Signed binary number 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011 converted to an integer in base ten

Signed binary 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 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.

0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 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:

0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011 = 011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011

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

      0
    • 252

      0
    • 251

      0
    • 250

      1
    • 249

      0
    • 248

      1
    • 247

      0
    • 246

      0
    • 245

      0
    • 244

      1
    • 243

      1
    • 242

      0
    • 241

      0
    • 240

      1
    • 239

      1
    • 238

      0
    • 237

      0
    • 236

      1
    • 235

      1
    • 234

      0
    • 233

      0
    • 232

      1
    • 231

      1
    • 230

      0
    • 229

      0
    • 228

      1
    • 227

      1
    • 226

      0
    • 225

      0
    • 224

      1
    • 223

      1
    • 222

      0
    • 221

      0
    • 220

      1
    • 219

      1
    • 218

      0
    • 217

      0
    • 216

      1
    • 215

      1
    • 214

      0
    • 213

      0
    • 212

      1
    • 211

      1
    • 210

      0
    • 29

      0
    • 28

      1
    • 27

      1
    • 26

      0
    • 25

      0
    • 24

      1
    • 23

      0
    • 22

      0
    • 21

      1
    • 20

      1

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

011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011(2) =


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


(0 + 2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 125 899 906 842 624 + 0 + 281 474 976 710 656 + 0 + 0 + 0 + 17 592 186 044 416 + 8 796 093 022 208 + 0 + 0 + 1 099 511 627 776 + 549 755 813 888 + 0 + 0 + 68 719 476 736 + 34 359 738 368 + 0 + 0 + 4 294 967 296 + 2 147 483 648 + 0 + 0 + 268 435 456 + 134 217 728 + 0 + 0 + 16 777 216 + 8 388 608 + 0 + 0 + 1 048 576 + 524 288 + 0 + 0 + 65 536 + 32 768 + 0 + 0 + 4 096 + 2 048 + 0 + 0 + 256 + 128 + 0 + 0 + 16 + 0 + 0 + 2 + 1)(10) =


(2 305 843 009 213 693 952 + 1 152 921 504 606 846 976 + 1 125 899 906 842 624 + 281 474 976 710 656 + 17 592 186 044 416 + 8 796 093 022 208 + 1 099 511 627 776 + 549 755 813 888 + 68 719 476 736 + 34 359 738 368 + 4 294 967 296 + 2 147 483 648 + 268 435 456 + 134 217 728 + 16 777 216 + 8 388 608 + 1 048 576 + 524 288 + 65 536 + 32 768 + 4 096 + 2 048 + 256 + 128 + 16 + 2 + 1)(10) =


3 460 200 036 201 765 267(10)

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

0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011(2) = 3 460 200 036 201 765 267(10)

Number 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011(2) converted from signed binary to an integer in decimal system (in base 10):
0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011(2) = 3 460 200 036 201 765 267(10)

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


More operations of this kind:

0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0010 = ?

0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 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)

0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011 = 3,460,200,036,201,765,267 Dec 02 23:03 UTC (GMT)
1000 0101 1000 0101 = -1,413 Dec 02 22:58 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 0001 1111 1111 1111 1111 1111 1111 1111 1111 = 8,589,934,591 Dec 02 22:58 UTC (GMT)
0000 0000 0000 0001 0101 0100 1001 0001 = 87,185 Dec 02 22:57 UTC (GMT)
0100 1100 = 76 Dec 02 22:56 UTC (GMT)
0001 0010 1001 1000 = 4,760 Dec 02 22:56 UTC (GMT)
1000 0000 0000 1111 0000 1111 0100 1000 = -986,952 Dec 02 22:56 UTC (GMT)
1111 0010 = -114 Dec 02 22:56 UTC (GMT)
1011 0001 1111 0001 = -12,785 Dec 02 22:56 UTC (GMT)
1100 1111 1000 0110 = -20,358 Dec 02 22:55 UTC (GMT)
0000 0000 0000 0000 0010 1000 0001 0101 = 10,261 Dec 02 22:55 UTC (GMT)
0101 0111 0001 1110 = 22,302 Dec 02 22:54 UTC (GMT)
0001 1001 0010 0110 = 6,438 Dec 02 22:54 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