Signed binary two's complement number 0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0001 converted to decimal system (base ten) signed integer

Signed binary two's complement 0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0001(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,

The first bit (the leftmost) indicates the sign,

1 = negative, 0 = positive.

0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0001 is the binary representation of a positive integer, on 64 bits (8 Bytes).


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

* Run this step only if the number is negative *

Note: 11 - 1 = 10; 10 - 1 = 1; 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 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:

* 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

      1

    • 260

      1

    • 259

      1

    • 258

      1

    • 257

      1

    • 256

      1

    • 255

      1

    • 254

      0

    • 253

      1

    • 252

      0

    • 251

      1

    • 250

      0

    • 249

      1

    • 248

      1

    • 247

      0

    • 246

      1

    • 245

      1

    • 244

      0

    • 243

      1

    • 242

      1

    • 241

      0

    • 240

      0

    • 239

      0

    • 238

      0

    • 237

      1

    • 236

      1

    • 235

      1

    • 234

      1

    • 233

      0

    • 232

      0

    • 231

      0

    • 230

      0

    • 229

      1

    • 228

      0

    • 227

      1

    • 226

      1

    • 225

      1

    • 224

      0

    • 223

      1

    • 222

      0

    • 221

      0

    • 220

      0

    • 219

      1

    • 218

      0

    • 217

      1

    • 216

      0

    • 215

      0

    • 214

      1

    • 213

      0

    • 212

      1

    • 211

      0

    • 210

      0

    • 29

      0

    • 28

      1

    • 27

      0

    • 26

      0

    • 25

      1

    • 24

      0

    • 23

      0

    • 22

      0

    • 21

      0

    • 20

      1

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

0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0001(2) =

(0 × 263 + 0 × 262 + 1 × 261 + 1 × 260 + 1 × 259 + 1 × 258 + 1 × 257 + 1 × 256 + 1 × 255 + 0 × 254 + 1 × 253 + 0 × 252 + 1 × 251 + 0 × 250 + 1 × 249 + 1 × 248 + 0 × 247 + 1 × 246 + 1 × 245 + 0 × 244 + 1 × 243 + 1 × 242 + 0 × 241 + 0 × 240 + 0 × 239 + 0 × 238 + 1 × 237 + 1 × 236 + 1 × 235 + 1 × 234 + 0 × 233 + 0 × 232 + 0 × 231 + 0 × 230 + 1 × 229 + 0 × 228 + 1 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 1 × 219 + 0 × 218 + 1 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 0 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 0 × 29 + 1 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 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 + 36 028 797 018 963 968 + 0 + 9 007 199 254 740 992 + 0 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 0 + 70 368 744 177 664 + 35 184 372 088 832 + 0 + 8 796 093 022 208 + 4 398 046 511 104 + 0 + 0 + 0 + 0 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 0 + 0 + 0 + 0 + 536 870 912 + 0 + 134 217 728 + 67 108 864 + 33 554 432 + 0 + 8 388 608 + 0 + 0 + 0 + 524 288 + 0 + 131 072 + 0 + 0 + 16 384 + 0 + 4 096 + 0 + 0 + 0 + 256 + 0 + 0 + 32 + 0 + 0 + 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 + 36 028 797 018 963 968 + 9 007 199 254 740 992 + 2 251 799 813 685 248 + 562 949 953 421 312 + 281 474 976 710 656 + 70 368 744 177 664 + 35 184 372 088 832 + 8 796 093 022 208 + 4 398 046 511 104 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 17 179 869 184 + 536 870 912 + 134 217 728 + 67 108 864 + 33 554 432 + 8 388 608 + 524 288 + 131 072 + 16 384 + 4 096 + 256 + 32 + 1)(10) =

4 587 879 651 141 636 385(10)

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

0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0001(2) = 4 587 879 651 141 636 385(10)

Number 0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0001(2) converted from signed binary two's complement representation to an integer in decimal system (in base 10):
0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0001(2) = 4 587 879 651 141 636 385(10)

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

More operations of this kind:

0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0000 converted from: signed binary two's complement representation, to signed integer = ?

0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0010 converted from: signed binary two's complement representation, to signed integer = ?


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)

0011 1111 1010 1011 0110 1100 0011 1100 0010 1110 1000 1010 0101 0001 0010 0001 converted from: signed binary two's complement representation, to signed integer = 4,587,879,651,141,636,385 May 29 15:07 UTC (GMT)
0010 1001 0101 1010 converted from: signed binary two's complement representation, to signed integer = 10,586 May 29 15:03 UTC (GMT)
0000 0000 0000 0011 1011 1100 0001 0110 converted from: signed binary two's complement representation, to signed integer = 244,758 May 29 15:03 UTC (GMT)
1000 0000 0100 0000 0000 0000 1100 1111 converted from: signed binary two's complement representation, to signed integer = -2,143,289,137 May 29 15:03 UTC (GMT)
0100 0000 0000 1111 converted from: signed binary two's complement representation, to signed integer = 16,399 May 29 15:03 UTC (GMT)
0000 0000 1001 0110 converted from: signed binary two's complement representation, to signed integer = 150 May 29 15:03 UTC (GMT)
1011 1101 1101 0111 1101 0110 1011 0110 converted from: signed binary two's complement representation, to signed integer = -1,109,928,266 May 29 15:03 UTC (GMT)
0000 0000 0000 0000 0000 0101 0101 0101 0000 0001 1111 1010 1011 1110 0000 1000 converted from: signed binary two's complement representation, to signed integer = 5,862,663,568,904 May 29 15:02 UTC (GMT)
1000 1011 0011 0011 1111 1111 1111 0100 converted from: signed binary two's complement representation, to signed integer = -1,959,526,412 May 29 15:01 UTC (GMT)
0110 1110 1100 1111 converted from: signed binary two's complement representation, to signed integer = 28,367 May 29 15:01 UTC (GMT)
0000 0000 0000 0010 0101 0010 1011 1110 converted from: signed binary two's complement representation, to signed integer = 152,254 May 29 15:00 UTC (GMT)
0000 0000 0000 1111 0010 1001 1101 1101 converted from: signed binary two's complement representation, to signed integer = 993,757 May 29 15:00 UTC (GMT)
0110 0110 1000 0001 1110 0010 1000 1000 converted from: signed binary two's complement representation, to signed integer = 1,719,788,168 May 29 14:59 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