Binary Arithmetic Operations

Binary Addition

Binary addition is similar to decimal addition but uses only two digits (0 and 1):

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (which is 0 with a carry of 1)

Binary Addition Example:

\[ \begin{array}{cccc} & 1 & 0 & 1 & 1 \\ + & 1 & 1 & 0 & 1 \\ \hline 1 & 1 & 0 & 0 & 0 \\ \end{array} \]

Explanation:

1. Align the numbers by their least significant bits (rightmost bits).

2. Start from the rightmost bit and add each pair of bits, carrying over any values as needed.

3. Write down the result for each bit position.

4. If there is a carry left over at the end, write it down as the most significant bit.

Binary Subtraction

Binary subtraction involves borrowing when a larger bit is subtracted from a smaller bit:

0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (with borrowing from the next higher bit)

Borrowing: When borrowing, the next higher bit is reduced by 1 and the current bit is treated as 10 (binary).

Binary Subtraction Example:

\[ \begin{array}{cccc} & 1 & 1 & 0 & 1 \\ - & 1 & 0 & 1 & 0 \\ \hline & 0 & 0 & 1 & 1 \\ \end{array} \]

Explanation:

1. Align the numbers:

1101

- 1010

2. Start from the rightmost bit:

- 1 - 0 = 1

3. Move to the next bit:

- 0 - 1 (need to borrow from the next bit)

- Borrow 1 from the next bit (making the next bit 0), convert 0 to 10.

- Now, 10 - 1 = 1

4. Continue with the next bit:

- Subtract 0 - 0 = 0 (after borrowing)

5. Finally:

- Subtract 1 - 1 = 0

6. Result:

- The result of the subtraction, after handling all borrowing, is 0011.

Binary Multiplication

Binary multiplication is similar to decimal multiplication but simpler, as it only involves multiplying by 0 or 1:

0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1

Process: Multiply each bit of the second number by each bit of the first number, and shift accordingly.

Binary Multiplication Example:

\[ \begin{array}{cccc} & & 1 & 0 & 1 \\ \times & & & 1 & 1 \\ \hline & & 1 & 0 & 1 \\ + & 1 & 0 & 1 & 0 \\ \hline & 1 & 1 & 1 & 1 \\ \end{array} \]

Explanation:

1. Multiply each bit of the first number by each bit of the second number.

2. Shift the results according to their bit positions.

3. Sum all the shifted results to get the final product.

Binary Division

Binary division is similar to decimal long division:

Align the divisor and dividend.
Subtract the divisor from the dividend, shifting and bringing down bits as needed.
Record the quotient and remainder.

Binary Division Example:

\[ \begin{array}{r|l} 1010 & 10 \\ \hline 0101 & \text{Quotient} \\ \underline{10} & \text{Subtract 10} \\ \hline 0010 & \text{Bring down the next bit} \\ \underline{00} & \text{Subtract 00} \\ \hline 010 & \text{Bring down the next bit} \\ \underline{10} & \text{Subtract 10} \\ \hline 00 & \text{Final remainder} \\ \end{array} \]

Explanation:

1. Align the divisor (10) with the leftmost bits of the dividend (1010).

2. Subtract the divisor (10) from the aligned portion of the dividend (10), resulting in 0010.

3. Bring down the next bit to get 010.

4. Subtract the divisor (10) from 010, resulting in 00.

5. The quotient is derived from the division steps, which is 101.

6. The remainder after the final subtraction is 00.

7. Therefore, the final result of the division is:

Show the result as:
Quotient: 0101
Final Result (Quotient with leading zeroes): 00101

The quotient written from top to bottom is: 101
The final result, including leading zeroes for clarity, is: 0101

Additional Explanations

Converting Binary to Decimal: To convert a binary number to decimal, sum the products of each bit and its positional value.

Binary Number: 1101

Positions: 3210

Values: 8421

Conversion to Decimal:

\[ 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 8 + 4 + 0 + 1 = 13 \]

Explanation:

1. Each binary digit corresponds to a power of 2 based on its position from right to left.

2. Calculate the value for each bit by multiplying it with \(2^{\text{position}}\).

3. Sum all these values to get the decimal representation.

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