0000 0000 1100 1001 1000 1001 0001 0111 0100 0001 0001 0101 0101 0101 0111 0001 Signed Binary Two's (2's) Complement Representation 64 Bit - To Decimal

How to convert 0000 0000 1100 1001 1000 1001 0001 0111 0100 0001 0001 0101 0101 0101 0111 0001(2), signed binary in two's (2's) complement representation, to decimal

What are the steps to convert the signed binary in two's (2's) complement representation to an integer in decimal system (in base ten)?

1. Is this a positive or a negative number?

0000 0000 1100 1001 1000 1001 0001 0111 0100 0001 0001 0101 0101 0101 0111 0001 is the binary representation of a positive integer, on 64 bits (8 Bytes).


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

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

* Run this step only if the number is negative

  • Note on binary subtraction rules:
  • 11 - 1 = 10; 10 - 1 = 01; 1 - 0 = 1; 1 - 1 = 0.

  • Subtract 1 from the initial binary number.

  • * Not the case - the number is positive


3. Get the binary representation of the positive (unsigned) number.

* Run this step only if the number is negative

Flip all the bits of the signed binary in 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 - the number is positive


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

  • 263

    0

  • 262

    0

  • 261

    0

  • 260

    0

  • 259

    0

  • 258

    0

  • 257

    0

  • 256

    0

  • 255

    1

  • 254

    1

  • 253

    0

  • 252

    0

  • 251

    1

  • 250

    0

  • 249

    0

  • 248

    1

  • 247

    1

  • 246

    0

  • 245

    0

  • 244

    0

  • 243

    1

  • 242

    0

  • 241

    0

  • 240

    1

  • 239

    0

  • 238

    0

  • 237

    0

  • 236

    1

  • 235

    0

  • 234

    1

  • 233

    1

  • 232

    1

  • 231

    0

  • 230

    1

  • 229

    0

  • 228

    0

  • 227

    0

  • 226

    0

  • 225

    0

  • 224

    1

  • 223

    0

  • 222

    0

  • 221

    0

  • 220

    1

  • 219

    0

  • 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

    1

  • 25

    1

  • 24

    1

  • 23

    0

  • 22

    0

  • 21

    0

  • 20

    1

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

0000 0000 1100 1001 1000 1001 0001 0111 0100 0001 0001 0101 0101 0101 0111 0001(2) =

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

(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 0 + 0 + 2 251 799 813 685 248 + 0 + 0 + 281 474 976 710 656 + 140 737 488 355 328 + 0 + 0 + 0 + 8 796 093 022 208 + 0 + 0 + 1 099 511 627 776 + 0 + 0 + 0 + 68 719 476 736 + 0 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 0 + 1 073 741 824 + 0 + 0 + 0 + 0 + 0 + 16 777 216 + 0 + 0 + 0 + 1 048 576 + 0 + 262 144 + 0 + 65 536 + 0 + 16 384 + 0 + 4 096 + 0 + 1 024 + 0 + 256 + 0 + 64 + 32 + 16 + 0 + 0 + 0 + 1)(10) =

(36 028 797 018 963 968 + 18 014 398 509 481 984 + 2 251 799 813 685 248 + 281 474 976 710 656 + 140 737 488 355 328 + 8 796 093 022 208 + 1 099 511 627 776 + 68 719 476 736 + 17 179 869 184 + 8 589 934 592 + 4 294 967 296 + 1 073 741 824 + 16 777 216 + 1 048 576 + 262 144 + 65 536 + 16 384 + 4 096 + 1 024 + 256 + 64 + 32 + 16 + 1)(10) =

56 727 203 288 012 145(10)

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

0000 0000 1100 1001 1000 1001 0001 0111 0100 0001 0001 0101 0101 0101 0111 0001(2) = 56 727 203 288 012 145(10)

The number 0000 0000 1100 1001 1000 1001 0001 0111 0100 0001 0001 0101 0101 0101 0111 0001(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
0000 0000 1100 1001 1000 1001 0001 0111 0100 0001 0001 0101 0101 0101 0111 0001(2) = 56 727 203 288 012 145(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 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.