Convert 1100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001, Base 2 Signed Binary Number on 64 Bit, To Base 10 Decimal System Integer

How to convert 1100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001(2), the base 2 signed binary number on 64 bit, to a base 10 decimal system integer

What are the steps to convert the base 2 signed binary number
1100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001(2) to a base 10 decimal system equivalent integer?

1. Is this a positive or a negative number?

1100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001 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 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001 = 100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001


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

    1

  • 256

    1

  • 255

    0

  • 254

    1

  • 253

    0

  • 252

    1

  • 251

    1

  • 250

    1

  • 249

    0

  • 248

    1

  • 247

    1

  • 246

    0

  • 245

    1

  • 244

    1

  • 243

    1

  • 242

    0

  • 241

    1

  • 240

    1

  • 239

    1

  • 238

    0

  • 237

    1

  • 236

    0

  • 235

    1

  • 234

    0

  • 233

    1

  • 232

    0

  • 231

    1

  • 230

    1

  • 229

    0

  • 228

    1

  • 227

    0

  • 226

    1

  • 225

    1

  • 224

    0

  • 223

    1

  • 222

    0

  • 221

    1

  • 220

    1

  • 219

    1

  • 218

    0

  • 217

    0

  • 216

    0

  • 215

    1

  • 214

    0

  • 213

    0

  • 212

    0

  • 211

    1

  • 210

    1

  • 29

    1

  • 28

    1

  • 27

    1

  • 26

    0

  • 25

    1

  • 24

    1

  • 23

    0

  • 22

    0

  • 21

    0

  • 20

    1

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

100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001(2) =

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

(4 611 686 018 427 387 904 + 0 + 0 + 0 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 0 + 281 474 976 710 656 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 0 + 137 438 953 472 + 0 + 34 359 738 368 + 0 + 8 589 934 592 + 0 + 2 147 483 648 + 1 073 741 824 + 0 + 268 435 456 + 0 + 67 108 864 + 33 554 432 + 0 + 8 388 608 + 0 + 2 097 152 + 1 048 576 + 524 288 + 0 + 0 + 0 + 32 768 + 0 + 0 + 0 + 2 048 + 1 024 + 512 + 256 + 128 + 0 + 32 + 16 + 0 + 0 + 0 + 1)(10) =

(4 611 686 018 427 387 904 + 288 230 376 151 711 744 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 281 474 976 710 656 + 140 737 488 355 328 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 137 438 953 472 + 34 359 738 368 + 8 589 934 592 + 2 147 483 648 + 1 073 741 824 + 268 435 456 + 67 108 864 + 33 554 432 + 8 388 608 + 2 097 152 + 1 048 576 + 524 288 + 32 768 + 2 048 + 1 024 + 512 + 256 + 128 + 32 + 16 + 1)(10) =

5 142 472 691 948 228 529(10)

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

1100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001(2) = -5 142 472 691 948 228 529(10)

1100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001(2), Base 2 signed binary number, converted and written as a base 10 decimal system equivalent integer:
1100 0111 0101 1101 1011 1011 1010 1010 1101 0110 1011 1000 1000 1111 1011 0001(2) = -5 142 472 691 948 228 529(10)

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

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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