Convert 1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 0001 Signed Binary in Two's (2's) Complement Representation on 64 Bit to Decimal

How to convert 1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 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?

1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 0001 is the binary representation of a negative 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.

  • 1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 0001 - 1 = 1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 0000


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:

!(1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 0000) = 0011 1101 1110 0010 1011 1101 1101 1100 0000 0011 1110 0000 0110 1000 1001 1111


4. 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

    0

  • 256

    1

  • 255

    1

  • 254

    1

  • 253

    1

  • 252

    0

  • 251

    0

  • 250

    0

  • 249

    1

  • 248

    0

  • 247

    1

  • 246

    0

  • 245

    1

  • 244

    1

  • 243

    1

  • 242

    1

  • 241

    0

  • 240

    1

  • 239

    1

  • 238

    1

  • 237

    0

  • 236

    1

  • 235

    1

  • 234

    1

  • 233

    0

  • 232

    0

  • 231

    0

  • 230

    0

  • 229

    0

  • 228

    0

  • 227

    0

  • 226

    0

  • 225

    1

  • 224

    1

  • 223

    1

  • 222

    1

  • 221

    1

  • 220

    0

  • 219

    0

  • 218

    0

  • 217

    0

  • 216

    0

  • 215

    0

  • 214

    1

  • 213

    1

  • 212

    0

  • 211

    1

  • 210

    0

  • 29

    0

  • 28

    0

  • 27

    1

  • 26

    0

  • 25

    0

  • 24

    1

  • 23

    1

  • 22

    1

  • 21

    1

  • 20

    1

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

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

(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 1 × 259 + 1 × 258 + 0 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 1 × 253 + 0 × 252 + 0 × 251 + 0 × 250 + 1 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 1 × 245 + 1 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 1 × 239 + 1 × 238 + 0 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 1 × 225 + 1 × 224 + 1 × 223 + 1 × 222 + 1 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 1 × 213 + 0 × 212 + 1 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 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 + 0 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 0 + 0 + 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 + 4 398 046 511 104 + 0 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 0 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 0 + 0 + 0 + 0 + 0 + 0 + 16 384 + 8 192 + 0 + 2 048 + 0 + 0 + 0 + 128 + 0 + 0 + 16 + 8 + 4 + 2 + 1)(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 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 9 007 199 254 740 992 + 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 + 4 398 046 511 104 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 2 097 152 + 16 384 + 8 192 + 2 048 + 128 + 16 + 8 + 4 + 2 + 1)(10) =

4 459 335 333 705 705 631(10)

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

1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 0001(2) = -4 459 335 333 705 705 631(10)

The number 1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 0001(2), signed binary in two's (2's) complement representation, converted and written as an integer in decimal system (base ten):
1100 0010 0001 1101 0100 0010 0010 0011 1111 1100 0001 1111 1001 0111 0110 0001(2) = -4 459 335 333 705 705 631(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.