Signed binary two's complement number 0011 1111 1011 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0000 converted to decimal system (base ten) signed integer

How to convert a signed binary two's complement:
0011 1111 1011 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0000(2)
to an integer in decimal system (in base 10)

1. Is this a positive or a negative number?


In a signed binary two's complement, first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.

0011 1111 1011 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0000 is the binary representation of a positive integer, on 64 bits (8 Bytes).


2. Get the binary representation in one's complement:


* Run this step only if the number is negative *

Subtract 1 from the binary initial number:

* Not the case *


2. Get the binary representation of the positive (unsigned) number:


* Run this step only if the number is negative *

Flip all the bits in the signed binary one's complement representation (reverse the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:

* Not the case *


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

    • 263

      0
    • 262

      0
    • 261

      1
    • 260

      1
    • 259

      1
    • 258

      1
    • 257

      1
    • 256

      1
    • 255

      1
    • 254

      0
    • 253

      1
    • 252

      1
    • 251

      1
    • 250

      1
    • 249

      1
    • 248

      1
    • 247

      0
    • 246

      1
    • 245

      1
    • 244

      1
    • 243

      1
    • 242

      1
    • 241

      0
    • 240

      0
    • 239

      1
    • 238

      1
    • 237

      1
    • 236

      0
    • 235

      1
    • 234

      1
    • 233

      0
    • 232

      1
    • 231

      1
    • 230

      0
    • 229

      0
    • 228

      1
    • 227

      0
    • 226

      0
    • 225

      0
    • 224

      1
    • 223

      0
    • 222

      1
    • 221

      1
    • 220

      0
    • 219

      1
    • 218

      0
    • 217

      0
    • 216

      0
    • 215

      0
    • 214

      1
    • 213

      1
    • 212

      1
    • 211

      0
    • 210

      0
    • 29

      1
    • 28

      0
    • 27

      1
    • 26

      0
    • 25

      1
    • 24

      1
    • 23

      0
    • 22

      0
    • 21

      0
    • 20

      0

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

0011 1111 1011 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0000(2) =


(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 1 × 259 + 1 × 258 + 1 × 257 + 1 × 256 + 1 × 255 + 0 × 254 + 1 × 253 + 1 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 1 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 0 × 240 + 1 × 239 + 1 × 238 + 1 × 237 + 0 × 236 + 1 × 235 + 1 × 234 + 0 × 233 + 1 × 232 + 1 × 231 + 0 × 230 + 0 × 229 + 1 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 1 × 224 + 0 × 223 + 1 × 222 + 1 × 221 + 0 × 220 + 1 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =


(0 + 0 + 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 + 0 + 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 + 0 + 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 + 0 + 0 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 0 + 34 359 738 368 + 17 179 869 184 + 0 + 4 294 967 296 + 2 147 483 648 + 0 + 0 + 268 435 456 + 0 + 0 + 0 + 16 777 216 + 0 + 4 194 304 + 2 097 152 + 0 + 524 288 + 0 + 0 + 0 + 0 + 16 384 + 8 192 + 4 096 + 0 + 0 + 512 + 0 + 128 + 0 + 32 + 16 + 0 + 0 + 0 + 0)(10) =


(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 + 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 + 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 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 34 359 738 368 + 17 179 869 184 + 4 294 967 296 + 2 147 483 648 + 268 435 456 + 16 777 216 + 4 194 304 + 2 097 152 + 524 288 + 16 384 + 8 192 + 4 096 + 512 + 128 + 32 + 16)(10) =


4 593 527 504 729 830 064(10)

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

0011 1111 1011 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0000(2) = 4 593 527 504 729 830 064(10)

Conclusion:

Number 0011 1111 1011 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0000(2) converted from signed binary two's complement representation to an integer in decimal system (in base 10):


0011 1111 1011 1111 0111 1100 1110 1101 1001 0001 0110 1000 0111 0010 1011 0000(2) = 4 593 527 504 729 830 064(10)

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

Convert signed binary two's complement numbers to decimal system (base ten) integers

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 in two's complement representation to an integer in base ten:

1) In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive.

2) Get the signed binary representation in one's complement, subtract 1 from the initial number.

3) Construct the unsigned binary number: flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s.

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

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

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

Latest binary numbers in two's complement representation converted to signed integers in decimal system (base ten)

How to convert signed binary numbers in two's complement representation from binary system to decimal

To understand how to convert a signed binary number in two's complement representation from the binary system to decimal (base ten), the easiest way is to do it by an example - convert binary, 1101 1110, to base ten:

  • In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
  • Get the signed binary representation in one's complement, subtract 1 from the initial number:
    1101 1110 - 1 = 1101 1101
  • Get the binary representation of the positive number, flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:
    !(1101 1101) = 0010 0010
  • Write bellow the positive binary number representation 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, increasing each corresonding power of 2 by exactly one unit:
  • powers of 2: 7 6 5 4 3 2 1 0
    digits: 0 0 1 0 0 0 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:

    0010 0010(2) =


    (0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =


    (0 + 0 + 32 + 0 + 0 + 0 + 2 + 0)(10) =


    (32 + 2)(10) =


    34(10)

  • Signed binary number in two's complement representation, 1101 1110 = -34(10), a signed negative integer in base 10