Signed binary two's complement number 1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 converted to decimal system (base ten) signed integer

Signed binary two's complement 1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111(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.

1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 is the binary representation of a negative 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:

1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 - 1 = 1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110


3. 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:

!(1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110) = 0011 1111 0111 1110 0101 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1001


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

      1
    • 256

      1
    • 255

      0
    • 254

      1
    • 253

      1
    • 252

      1
    • 251

      1
    • 250

      1
    • 249

      1
    • 248

      0
    • 247

      0
    • 246

      1
    • 245

      0
    • 244

      1
    • 243

      0
    • 242

      1
    • 241

      0
    • 240

      0
    • 239

      0
    • 238

      0
    • 237

      0
    • 236

      0
    • 235

      0
    • 234

      0
    • 233

      0
    • 232

      0
    • 231

      0
    • 230

      0
    • 229

      0
    • 228

      0
    • 227

      0
    • 226

      0
    • 225

      0
    • 224

      0
    • 223

      0
    • 222

      0
    • 221

      0
    • 220

      0
    • 219

      0
    • 218

      0
    • 217

      0
    • 216

      0
    • 215

      0
    • 214

      0
    • 213

      0
    • 212

      0
    • 211

      0
    • 210

      0
    • 29

      0
    • 28

      0
    • 27

      0
    • 26

      0
    • 25

      0
    • 24

      0
    • 23

      1
    • 22

      0
    • 21

      0
    • 20

      1

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

0011 1111 0111 1110 0101 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 1001(2) =


(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 1 × 259 + 1 × 258 + 1 × 257 + 1 × 256 + 0 × 255 + 1 × 254 + 1 × 253 + 1 × 252 + 1 × 251 + 1 × 250 + 1 × 249 + 0 × 248 + 0 × 247 + 1 × 246 + 0 × 245 + 1 × 244 + 0 × 243 + 1 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 0 × 237 + 0 × 236 + 0 × 235 + 0 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 0 × 226 + 0 × 225 + 0 × 224 + 0 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 0 × 219 + 0 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 0 × 214 + 0 × 213 + 0 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 0 × 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 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 0 + 18 014 398 509 481 984 + 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 + 0 + 0 + 70 368 744 177 664 + 0 + 17 592 186 044 416 + 0 + 4 398 046 511 104 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 + 0 + 0 + 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 + 144 115 188 075 855 872 + 72 057 594 037 927 936 + 18 014 398 509 481 984 + 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 + 70 368 744 177 664 + 17 592 186 044 416 + 4 398 046 511 104 + 8 + 1)(10) =


4 575 186 630 431 735 817(10)

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

1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111(2) = -4 575 186 630 431 735 817(10)

Number 1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111(2) converted from signed binary two's complement representation to an integer in decimal system (in base 10):
1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111(2) = -4 575 186 630 431 735 817(10)

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


More operations of this kind:

1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0110 = ?

1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1000 = ?


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)

1100 0000 1000 0001 1010 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 = -4,575,186,630,431,735,817 Oct 28 11:17 UTC (GMT)
1111 1111 1111 1100 0000 0100 0111 1000 = -261,000 Oct 28 11:17 UTC (GMT)
0000 0000 0011 0110 0010 1010 1100 0110 = 3,549,894 Oct 28 11:17 UTC (GMT)
1011 1100 1100 0000 0000 0000 0001 0000 = -1,128,267,760 Oct 28 11:16 UTC (GMT)
0001 0110 0110 0011 = 5,731 Oct 28 11:15 UTC (GMT)
0010 0000 0110 0011 0001 0101 0110 1101 = 543,364,461 Oct 28 11:14 UTC (GMT)
1111 1111 1111 1111 1111 1110 1110 0110 = -282 Oct 28 11:14 UTC (GMT)
0010 1001 = 41 Oct 28 11:13 UTC (GMT)
0111 0010 0010 1110 = 29,230 Oct 28 11:13 UTC (GMT)
1001 1011 = -101 Oct 28 11:13 UTC (GMT)
0000 0000 0000 0000 1110 0000 0000 0000 0000 0111 1110 1111 1111 1111 1110 0100 = 246,290,737,790,948 Oct 28 11:13 UTC (GMT)
1101 1010 = -38 Oct 28 11:13 UTC (GMT)
1111 1100 0001 0110 = -1,002 Oct 28 11:12 UTC (GMT)
All the converted signed binary two's complement numbers

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