Understanding Binary Numbers and Decimal Conversion

The question asks us to find the sum of two binary numbers, (1101101)$_2$ and (1011011)$_2$, and express the result in the decimal system (base 10).

There are two main ways to solve this problem:

1. Convert each binary number to decimal first, and then add the decimal numbers.
2. Add the binary numbers directly, and then convert the resulting binary sum to decimal.

Method 1: Convert to Decimal First

To convert a binary number to a decimal number, we multiply each digit by the power of 2 corresponding to its position, starting from the rightmost digit (position 0) and moving left.

Let's convert the first binary number, (1101101)$_2$, to decimal:

$ (1101101)_2 = 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 $

$ = 1 \times 64 + 1 \times 32 + 0 \times 16 + 1 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1 $

$ = 64 + 32 + 0 + 8 + 4 + 0 + 1 $

$ = (109)_{10} $

Now, let's convert the second binary number, (1011011)$_2$, to decimal:

$ (1011011)_2 = 1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 $

$ = 1 \times 64 + 0 \times 32 + 1 \times 16 + 1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 $

$ = 64 + 0 + 16 + 8 + 0 + 2 + 1 $

$ = (91)_{10} $

Now, add the two decimal numbers:

$ 109 + 91 = 200 $

So, the sum in decimal is (200)$_{10}$.

Method 2: Perform Binary Addition First

Binary addition follows these rules:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (which is 0 with a carry of 1 to the next position)
• 1 + 1 + 1 = 11 (which is 1 with a carry of 1 to the next position)

Let's add the two binary numbers (1101101)$_2$ and (1011011)$_2$:

Starting from the rightmost column:

• 1 + 1 = 10$_2$. Write down 0, carry over 1.
• 0 + 1 + carry 1 = 10$_2$. Write down 0, carry over 1.
• 1 + 0 + carry 1 = 10$_2$. Write down 0, carry over 1.
• 1 + 1 + carry 1 = 11$_2$. Write down 1, carry over 1.
• 0 + 1 + carry 1 = 10$_2$. Write down 0, carry over 1.
• 1 + 0 + carry 1 = 10$_2$. Write down 0, carry over 1.
• 1 + 1 + carry 1 = 11$_2$. Write down 1, carry over 1.
• The final carry is 1.

The resulting binary sum is (11001000)$_2$.

Now, convert the binary sum (11001000)$_2$ to decimal:

$ (11001000)_2 = 1 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 $

$ = 1 \times 128 + 1 \times 64 + 0 \times 32 + 0 \times 16 + 1 \times 8 + 0 \times 4 + 0 \times 2 + 0 \times 1 $

$ = 128 + 64 + 0 + 0 + 8 + 0 + 0 + 0 $

$ = (200)_{10} $

Final Result in Decimal System

Both methods show that the sum of (1101101)$_2$ and (1011011)$_2$ is (200)$_{10}$.

Revision Table: Binary and Decimal Conversion

Additional Information on Number Systems

Number systems are ways of representing numbers using a set of digits and a base (or radix). The base determines the number of unique digits available and the value of each digit based on its position.

• Decimal System (Base 10): This is the most common system we use daily. It has 10 digits (0-9) and place values are powers of 10.
• Binary System (Base 2): Used extensively in computers and digital electronics. It has only 2 digits (0 and 1) and place values are powers of 2.
• Octal System (Base 8): Uses 8 digits (0-7) and place values are powers of 8. Often used as a compact way to represent binary numbers.
• Hexadecimal System (Base 16): Uses 16 symbols (0-9 and A-F, where A=10, B=11, C=12, D=13, E=14, F=15) and place values are powers of 16. Commonly used in computing and digital electronics.

Understanding how to convert between different number systems and perform basic arithmetic operations like binary addition is fundamental in computer science and digital logic.