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

Signed binary 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 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.

0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 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:

0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 = 000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000

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

      0
    • 250

      0
    • 249

      0
    • 248

      0
    • 247

      0
    • 246

      0
    • 245

      0
    • 244

      0
    • 243

      0
    • 242

      0
    • 241

      0
    • 240

      0
    • 239

      0
    • 238

      0
    • 237

      0
    • 236

      0
    • 235

      0
    • 234

      0
    • 233

      0
    • 232

      1
    • 231

      0
    • 230

      0
    • 229

      0
    • 228

      0
    • 227

      0
    • 226

      0
    • 225

      0
    • 224

      0
    • 223

      0
    • 222

      0
    • 221

      0
    • 220

      0
    • 219

      0
    • 218

      0
    • 217

      0
    • 216

      0
    • 215

      0
    • 214

      0
    • 213

      0
    • 212

      0
    • 211

      0
    • 210

      0
    • 29

      0
    • 28

      0
    • 27

      0
    • 26

      0
    • 25

      0
    • 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:

000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000(2) =


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


(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 4 294 967 296 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0)(10) =


4 294 967 296(10)

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

0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000(2) = 4 294 967 296(10)

Number 0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000(2) converted from signed binary to an integer in decimal system (in base 10):
0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000(2) = 4 294 967 296(10)

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


More operations of this kind:

0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 = ?

0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 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)

0000 0000 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 = 4,294,967,296 May 18 00:55 UTC (GMT)
1011 0001 1111 1010 = -12,794 May 18 00:55 UTC (GMT)
0000 0000 0000 0001 0000 1101 0011 1111 = 68,927 May 18 00:55 UTC (GMT)
0000 0000 0001 1000 1110 0000 0110 1000 = 1,630,312 May 18 00:55 UTC (GMT)
0000 0000 0110 1001 1111 1101 1001 0001 = 6,946,193 May 18 00:55 UTC (GMT)
1111 1111 0100 0000 0001 0111 1111 1010 = -2,134,906,874 May 18 00:55 UTC (GMT)
1000 0000 0110 0110 1111 1111 1111 0001 = -6,750,193 May 18 00:54 UTC (GMT)
1011 0011 1001 0001 1101 1110 0000 1110 = -865,197,582 May 18 00:54 UTC (GMT)
0000 0000 0000 0001 1110 0010 0110 0110 = 123,494 May 18 00:54 UTC (GMT)
1101 1100 1100 1111 = -23,759 May 18 00:54 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 0000 0010 0010 0001 0001 0001 0111 0111 1011 = 571,545,467 May 18 00:54 UTC (GMT)
0100 0011 0010 1011 1010 0000 0000 0101 = 1,126,932,485 May 18 00:53 UTC (GMT)
0111 1101 1111 0001 = 32,241 May 18 00:53 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