Signed binary one's complement number 1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1011 converted to decimal system (base ten) signed integer

How to convert a signed binary one's complement:
1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1011(2)
to an integer in decimal system (in base 10)

1. Is this a positive or a negative number?


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

1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1011 is the binary representation of a negative integer, on 64 bits (8 Bytes).


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:

!(1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1011) = 0101 0100 1100 1010 1011 1011 1010 0101 0010 1011 0111 0110 1010 1010 1001 0100


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

    • 263

      0
    • 262

      1
    • 261

      0
    • 260

      1
    • 259

      0
    • 258

      1
    • 257

      0
    • 256

      0
    • 255

      1
    • 254

      1
    • 253

      0
    • 252

      0
    • 251

      1
    • 250

      0
    • 249

      1
    • 248

      0
    • 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

      0
    • 234

      1
    • 233

      0
    • 232

      1
    • 231

      0
    • 230

      0
    • 229

      1
    • 228

      0
    • 227

      1
    • 226

      0
    • 225

      1
    • 224

      1
    • 223

      0
    • 222

      1
    • 221

      1
    • 220

      1
    • 219

      0
    • 218

      1
    • 217

      1
    • 216

      0
    • 215

      1
    • 214

      0
    • 213

      1
    • 212

      0
    • 211

      1
    • 210

      0
    • 29

      1
    • 28

      0
    • 27

      1
    • 26

      0
    • 25

      0
    • 24

      1
    • 23

      0
    • 22

      1
    • 21

      0
    • 20

      0

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

0101 0100 1100 1010 1011 1011 1010 0101 0010 1011 0111 0110 1010 1010 1001 0100(2) =


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


(0 + 4 611 686 018 427 387 904 + 0 + 1 152 921 504 606 846 976 + 0 + 288 230 376 151 711 744 + 0 + 0 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 0 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 0 + 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 + 0 + 17 179 869 184 + 0 + 4 294 967 296 + 0 + 0 + 536 870 912 + 0 + 134 217 728 + 0 + 33 554 432 + 16 777 216 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 0 + 262 144 + 131 072 + 0 + 32 768 + 0 + 8 192 + 0 + 2 048 + 0 + 512 + 0 + 128 + 0 + 0 + 16 + 0 + 4 + 0 + 0)(10) =


(4 611 686 018 427 387 904 + 1 152 921 504 606 846 976 + 288 230 376 151 711 744 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 2 251 799 813 685 248 + 562 949 953 421 312 + 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 + 17 179 869 184 + 4 294 967 296 + 536 870 912 + 134 217 728 + 33 554 432 + 16 777 216 + 4 194 304 + 2 097 152 + 1 048 576 + 262 144 + 131 072 + 32 768 + 8 192 + 2 048 + 512 + 128 + 16 + 4)(10) =


6 109 902 162 554 694 292(10)

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

1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1011(2) = -6 109 902 162 554 694 292(10)

Conclusion:
Number 1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1011(2) converted from signed binary one's complement representation to an integer in decimal system (in base 10):


1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1011(2) = -6 109 902 162 554 694 292(10)

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


More operations of this kind:

1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1010 = ?

1010 1011 0011 0101 0100 0100 0101 1010 1101 0100 1000 1001 0101 0101 0110 1100 = ?


Convert signed binary one'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 one's complement representation to an integer in base ten:

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

2) 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.

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

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

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

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

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

To understand how to convert a signed binary number in one's complement representation from binary system to decimal (base ten), the easiest way is to do it through an example - convert binary, 1001 1101, to base ten:

  • In a signed binary one's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
  • 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:
    !(1001 1101) = 0110 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 by increasing each corresonding power of 2 by exactly one unit:
  • powers of 2: 7 6 5 4 3 2 1 0
    digits: 0 1 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:

    0110 0010(2) =


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


    (0 + 64 + 32 + 0 + 0 + 0 + 2 + 0)(10) =


    (64 + 32 + 2)(10) =


    98(10)

  • Signed binary number in one's complement representation, 1001 1110 = -98(10), a signed negative integer in base 10