Signed binary number 0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011 converted to an integer in base ten

Signed binary 0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011(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.

0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011 is the binary representation of a positive integer, on 64 bits (8 Bytes).


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

0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011 = 000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011

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

    • 262

      0
    • 261

      0
    • 260

      0
    • 259

      0
    • 258

      0
    • 257

      0
    • 256

      0
    • 255

      0
    • 254

      0
    • 253

      0
    • 252

      0
    • 251

      1
    • 250

      0
    • 249

      0
    • 248

      1
    • 247

      1
    • 246

      1
    • 245

      0
    • 244

      0
    • 243

      0
    • 242

      1
    • 241

      1
    • 240

      1
    • 239

      1
    • 238

      1
    • 237

      1
    • 236

      0
    • 235

      0
    • 234

      1
    • 233

      1
    • 232

      0
    • 231

      1
    • 230

      1
    • 229

      1
    • 228

      0
    • 227

      1
    • 226

      0
    • 225

      1
    • 224

      0
    • 223

      0
    • 222

      1
    • 221

      1
    • 220

      1
    • 219

      0
    • 218

      1
    • 217

      1
    • 216

      1
    • 215

      1
    • 214

      1
    • 213

      1
    • 212

      0
    • 211

      0
    • 210

      1
    • 29

      1
    • 28

      1
    • 27

      1
    • 26

      0
    • 25

      0
    • 24

      0
    • 23

      0
    • 22

      0
    • 21

      1
    • 20

      1

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

000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011(2) =


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


(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 251 799 813 685 248 + 0 + 0 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 0 + 0 + 0 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 0 + 0 + 17 179 869 184 + 8 589 934 592 + 0 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 0 + 134 217 728 + 0 + 33 554 432 + 0 + 0 + 4 194 304 + 2 097 152 + 1 048 576 + 0 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 0 + 0 + 1 024 + 512 + 256 + 128 + 0 + 0 + 0 + 0 + 0 + 2 + 1)(10) =


(2 251 799 813 685 248 + 281 474 976 710 656 + 140 737 488 355 328 + 70 368 744 177 664 + 4 398 046 511 104 + 2 199 023 255 552 + 1 099 511 627 776 + 549 755 813 888 + 274 877 906 944 + 137 438 953 472 + 17 179 869 184 + 8 589 934 592 + 2 147 483 648 + 1 073 741 824 + 536 870 912 + 134 217 728 + 33 554 432 + 4 194 304 + 2 097 152 + 1 048 576 + 262 144 + 131 072 + 65 536 + 32 768 + 16 384 + 8 192 + 1 024 + 512 + 256 + 128 + 2 + 1)(10) =


2 753 069 380 528 003(10)

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

0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011(2) = 2 753 069 380 528 003(10)

Number 0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011(2) converted from signed binary to an integer in decimal system (in base 10):
0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011(2) = 2 753 069 380 528 003(10)

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


More operations of this kind:

0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0010 = ?

0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0100 = ?


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)

0000 0000 0000 1001 1100 0111 1110 0110 1110 1010 0111 0111 1110 0111 1000 0011 = 2,753,069,380,528,003 Jul 24 11:55 UTC (GMT)
0000 0000 1110 1110 1111 1110 1000 0101 = 15,662,725 Jul 24 11:55 UTC (GMT)
0000 0000 0000 0000 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0010 = 35,184,372,088,818 Jul 24 11:55 UTC (GMT)
1011 0011 0001 0110 = -13,078 Jul 24 11:55 UTC (GMT)
1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0100 1101 = -9,223,372,036,854,775,629 Jul 24 11:55 UTC (GMT)
1011 1110 0100 0110 = -15,942 Jul 24 11:55 UTC (GMT)
0000 1000 1010 0100 = 2,212 Jul 24 11:55 UTC (GMT)
1000 0000 0000 0000 1010 1101 1100 1101 = -44,493 Jul 24 11:55 UTC (GMT)
1101 1001 = -89 Jul 24 11:54 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 1110 0111 1000 1100 0111 0010 0010 1010 0111 = 62,155,858,599 Jul 24 11:54 UTC (GMT)
1111 1010 0100 1101 = -31,309 Jul 24 11:54 UTC (GMT)
0000 0000 0000 0001 0110 0011 1000 0010 = 91,010 Jul 24 11:54 UTC (GMT)
1001 0001 1100 0111 = -4,551 Jul 24 11:54 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