Signed binary number 0000 0000 0000 0000 0000 0000 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101 converted to an integer in base ten

Signed binary 0000 0000 0000 0000 0000 0000 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101(2) to an integer in decimal system (in base 10) = ?

1. Is this a positive or a negative number?

In a signed binary, the 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 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101 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 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101 = 000 0000 0000 0000 0000 0000 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101


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

      1

    • 236

      1

    • 235

      0

    • 234

      1

    • 233

      1

    • 232

      0

    • 231

      0

    • 230

      1

    • 229

      0

    • 228

      0

    • 227

      0

    • 226

      1

    • 225

      1

    • 224

      0

    • 223

      1

    • 222

      1

    • 221

      0

    • 220

      0

    • 219

      1

    • 218

      1

    • 217

      1

    • 216

      0

    • 215

      1

    • 214

      1

    • 213

      1

    • 212

      0

    • 211

      0

    • 210

      1

    • 29

      0

    • 28

      0

    • 27

      1

    • 26

      1

    • 25

      1

    • 24

      0

    • 23

      0

    • 22

      1

    • 21

      0

    • 20

      1

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

000 0000 0000 0000 0000 0000 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101(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 + 1 × 237 + 1 × 236 + 0 × 235 + 1 × 234 + 1 × 233 + 0 × 232 + 0 × 231 + 1 × 230 + 0 × 229 + 0 × 228 + 0 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 1 × 223 + 1 × 222 + 0 × 221 + 0 × 220 + 1 × 219 + 1 × 218 + 1 × 217 + 0 × 216 + 1 × 215 + 1 × 214 + 1 × 213 + 0 × 212 + 0 × 211 + 1 × 210 + 0 × 29 + 0 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 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 + 137 438 953 472 + 68 719 476 736 + 0 + 17 179 869 184 + 8 589 934 592 + 0 + 0 + 1 073 741 824 + 0 + 0 + 0 + 67 108 864 + 33 554 432 + 0 + 8 388 608 + 4 194 304 + 0 + 0 + 524 288 + 262 144 + 131 072 + 0 + 32 768 + 16 384 + 8 192 + 0 + 0 + 1 024 + 0 + 0 + 128 + 64 + 32 + 0 + 0 + 4 + 0 + 1)(10) =

(137 438 953 472 + 68 719 476 736 + 17 179 869 184 + 8 589 934 592 + 1 073 741 824 + 67 108 864 + 33 554 432 + 8 388 608 + 4 194 304 + 524 288 + 262 144 + 131 072 + 32 768 + 16 384 + 8 192 + 1 024 + 128 + 64 + 32 + 4 + 1)(10) =

233 116 198 117(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 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101(2) = 233 116 198 117(10)

Number 0000 0000 0000 0000 0000 0000 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101(2) converted from signed binary to an integer in decimal system (in base 10):
0000 0000 0000 0000 0000 0000 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101(2) = 233 116 198 117(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 0011 0110 0100 0110 1100 1110 1110 0100 1110 0100 converted from: signed binary, to signed integer = ?

0000 0000 0000 0000 0000 0000 0011 0110 0100 0110 1100 1110 1110 0100 1110 0110 converted from: signed binary, to signed integer = ?


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

The 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 0011 0110 0100 0110 1100 1110 1110 0100 1110 0101 converted from: signed binary, to signed integer = 233,116,198,117 May 29 15:34 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 1000 1111 0010 0111 0101 0100 1000 1001 0111 converted from: signed binary, to signed integer = 38,427,510,935 May 29 15:34 UTC (GMT)
0100 1011 converted from: signed binary, to signed integer = 75 May 29 15:32 UTC (GMT)
1000 0000 0000 0000 0000 0000 1110 1000 1101 0100 1010 0101 0001 0000 0000 0111 converted from: signed binary, to signed integer = -1,000,000,000,007 May 29 15:31 UTC (GMT)
0000 0000 1011 1111 1110 1010 0011 1100 converted from: signed binary, to signed integer = 12,577,340 May 29 15:31 UTC (GMT)
1111 1100 0111 0001 1011 1001 0001 0100 converted from: signed binary, to signed integer = -2,087,827,732 May 29 15:31 UTC (GMT)
0000 0000 0000 0001 0000 0011 1110 0101 1111 1000 0010 0010 1100 0101 1000 1100 converted from: signed binary, to signed integer = 285,761,222,133,132 May 29 15:30 UTC (GMT)
0000 0000 0000 0000 0000 0000 0000 1000 1011 0110 1001 0100 0100 1101 0011 0010 converted from: signed binary, to signed integer = 37,422,910,770 May 29 15:29 UTC (GMT)
0000 0000 0000 0000 0000 0001 0000 0011 1010 0001 1101 1110 0111 1001 1100 1000 converted from: signed binary, to signed integer = 1,115,112,241,608 May 29 15:29 UTC (GMT)
1110 0010 1001 0100 converted from: signed binary, to signed integer = -25,236 May 29 15:28 UTC (GMT)
0000 0001 0000 0000 converted from: signed binary, to signed integer = 256 May 29 15:27 UTC (GMT)
1111 1111 1111 0111 1111 1111 1011 1111 0000 1000 0000 1100 0001 0000 0110 1011 converted from: signed binary, to signed integer = -9,221,119,958,003,224,683 May 29 15:27 UTC (GMT)
0110 0101 0110 0111 converted from: signed binary, to signed integer = 25,959 May 29 15:27 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