Signed binary number 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000 converted to an integer in base ten

Signed binary 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000(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.

0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000 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:

0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000 = 011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000

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

    • 262

      0
    • 261

      1
    • 260

      1
    • 259

      0
    • 258

      0
    • 257

      0
    • 256

      0
    • 255

      0
    • 254

      0
    • 253

      1
    • 252

      1
    • 251

      0
    • 250

      0
    • 249

      0
    • 248

      0
    • 247

      0
    • 246

      0
    • 245

      1
    • 244

      1
    • 243

      0
    • 242

      0
    • 241

      0
    • 240

      0
    • 239

      0
    • 238

      0
    • 237

      1
    • 236

      1
    • 235

      0
    • 234

      0
    • 233

      0
    • 232

      0
    • 231

      0
    • 230

      0
    • 229

      1
    • 228

      1
    • 227

      0
    • 226

      0
    • 225

      0
    • 224

      0
    • 223

      0
    • 222

      0
    • 221

      1
    • 220

      1
    • 219

      0
    • 218

      0
    • 217

      0
    • 216

      0
    • 215

      0
    • 214

      0
    • 213

      1
    • 212

      1
    • 211

      0
    • 210

      0
    • 29

      0
    • 28

      0
    • 27

      0
    • 26

      0
    • 25

      1
    • 24

      0
    • 23

      0
    • 22

      0
    • 21

      0
    • 20

      0

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

011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000(2) =


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


(0 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 0 + 0 + 0 + 0 + 0 + 0 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 0 + 0 + 0 + 0 + 0 + 0 + 35 184 372 088 832 + 17 592 186 044 416 + 0 + 0 + 0 + 0 + 0 + 0 + 137 438 953 472 + 68 719 476 736 + 0 + 0 + 0 + 0 + 0 + 0 + 536 870 912 + 268 435 456 + 0 + 0 + 0 + 0 + 0 + 0 + 2 097 152 + 1 048 576 + 0 + 0 + 0 + 0 + 0 + 0 + 8 192 + 4 096 + 0 + 0 + 0 + 0 + 0 + 0 + 32 + 0 + 0 + 0 + 0 + 0)(10) =


(2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 9 007 199 254 740 992 + 4 503 599 627 370 496 + 35 184 372 088 832 + 17 592 186 044 416 + 137 438 953 472 + 68 719 476 736 + 536 870 912 + 268 435 456 + 2 097 152 + 1 048 576 + 8 192 + 4 096 + 32)(10) =


3 472 328 296 227 680 288(10)

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

0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000(2) = 3 472 328 296 227 680 288(10)

Number 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000(2) converted from signed binary to an integer in decimal system (in base 10):
0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000(2) = 3 472 328 296 227 680 288(10)

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


More operations of this kind:

0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0001 1111 = ?

0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0001 = ?


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)

0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0011 0000 0010 0000 = 3,472,328,296,227,680,288 Dec 02 21:57 UTC (GMT)
1111 1111 = -127 Dec 02 21:56 UTC (GMT)
0000 0000 1111 1111 1001 1100 0110 1100 = 16,751,724 Dec 02 21:55 UTC (GMT)
0000 0000 0000 1000 1000 1000 0111 0111 = 559,223 Dec 02 21:55 UTC (GMT)
0100 1000 0000 1000 = 18,440 Dec 02 21:55 UTC (GMT)
1111 1111 1111 1111 = -32,767 Dec 02 21:51 UTC (GMT)
0000 0000 0000 0001 1111 1010 0111 1100 = 129,660 Dec 02 21:49 UTC (GMT)
1111 1111 = -127 Dec 02 21:49 UTC (GMT)
1010 1010 = -42 Dec 02 21:48 UTC (GMT)
0000 0000 0000 0000 1000 1110 1000 1011 = 36,491 Dec 02 21:48 UTC (GMT)
0000 0000 0000 0000 0111 1111 1111 1111 1111 1111 1111 1111 1110 0101 1110 1001 = 140,737,488,348,649 Dec 02 21:47 UTC (GMT)
0001 1111 = 31 Dec 02 21:47 UTC (GMT)
1000 1000 1001 0111 = -2,199 Dec 02 21:47 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