Question

The addition of two binary quantities \((10011)_2\) and \((1111)_2\) will be

• A. 100010
• B. 110010
• C. 100011
• D. 101011

Hint:

To verify binary arithmetic quickly, convert to decimal, add the numbers, and convert the result back to binary. This double-check takes only a few seconds and prevents carrying errors.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept

Binary addition is the process of adding numbers in the base-\(2\) number system. It follows rules similar to decimal addition, but a carry is generated whenever the sum is \(2\) or more.

Basic Rules of Binary Addition:

• \(0+0=0\)
• \(0+1=1\)
• \(1+0=1\)
• \(1+1=10_2\), which means write \(0\) and carry \(1\)
• \(1+1+1=11_2\), which means write \(1\) and carry \(1\)

Step 2: Detailed Explanation

We need to add

\[ (10011)_2+(1111)_2. \]

Method 1: Direct Binary Addition

First, align both binary numbers from the right:

\[ \begin{array}{r} \phantom{+}10011\\ +01111\\ \hline 100010 \end{array} \]

Let us add the digits column by column, starting from the right.

• First column: \[ 1+1=10_2 \] Write \(0\) and carry \(1\).
• Second column: \[ 1+1+1=11_2 \] Write \(1\) and carry \(1\).
• Third column: \[ 0+1+1=10_2 \] Write \(0\) and carry \(1\).
• Fourth column: \[ 0+1+1=10_2 \] Write \(0\) and carry \(1\).
• Fifth column: \[ 1+0+1=10_2 \] Write \(0\) and carry \(1\).
• Final carry: The remaining carry \(1\) is written in the leftmost position.

Therefore,

\[ (10011)_2+(1111)_2=(100010)_2. \]

Method 2: Verification Using Decimal Conversion

Convert \((10011)_2\) into decimal:

\[ \begin{aligned} (10011)_2 &= 1\cdot2^4+0\cdot2^3+0\cdot2^2+1\cdot2^1+1\cdot2^0\\ &=16+0+0+2+1\\ &=19_{10}. \end{aligned} \]

Convert \((1111)_2\) into decimal:

\[ \begin{aligned} (1111)_2 &= 1\cdot2^3+1\cdot2^2+1\cdot2^1+1\cdot2^0\\ &=8+4+2+1\\ &=15_{10}. \end{aligned} \]

Now add the decimal values:

\[ 19+15=34. \]

Convert \(34\) back into binary:

\[ 34=32+2=2^5+2^1. \]

Therefore,

\[ 34_{10}=(100010)_2. \]

Both methods give the same result.

Step 3: Final Answer

\[ \boxed{(10011)_2+(1111)_2=(100010)_2} \]

Hence, the correct answer is Option (A).