Signed: Binary -> Integer: 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

The signed binary (in base two) 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011(2) to an integer (with sign) in decimal system (in base ten) = ?

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.

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)

The number 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011(2) converted from a signed binary (base two) and written as an integer in decimal system (base ten):
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 were used to group digits: for binary, by 4, for decimal, by 3.

The signed binary number 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0010 (in base two), converted and written as an integer in decimal system (in base ten) = ?

The signed binary number 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0100 (in base two), converted and written as an integer in decimal system (in base ten) = ?

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

The first bit (the leftmost) is reserved for the sign (1 = negative, 0 = positive) and it doesn't count when calculating the absolute value.

Binary number's length must be: 2, 4, 8, 16, 32, 64 - or else extra bits on 0 are 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.

The latest signed binary numbers converted and written as signed integers in decimal system (in base ten)

Convert the signed binary number 0011 0000 0000 0101 0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 0011, write it as a decimal system integer number (written in base ten) Nov 30 17:41 UTC (GMT)
Convert the signed binary number 1111 1111 1011 1100 0011 1011 0000 0010, write it as a decimal system integer number (written in base ten) Nov 30 17:41 UTC (GMT)
Convert the signed binary number 0000 1100 0000 1010 0010 0011 1100 0100, write it as a decimal system integer number (written in base ten) Nov 30 17:41 UTC (GMT)
Convert the signed binary number 1010 1100 0000 0000, write it as a decimal system integer number (written in base ten) Nov 30 17:41 UTC (GMT)
Convert the signed binary number 1000 0101 0100 0001 0000 0000 0011 0101, write it as a decimal system integer number (written in base ten) Nov 30 17:40 UTC (GMT)
Convert the signed binary number 0110 1011, write it as a decimal system integer number (written in base ten) Nov 30 17:40 UTC (GMT)
Convert the signed binary number 1100 0000 1000 1111 1111 1111 1100 0101, write it as a decimal system integer number (written in base ten) Nov 30 17:40 UTC (GMT)
Convert the signed binary number 0000 1101 1111 1111 1111 1111 1110 0001, write it as a decimal system integer number (written in base ten) Nov 30 17:40 UTC (GMT)
Convert the signed binary number 1000 0001 0001 0000 0000 0000 0011 1101, write it as a decimal system integer number (written in base ten) Nov 30 17:40 UTC (GMT)
Convert the signed binary number 0001 0110 0000 0000 0000 0000 0011 0100, write it as a decimal system integer number (written in base ten) Nov 30 17:40 UTC (GMT)
All the signed binary numbers converted to integers in decimal system (written 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