Signed binary number 1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110 converted to an integer in base ten

Signed binary 1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110(2) to an integer in decimal system (in base 10) = ?

1. Is this a positive or a negative number?


In a signed binary, first bit (the leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110 is the binary representation of a negative integer, on 64 bits (8 Bytes).


2. Construct the unsigned binary number, exclude the first bit (the leftmost), that is reserved for the sign:

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110 = 100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110

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

    • 262

      1
    • 261

      0
    • 260

      0
    • 259

      0
    • 258

      0
    • 257

      0
    • 256

      0
    • 255

      0
    • 254

      1
    • 253

      0
    • 252

      1
    • 251

      1
    • 250

      0
    • 249

      1
    • 248

      1
    • 247

      1
    • 246

      0
    • 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

      1
    • 232

      0
    • 231

      0
    • 230

      1
    • 229

      0
    • 228

      0
    • 227

      0
    • 226

      1
    • 225

      0
    • 224

      1
    • 223

      1
    • 222

      0
    • 221

      1
    • 220

      0
    • 219

      0
    • 218

      0
    • 217

      0
    • 216

      0
    • 215

      0
    • 214

      1
    • 213

      0
    • 212

      1
    • 211

      1
    • 210

      0
    • 29

      1
    • 28

      0
    • 27

      1
    • 26

      1
    • 25

      1
    • 24

      1
    • 23

      0
    • 22

      1
    • 21

      1
    • 20

      0

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

100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110(2) =


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


(4 611 686 018 427 387 904 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 18 014 398 509 481 984 + 0 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 0 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 0 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 0 + 8 589 934 592 + 0 + 0 + 1 073 741 824 + 0 + 0 + 0 + 67 108 864 + 0 + 16 777 216 + 8 388 608 + 0 + 2 097 152 + 0 + 0 + 0 + 0 + 0 + 0 + 16 384 + 0 + 4 096 + 2 048 + 0 + 512 + 0 + 128 + 64 + 32 + 16 + 0 + 4 + 2 + 0)(10) =


(4 611 686 018 427 387 904 + 18 014 398 509 481 984 + 4 503 599 627 370 496 + 2 251 799 813 685 248 + 562 949 953 421 312 + 281 474 976 710 656 + 140 737 488 355 328 + 35 184 372 088 832 + 17 592 186 044 416 + 8 796 093 022 208 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 137 438 953 472 + 68 719 476 736 + 34 359 738 368 + 8 589 934 592 + 1 073 741 824 + 67 108 864 + 16 777 216 + 8 388 608 + 2 097 152 + 16 384 + 4 096 + 2 048 + 512 + 128 + 64 + 32 + 16 + 4 + 2)(10) =


4 637 506 650 014 505 718(10)

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

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110(2) = -4 637 506 650 014 505 718(10)

Number 1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110(2) converted from signed binary to an integer in decimal system (in base 10):
1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110(2) = -4 637 506 650 014 505 718(10)

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


More operations of this kind:

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0101 = ?

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0111 = ?


Convert signed binary numbers to integers in decimal system (base 10)

First bit (the leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

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 to an integer in base ten:

1) Construct the unsigned binary number: exclude the first bit (the leftmost); this bit is reserved for the sign, 1 = negative, 0 = positive and does not count when calculating the absolute value (without sign).

2) Multiply each bit of the binary number by its corresponding power of 2 that its place value represents.

3) Add all the terms up to get the positive integer number in base ten.

4) Adjust the sign of the integer number by the first bit of the initial binary number.

Latest signed binary numbers converted to signed integers in decimal system (base ten)

1100 0000 0101 1011 1011 1011 1011 1010 0100 0101 1010 0000 0101 1010 1111 0110 = -4,637,506,650,014,505,718 Jun 13 22:49 UTC (GMT)
0111 0100 0100 1000 = 29,768 Jun 13 22:48 UTC (GMT)
0011 0011 1000 0111 = 13,191 Jun 13 22:48 UTC (GMT)
0000 0000 0000 0001 0111 1111 0000 0000 0000 0000 0000 0000 0001 0001 0100 1000 = 421,112,953,442,632 Jun 13 22:48 UTC (GMT)
0011 0000 1010 1111 1111 1111 1100 1010 = 816,840,650 Jun 13 22:48 UTC (GMT)
0011 1010 1011 1110 = 15,038 Jun 13 22:48 UTC (GMT)
0011 1001 1101 1000 = 14,808 Jun 13 22:47 UTC (GMT)
1000 1100 1010 0101 = -3,237 Jun 13 22:47 UTC (GMT)
0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011 = 2,753,069,380,528,003 Jun 13 22:47 UTC (GMT)
1100 0001 1011 1000 0000 0000 0000 1111 = -1,102,577,679 Jun 13 22:47 UTC (GMT)
0000 0000 0000 1010 0101 1010 1000 0101 = 678,533 Jun 13 22:46 UTC (GMT)
0111 0101 1100 0100 = 30,148 Jun 13 22:46 UTC (GMT)
1111 1101 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1001 = -9,079,256,848,778,919,929 Jun 13 22:45 UTC (GMT)
All the converted signed binary numbers to integers in base ten

How to convert signed binary numbers from binary system to decimal (base ten)

To understand how to convert a signed binary number from binary system to decimal (base ten), the easiest way is to do it through an example - convert the binary number, 1001 1110, to base ten:

  • In a signed binary, the first bit (leftmost) is reserved for the sign, 1 = negative, 0 = positive. This bit does not count when calculating the absolute value (value without sign). The first bit is 1, so our number is negative.
  • Write bellow the binary number 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 and increasing each corresonding power of 2 by exactly one unit, but ignoring the very first bit (the leftmost, the one representing the sign):
  • powers of 2:   6 5 4 3 2 1 0
    digits: 1 0 0 1 1 1 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, but also taking care of the number sign:

    1001 1110 =


    - (0 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20)(10) =


    - (0 + 0 + 16 + 8 + 4 + 2 + 0)(10) =


    - (16 + 8 + 4 + 2)(10) =


    -30(10)

  • Binary signed number, 1001 1110 = -30(10), signed negative integer in base 10