Two's Complement: Binary -> Integer: 0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0000 Signed Binary Number in Two's Complement Representation, Converted and Written as a Decimal System Integer (in Base Ten)

Signed binary in two's complement representation 0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0000(2) converted to an integer in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

In a signed binary in two's complement representation, the first bit (the leftmost) indicates the sign, 1 = negative, 0 = positive.

0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0000 is the binary representation of a positive integer, on 64 bits (8 Bytes).


2. Get the binary representation in one's complement.

* Run this step only if the number is negative *

Note on binary subtraction rules:

11 - 1 = 10; 10 - 1 = 1; 1 - 0 = 1; 1 - 1 = 0.

Subtract 1 from the initial binary number.

* Not the case - the number is positive *


3. Get the binary representation of the positive (unsigned) number.

* Run this step only if the number is negative *

Flip all the bits of the signed binary in one's complement representation (reverse the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:

* Not the case - the number is positive *


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

    • 263

      0

    • 262

      0

    • 261

      0

    • 260

      0

    • 259

      0

    • 258

      0

    • 257

      0

    • 256

      0

    • 255

      1

    • 254

      0

    • 253

      0

    • 252

      0

    • 251

      0

    • 250

      1

    • 249

      0

    • 248

      0

    • 247

      1

    • 246

      0

    • 245

      1

    • 244

      0

    • 243

      1

    • 242

      1

    • 241

      1

    • 240

      0

    • 239

      1

    • 238

      0

    • 237

      0

    • 236

      1

    • 235

      0

    • 234

      1

    • 233

      0

    • 232

      0

    • 231

      1

    • 230

      0

    • 229

      1

    • 228

      0

    • 227

      0

    • 226

      1

    • 225

      1

    • 224

      0

    • 223

      1

    • 222

      0

    • 221

      0

    • 220

      0

    • 219

      1

    • 218

      1

    • 217

      0

    • 216

      0

    • 215

      0

    • 214

      1

    • 213

      1

    • 212

      1

    • 211

      0

    • 210

      0

    • 29

      1

    • 28

      0

    • 27

      1

    • 26

      0

    • 25

      0

    • 24

      1

    • 23

      0

    • 22

      0

    • 21

      0

    • 20

      0

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

0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0000(2) =

(0 × 263 + 0 × 262 + 0 × 261 + 0 × 260 + 0 × 259 + 0 × 258 + 0 × 257 + 0 × 256 + 1 × 255 + 0 × 254 + 0 × 253 + 0 × 252 + 0 × 251 + 1 × 250 + 0 × 249 + 0 × 248 + 1 × 247 + 0 × 246 + 1 × 245 + 0 × 244 + 1 × 243 + 1 × 242 + 1 × 241 + 0 × 240 + 1 × 239 + 0 × 238 + 0 × 237 + 1 × 236 + 0 × 235 + 1 × 234 + 0 × 233 + 0 × 232 + 1 × 231 + 0 × 230 + 1 × 229 + 0 × 228 + 0 × 227 + 1 × 226 + 1 × 225 + 0 × 224 + 1 × 223 + 0 × 222 + 0 × 221 + 0 × 220 + 1 × 219 + 1 × 218 + 0 × 217 + 0 × 216 + 0 × 215 + 1 × 214 + 1 × 213 + 1 × 212 + 0 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =

(0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 36 028 797 018 963 968 + 0 + 0 + 0 + 0 + 1 125 899 906 842 624 + 0 + 0 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 0 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 0 + 549 755 813 888 + 0 + 0 + 68 719 476 736 + 0 + 17 179 869 184 + 0 + 0 + 2 147 483 648 + 0 + 536 870 912 + 0 + 0 + 67 108 864 + 33 554 432 + 0 + 8 388 608 + 0 + 0 + 0 + 524 288 + 262 144 + 0 + 0 + 0 + 16 384 + 8 192 + 4 096 + 0 + 0 + 512 + 0 + 128 + 0 + 0 + 16 + 0 + 0 + 0 + 0)(10) =

(36 028 797 018 963 968 + 1 125 899 906 842 624 + 140 737 488 355 328 + 35 184 372 088 832 + 8 796 093 022 208 + 4 398 046 511 104 + 2 199 023 255 552 + 549 755 813 888 + 68 719 476 736 + 17 179 869 184 + 2 147 483 648 + 536 870 912 + 67 108 864 + 33 554 432 + 8 388 608 + 524 288 + 262 144 + 16 384 + 8 192 + 4 096 + 512 + 128 + 16)(10) =

37 346 650 398 421 648(10)

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

0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0000(2) = 37 346 650 398 421 648(10)

The signed binary number in two's complement representation 0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0000(2) converted and written as an integer in decimal system (base ten):
0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0000(2) = 37 346 650 398 421 648(10)

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

0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1000 1111 The signed binary in two's complement representation converted and written as an integer number in decimal system (in base ten) = ?

