Use your knowledge of number systems to solve the problem given below (Please show all intermediate steps)

Use your knowledge of number systems to solve the problem given below (Please show all intermediate steps):

1. Convert following Binary Numbers into Decimal
   (i) (11010), = ( 10
   (ii) (110101.11) = ( 10

The Correct Answer and Explanation is :

Let’s break down the conversion process for both binary numbers step-by-step:

Part (i): Convert binary number ( (11010)_2 ) to decimal:

Binary is a base-2 number system, where each digit represents a power of 2. The rightmost digit represents ( 2^0 ), the next represents ( 2^1 ), and so on.

The binary number is: ( (11010)_2 ).

1. Write the binary digits and their corresponding powers of 2:

[
\begin{array}{c|c}
\text{Binary Digit} & \text{Power of 2} \
\hline
1 & 2^4 = 16 \
1 & 2^3 = 8 \
0 & 2^2 = 4 \
1 & 2^1 = 2 \
0 & 2^0 = 1 \
\end{array}
]

2. Now, multiply each binary digit by its corresponding power of 2:

[
1 \times 16 = 16
]
[
1 \times 8 = 8
]
[
0 \times 4 = 0
]
[
1 \times 2 = 2
]
[
0 \times 1 = 0
]

3. Add the results together:

[
16 + 8 + 0 + 2 + 0 = 26
]

Thus, ( (11010)2 = (26){10} ).

---

Part (ii): Convert binary number ( (110101.11)_2 ) to decimal:

In this binary number, there is both an integer part and a fractional part. We will treat the integer and fractional parts separately.

Integer Part Conversion:

The integer part is ( 110101_2 ). We apply the same method as in Part (i):

1. Write the binary digits and their corresponding powers of 2:

[
\begin{array}{c|c}
\text{Binary Digit} & \text{Power of 2} \
\hline
1 & 2^5 = 32 \
1 & 2^4 = 16 \
0 & 2^3 = 8 \
1 & 2^2 = 4 \
0 & 2^1 = 2 \
1 & 2^0 = 1 \
\end{array}
]

2. Multiply each binary digit by its corresponding power of 2:

[
1 \times 32 = 32
]
[
1 \times 16 = 16
]
[
0 \times 8 = 0
]
[
1 \times 4 = 4
]
[
0 \times 2 = 0
]
[
1 \times 1 = 1
]

3. Add the results together:

[
32 + 16 + 0 + 4 + 0 + 1 = 53
]

Thus, the integer part ( (110101)2 = (53){10} ).

Fractional Part Conversion:

Now, the fractional part is ( 11_2 ) (which is after the decimal point). For binary fractions, each digit after the decimal point represents a negative power of 2.

1. Write the binary digits and their corresponding powers of 2:

[
\begin{array}{c|c}
\text{Binary Digit} & \text{Power of 2} \
\hline
1 & 2^{-1} = 0.5 \
1 & 2^{-2} = 0.25 \
\end{array}
]

2. Multiply each binary digit by its corresponding negative power of 2:

[
1 \times 0.5 = 0.5
]
[
1 \times 0.25 = 0.25
]

3. Add the results together:

[
0.5 + 0.25 = 0.75
]

Thus, the fractional part ( (11)2 = (0.75){10} ).

---

Final Result:

Now combine the integer and fractional parts:

[
(110101.11)_2 = 53 + 0.75 = 53.75
]

Thus, ( (110101.11)2 = (53.75){10} ).

---

Explanation:

The binary system is a base-2 number system, where each digit is either a 0 or a 1. Each position represents a power of 2. By converting binary to decimal, we multiply each binary digit by its corresponding power of 2, then sum the results.

For whole binary numbers, we sum the products of each digit and its corresponding power of 2. For fractional binary numbers, we sum the products of each digit and the negative powers of 2. This method allows us to convert binary numbers into the decimal (base-10) system, which is commonly used in everyday life.