 1)(10) =


(4 611 686 018 427 387 904 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 33 554 432 + 4 194 304 + 2 097 152 + 1 048 576 + 524 288 + 131 072 + 2 048 + 128 + 16 + 4 + 1)(10) =


9 223 372 036 762 110 101(10)

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

1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101(2) = -9 223 372 036 762 110 101(10)

Conclusion:

Number 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101(2) converted from signed binary to an integer in decimal system (in base 10):


1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 0111 1010 0000 1000 1001 0101(2) = -9 223 372 036 762 110 101(10)

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

Convert signed binary numbers to integers in decimal system (base 10)

First bit (the leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

Entered binary number length must be: 2, 4, 8, 16, 32, or 64 - otherwise extra bits on 0 will be added in front (to the left).

How to convert a signed binary number to an integer in base ten:

1) Construct the unsigned binary number: exclude the first bit (the leftmost); this bit is reserved for the sign, 1 = negative, 0 = positive and does not count when calculating the absolute value (without sign).

2) Multiply each bit of the binary number by its corresponding power of 2 that its place value represents.

3) Add all the terms up to get the positive integer number in base ten.

4) Adjust the sign of the integer number by the first bit of the initial binary number.

Latest signed binary numbers converted to signed integers in decimal system (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