0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0001 The signed binary in two's complement representation converted and written as an integer number in decimal system (in base ten) = ?

Convert signed binary numbers in two's complement representation to decimal system (base ten) integers

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 in two's complement representation to an integer in base ten:

1) In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive.

2) Get the signed binary representation in one's complement, subtract 1 from the initial number.

3) Construct the unsigned binary number: flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s.

4) Multiply each bit of the binary number by its corresponding power of 2 that its place value represents.

5) Add all the terms up to get the positive integer number in base ten.

6) Adjust the sign of the integer number by the first bit of the initial binary number.

The latest binary numbers written in two\'s complement representation converted to signed integers written in decimal system (in base ten)

Convert the signed binary number written in two's complement representation 0000 0000 1000 0100 1010 1110 1001 0100 1010 0110 1000 1100 0111 0010 1001 0000, write it as a decimal system (base ten) integer Nov 30 17:39 UTC (GMT)
Convert the signed binary number written in two's complement representation 1010 1011 0001 0010 0011 0100 0101 0000, write it as a decimal system (base ten) integer Nov 30 17:39 UTC (GMT)
Convert the signed binary number written in two's complement representation 0000 0000 0000 0000 0000 0000 0000 0001 1110 1110 1110 1110 1110 1111 1111 1011, write it as a decimal system (base ten) integer Nov 30 17:39 UTC (GMT)
Convert the signed binary number written in two's complement representation 1111 1111 1111 1101 0000 1001 0011 0001, write it as a decimal system (base ten) integer Nov 30 17:39 UTC (GMT)
Convert the signed binary number written in two's complement representation 1111 1011 0110 0000, write it as a decimal system (base ten) integer Nov 30 17:39 UTC (GMT)
Convert the signed binary number written in two's complement representation 0000 0000 1111 1111 1111 1001 1111 1111, write it as a decimal system (base ten) integer Nov 30 17:38 UTC (GMT)
Convert the signed binary number written in two's complement representation 0000 1010 1001 1010 1001 0101 0111 0010, write it as a decimal system (base ten) integer Nov 30 17:38 UTC (GMT)
Convert the signed binary number written in two's complement representation 1111 1111 1100 0010 1111 0111 0100 0101, write it as a decimal system (base ten) integer Nov 30 17:38 UTC (GMT)
Convert the signed binary number written in two's complement representation 0101 1001 1001 1001 0110 0110 1010 1010 0101 1010 1010 1010 0110 0101 1101 0100, write it as a decimal system (base ten) integer Nov 30 17:38 UTC (GMT)
Convert the signed binary number written in two's complement representation 1101 1100 0111 0100, write it as a decimal system (base ten) integer Nov 30 17:38 UTC (GMT)
All the signed binary numbers written in two's complement representation converted to decimal system (base ten) integers

How to convert signed binary numbers in two's complement representation from binary system to decimal

To understand how to convert a signed binary number in two's complement representation from the binary system to decimal (base ten), the easiest way is to do it by an example - convert binary, 1101 1110, to base ten:

  • In a signed binary two's complement, first bit (leftmost) indicates the sign, 1 = negative, 0 = positive. The first bit is 1, so our number is negative.
  • Get the signed binary representation in one's complement, subtract 1 from the initial number:
    1101 1110 - 1 = 1101 1101
  • Get the binary representation of the positive number, flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:
    !(1101 1101) = 0010 0010
  • Write bellow the positive binary number representation 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, increasing each corresonding power of 2 by exactly one unit:
  • powers of 2: 7 6 5 4 3 2 1 0
    digits: 0 0 1 0 0 0 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:

    0010 0010(2) =

    (0 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20)(10) =

    (0 + 0 + 32 + 0 + 0 + 0 + 2 + 0)(10) =

    (32 + 2)(10) =

    34(10)

  • Signed binary number in two's complement representation, 1101 1110 = -34(10), a signed negative integer in base 10.