Two's Complement: Binary ↘ Integer: 1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 1000 Signed Binary Number in Two's Complement Representation, Converted and Written as a Decimal System Integer (in Base Ten)

Signed binary in two's complement representation 1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 1000(2) converted to an integer in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 1000 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 = 1; 1 - 0 = 1; 1 - 1 = 0.

Subtract 1 from the initial binary number.

1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 1000 - 1 = 1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0111


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 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0111) = 0011 1101 1101 1100 1100 0101 1101 1111 1100 0110 0100 0111 1011 0101 0101 1000


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

    0

  • 252

    1

  • 251

    1

  • 250

    1

  • 249

    0

  • 248

    0

  • 247

    1

  • 246

    1

  • 245

    0

  • 244

    0

  • 243

    0

  • 242

    1

  • 241

    0

  • 240

    1

  • 239

    1

  • 238

    1

  • 237

    0

  • 236

    1

  • 235

    1

  • 234

    1

  • 233

    1

  • 232

    1

  • 231

    1

  • 230

    1

  • 229

    0

  • 228

    0

  • 227

    0

  • 226

    1

  • 225

    1

  • 224

    0

  • 223

    0

  • 222

    1

  • 221

    0

  • 220

    0

  • 219

    0

  • 218

    1

  • 217

    1

  • 216

    1

  • 215

    1

  • 214

    0

  • 213

    1

  • 212

    1

  • 211

    0

  • 210

    1

  • 29

    0

  • 28

    1

  • 27

    0

  • 26

    1

  • 25

    0

  • 24

    1

  • 23

    1

  • 22

    0

  • 21

    0

  • 20

    0

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

0011 1101 1101 1100 1100 0101 1101 1111 1100 0110 0100 0111 1011 0101 0101 1000(2) =

(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 1 × 259 + 1 × 258 + 0 × 257 + 1 × 256 + 1 × 255 + 1 × 254 + 0 × 253 + 1 × 252 + 1 × 251 + 1 × 250 + 0 × 249 + 0 × 248 + 1 × 247 + 1 × 246 + 0 × 245 + 0 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 1 × 240 + 1 × 239 + 1 × 238 + 0 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 1 × 233 + 1 × 232 + 1 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 0 × 223 + 1 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 1 × 218 + 1 × 217 + 1 × 216 + 1 × 215 + 0 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 1 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 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 + 0 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 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 + 0 + 140 737 488 355 328 + 70 368 744 177 664 + 0 + 0 + 0 + 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 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 0 + 0 + 0 + 67 108 864 + 33 554 432 + 0 + 0 + 4 194 304 + 0 + 0 + 0 + 262 144 + 131 072 + 65 536 + 32 768 + 0 + 8 192 + 4 096 + 0 + 1 024 + 0 + 256 + 0 + 64 + 0 + 16 + 8 + 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 + 72 057 594 037 927 936 + 36 028 797 018 963 968 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 1 125 899 906 842 624 + 140 737 488 355 328 + 70 368 744 177 664 + 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 + 8 589 934 592 + 4 294 967 296 + 2 147 483 648 + 1 073 741 824 + 67 108 864 + 33 554 432 + 4 194 304 + 262 144 + 131 072 + 65 536 + 32 768 + 8 192 + 4 096 + 1 024 + 256 + 64 + 16 + 8)(10) =

4 457 655 296 084 915 544(10)

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

1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 1000(2) = -4 457 655 296 084 915 544(10)

The signed binary number in two's complement representation 1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 1000(2) converted and written as an integer in decimal system (base ten):
1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 1000(2) = -4 457 655 296 084 915 544(10)

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

More operaions on signed binary in two's complement representation:

» 1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 0111 The signed binary in two's complement representation converted and written as an integer number in decimal system (in base ten) = ?

» 1100 0010 0010 0011 0011 1010 0010 0000 0011 1001 1011 1000 0100 1010 1010 1001 The signed binary in two's complement representation converted and written as an integer number in decimal system (in base ten) = ?

The latest binary numbers written in two\'s complement representation converted to signed integers written in decimal system (in base ten)

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

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.