Two's Complement: Binary ↘ Integer: 0000 1100 1101 1101 0011 1010 0101 1000 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 1100 1101 1101 0011 1010 0101 1000(2) converted to an integer in decimal system (in base ten) = ?

1. Is this a positive or a negative number?

0000 1100 1101 1101 0011 1010 0101 1000 is the binary representation of a positive integer, on 32 bits (4 Bytes).


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


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:

  • 231

    0
  • 230

    0
  • 229

    0
  • 228

    0
  • 227

    1
  • 226

    1
  • 225

    0
  • 224

    0
  • 223

    1
  • 222

    1
  • 221

    0
  • 220

    1
  • 219

    1
  • 218

    1
  • 217

    0
  • 216

    1
  • 215

    0
  • 214

    0
  • 213

    1
  • 212

    1
  • 211

    1
  • 210

    0
  • 29

    1
  • 28

    0
  • 27

    0
  • 26

    1
  • 25

    0
  • 24

    1
  • 23

    1
  • 22

    0
  • 21

    0
  • 20

    0

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

0000 1100 1101 1101 0011 1010 0101 1000(2) =


(0 × 231 + 0 × 230 + 0 × 229 + 0 × 228 + 1 × 227 + 1 × 226 + 0 × 225 + 0 × 224 + 1 × 223 + 1 × 222 + 0 × 221 + 1 × 220 + 1 × 219 + 1 × 218 + 0 × 217 + 1 × 216 + 0 × 215 + 0 × 214 + 1 × 213 + 1 × 212 + 1 × 211 + 0 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20)(10) =


(0 + 0 + 0 + 0 + 134 217 728 + 67 108 864 + 0 + 0 + 8 388 608 + 4 194 304 + 0 + 1 048 576 + 524 288 + 262 144 + 0 + 65 536 + 0 + 0 + 8 192 + 4 096 + 2 048 + 0 + 512 + 0 + 0 + 64 + 0 + 16 + 8 + 0 + 0 + 0)(10) =


(134 217 728 + 67 108 864 + 8 388 608 + 4 194 304 + 1 048 576 + 524 288 + 262 144 + 65 536 + 8 192 + 4 096 + 2 048 + 512 + 64 + 16 + 8)(10) =


215 824 984(10)

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

0000 1100 1101 1101 0011 1010 0101 1000(2) = 215 824 984(10)

The signed binary number in two's complement representation 0000 1100 1101 1101 0011 1010 0101 1000(2) converted and written as an integer in decimal system (base ten):
0000 1100 1101 1101 0011 1010 0101 1000(2) = 215 824 984(10)

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

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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.

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