Part 1: Convert the binary number $(1101.011)_2$ to decimal

In a binary number, each digit represents a power of 2, with the decimal point separating the positive and negative powers. Let's break down the conversion:

Step 1: Convert the integer part $1101_2$

The binary number $1101_2$ is converted to decimal by multiplying each bit by $2^n$, where $n$ is the position of the bit from the right (starting at 0):

$1101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$

This simplifies to:

$= 1 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1 = 8 + 4 + 0 + 1 = 13$

Step 2: Convert the fractional part $.011_2$

The fractional part is converted by multiplying each bit by $2^{-n}$, where $n$ is the position of the bit from the left of the decimal point (starting at 1):

$.011_2 = 0 \times 2^{-1} + 1 \times 2^{-2} + 1 \times 2^{-3}$

This simplifies to:

$= 0 + \frac{1}{4} + \frac{1}{8} = \frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8} = 0.375$

Step 3: Combine integer and fractional parts

The decimal value is:

$1101.011_2 = 13 + 0.375 = 13.375_{10}$

So, $(1101.011)_2$ in decimal is $13.375$.

---

Part 2: Perform the binary multiplication $(110)_2 \times (101)_2$

Let's multiply the two binary numbers step by step.

Step 1: Write the binary numbers

$110_2 = 6_{10}, \quad 101_2 = 5_{10}$

We will perform the multiplication manually in binary:

$\ \ \ \ \ 110 \ \ (\text{This is 6 in binary})$ $\times \ \ 101 \ \ (\text{This is 5 in binary})$

Step 2: Multiply each bit

• Multiply $110 \times 1$ (shift by 0 positions): $110$

• Multiply $110 \times 0$ (shift by 1 position): $000$

• Multiply $110 \times 1$ (shift by 2 positions): $11000$

Step 3: Add the results together:

$110 + 0000 + 11000 = 100110_2$

Thus, $(110)_2 \times (101)_2 = 100110_2$.

Step 4: Convert the result to decimal

$100110_2 = 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0$ $= 32 + 0 + 0 + 4 + 2 + 0 = 38$

Thus, $100110_2 = 38_{10}$.

---

Final Results:

1. $(1101.011)_2 = 13.375_{10}$
2. $(110)_2 \times (101)_2 = 100110_2 = 38_{10}$

Would you like more details on any step, or do you have any questions?

Related Questions:

1. How do you convert decimal numbers to binary?
2. What is the process of dividing two binary numbers?
3. How does binary addition differ from decimal addition?
4. How would you convert a binary number to hexadecimal?
5. How are fractional binary numbers handled in multiplication?

Tip:

When multiplying binary numbers, think of it like long multiplication in decimal, with shifts instead of carrying.