Converter of Signed Binary Numbers in Two's complement Representation: Write Them as Decimal System Integer Numbers (in Base Ten). All the Steps Explained in Detail

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.

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₎ =

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

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

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  (32 + 2)₍₁₀₎ =

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  34₍₁₀₎

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

Conversions Between Decimal System Numbers (Written in Base Ten) and Binary System Numbers (Base Two and Computer Representation):

1. Integer -> Binary

• 1.1. Unsigned integer -> Unsigned binary

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  1.2. Signed integer -> Signed binary

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  1.3. Signed integer -> Signed binary one's complement

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  1.4. Signed integer -> Signed binary two's complement

2. Decimal -> Binary

• 2.1. Decimal -> 32bit single precision IEEE 754 binary floating point

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  2.2. Decimal -> 64bit double precision IEEE 754 binary floating point

3. Binary -> Integer

• 3.1. Unsigned binary -> Unsigned integer

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  3.2. Signed binary -> Signed integer

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  3.3. Signed binary one's complement -> Signed integer

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  3.4. Signed binary two's complement -> Signed integer

4. Binary -> Decimal

• 4.1. 32bit single precision IEEE 754 binary floating point -> Float

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  4.2. 64bit double precision IEEE 754 binary floating point -> Double