Convert the following binary numbers to decimals.

a) 110011

b) 101010

c) 10000

d) 11001100

e) 100000

f) 111000

g) 100100

h) 10001

To convert binary numbers to decimal, we use the positional value of each digit in the binary system, which is base 2. Each digit represents a power of 2, starting from the rightmost digit, which is 2^0. Let's break down the conversion process for each of the binary numbers you've provided.

Understanding Binary to Decimal Conversion

In binary, each digit (bit) can either be a 0 or a 1. The position of each bit determines its value. For example, in the binary number 110011:

• The rightmost bit is 1 (2^0 = 1)
• The next bit is 1 (2^1 = 2)
• The next bit is 0 (2^2 = 0)
• The next bit is 0 (2^3 = 0)
• The next bit is 1 (2^4 = 16)
• The leftmost bit is 1 (2^5 = 32)

To find the decimal value, we sum the values of the bits that are set to 1:

Conversions

Now, let's convert each of your binary numbers to decimal:

a) 110011

Calculating the value:

• 1 × 2^5 = 32
• 1 × 2^4 = 16
• 0 × 2^3 = 0
• 0 × 2^2 = 0
• 1 × 2^1 = 2
• 1 × 2^0 = 1

Sum: 32 + 16 + 0 + 0 + 2 + 1 = 51

b) 101010

Calculating the value:

• 1 × 2^5 = 32
• 0 × 2^4 = 0
• 1 × 2^3 = 8
• 0 × 2^2 = 0
• 1 × 2^1 = 2
• 0 × 2^0 = 0

Sum: 32 + 0 + 8 + 0 + 2 + 0 = 42

c) 10000

Calculating the value:

• 1 × 2^4 = 16
• 0 × 2^3 = 0
• 0 × 2^2 = 0
• 0 × 2^1 = 0
• 0 × 2^0 = 0

Sum: 16 + 0 + 0 + 0 + 0 = 16

d) 11001100

Calculating the value:

• 1 × 2^7 = 128
• 1 × 2^6 = 64
• 0 × 2^5 = 0
• 0 × 2^4 = 0
• 1 × 2^3 = 8
• 1 × 2^2 = 4
• 0 × 2^1 = 0
• 0 × 2^0 = 0

Sum: 128 + 64 + 0 + 0 + 8 + 4 + 0 + 0 = 196

e) 100000

Calculating the value:

• 1 × 2^5 = 32
• 0 × 2^4 = 0
• 0 × 2^3 = 0
• 0 × 2^2 = 0
• 0 × 2^1 = 0
• 0 × 2^0 = 0

Sum: 32 + 0 + 0 + 0 + 0 + 0 = 32

f) 111000

Calculating the value:

• 1 × 2^5 = 32
• 1 × 2^4 = 16
• 1 × 2^3 = 8
• 0 × 2^2 = 0
• 0 × 2^1 = 0
• 0 × 2^0 = 0

Sum: 32 + 16 + 8 + 0 + 0 + 0 = 56

g) 100100

Calculating the value:

• 1 × 2^5 = 32
• 0 × 2^4 = 0
• 0 × 2^3 = 0
• 1 × 2^2 = 4
• 0 × 2^1 = 0
• 0 × 2^0 = 0

Sum: 32 + 0 + 0 + 4 + 0 + 0 = 36

h) 10001

Calculating the value:

• 1 × 2^4 = 16
• 0 × 2^3 = 0
• 0 × 2^2 = 0
• 0 × 2^1 = 0
• 1 × 2^0 = 1

Sum: 16 + 0 + 0 +