Signed Binary Number 1011 0101 0101 0110 Converted and Written as a Decimal System Integer Number (in Base Ten). All the Steps Explained in Detail

The signed binary number 1011 0101 0101 0101 (in base two), converted and written as an integer in decimal system (in base ten) = ?

The signed binary number 1011 0101 0101 0111 (in base two), converted and written as an integer in decimal system (in base ten) = ?

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.

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 =

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

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  - (0 + 0 + 16 + 8 + 4 + 2 + 0)₍₁₀₎ =

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  - (16 + 8 + 4 + 2)₍₁₀₎ =

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  -30₍₁₀₎

• Binary signed number, 1001 1110 = -30₍₁₀₎, 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