Signed binary two's complement number 1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1000 converted to decimal system (base ten) signed integer

Signed binary two's complement 1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1000(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.

1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1000 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:

1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1000 - 1 = 1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 0111


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:

!(1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 0111) = 0101 1010 1001 1010 1010 1010 1011 0100 0110 1011 1101 0111 0101 1010 1100 1000


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

    • 263

      0
    • 262

      1
    • 261

      0
    • 260

      1
    • 259

      1
    • 258

      0
    • 257

      1
    • 256

      0
    • 255

      1
    • 254

      0
    • 253

      0
    • 252

      1
    • 251

      1
    • 250

      0
    • 249

      1
    • 248

      0
    • 247

      1
    • 246

      0
    • 245

      1
    • 244

      0
    • 243

      1
    • 242

      0
    • 241

      1
    • 240

      0
    • 239

      1
    • 238

      0
    • 237

      1
    • 236

      1
    • 235

      0
    • 234

      1
    • 233

      0
    • 232

      0
    • 231

      0
    • 230

      1
    • 229

      1
    • 228

      0
    • 227

      1
    • 226

      0
    • 225

      1
    • 224

      1
    • 223

      1
    • 222

      1
    • 221

      0
    • 220

      1
    • 219

      0
    • 218

      1
    • 217

      1
    • 216

      1
    • 215

      0
    • 214

      1
    • 213

      0
    • 212

      1
    • 211

      1
    • 210

      0
    • 29

      1
    • 28

      0
    • 27

      1
    • 26

      1
    • 25

      0
    • 24

      0
    • 23

      1
    • 22

      0
    • 21

      0
    • 20

      0

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

0101 1010 1001 1010 1010 1010 1011 0100 0110 1011 1101 0111 0101 1010 1100 1000(2) =


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


(0 + 4 611 686 018 427 387 904 + 0 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 0 + 144 115 188 075 855 872 + 0 + 36 028 797 018 963 968 + 0 + 0 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 0 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 0 + 549 755 813 888 + 0 + 137 438 953 472 + 68 719 476 736 + 0 + 17 179 869 184 + 0 + 0 + 0 + 1 073 741 824 + 536 870 912 + 0 + 134 217 728 + 0 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 0 + 1 048 576 + 0 + 262 144 + 131 072 + 65 536 + 0 + 16 384 + 0 + 4 096 + 2 048 + 0 + 512 + 0 + 128 + 64 + 0 + 0 + 8 + 0 + 0 + 0)(10) =


(4 611 686 018 427 387 904 + 1 152 921 504 606 846 976 + 576 460 752 303 423 488 + 144 115 188 075 855 872 + 36 028 797 018 963 968 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 562 949 953 421 312 + 140 737 488 355 328 + 35 184 372 088 832 + 8 796 093 022 208 + 2 199 023 255 552 + 549 755 813 888 + 137 438 953 472 + 68 719 476 736 + 17 179 869 184 + 1 073 741 824 + 536 870 912 + 134 217 728 + 33 554 432 + 16 777 216 + 8 388 608 + 4 194 304 + 1 048 576 + 262 144 + 131 072 + 65 536 + 16 384 + 4 096 + 2 048 + 512 + 128 + 64 + 8)(10) =


6 528 718 301 707 066 056(10)

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

1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1000(2) = -6 528 718 301 707 066 056(10)

Number 1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1000(2) converted from signed binary two's complement representation to an integer in decimal system (in base 10):
1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1000(2) = -6 528 718 301 707 066 056(10)

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


More operations of this kind:

1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 0111 = ?

1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1001 = ?


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)

1010 0101 0110 0101 0101 0101 0100 1011 1001 0100 0010 1000 1010 0101 0011 1000 = -6,528,718,301,707,066,056 Sep 20 02:52 UTC (GMT)
1100 1011 = -53 Sep 20 02:52 UTC (GMT)
0000 0000 0000 0000 0000 0000 1111 1111 1000 0001 0000 0010 0010 1010 0000 0001 = 1,097,381,063,169 Sep 20 02:51 UTC (GMT)
0000 0000 0000 0000 0111 1111 1110 1011 1010 1001 1101 1101 1000 0000 0000 0100 = 140,650,143,907,844 Sep 20 02:51 UTC (GMT)
0000 1010 1101 0001 0000 0000 0000 0111 = 181,469,191 Sep 20 02:51 UTC (GMT)
0110 0010 1010 1111 = 25,263 Sep 20 02:51 UTC (GMT)
1111 0011 0101 0001 = -3,247 Sep 20 02:51 UTC (GMT)
0101 = 5 Sep 20 02:50 UTC (GMT)
0101 1010 = 90 Sep 20 02:50 UTC (GMT)
0000 0001 0000 0001 0000 0111 0001 1010 = 16,844,570 Sep 20 02:50 UTC (GMT)
0000 0000 0000 0000 0001 1111 1101 0011 = 8,147 Sep 20 02:50 UTC (GMT)
1011 1101 = -67 Sep 20 02:50 UTC (GMT)
1001 1010 = -102 Sep 20 02:49 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