Problem 26 - Discrete Mathematics with Applications

Question

In Exercises \(21-28,\) perform the indicated operations. $$ \begin{array}{r}{10111_{\mathrm{tw} 0}} \\ { \times 1101_{\mathrm{two}}}\end{array} $$

Step-by-Step Solution

Answer

The multiplication of the two binary numbers 10111 and 1101 is 100011111.

Step 1: Understand Binary Multiplication

The process of binary multiplication is similar to that of decimal multiplication. The primary difference is that we only need to consider 0s and 1s as the digits in binary numbers.

Step 2: Arrange the Numbers

Arrange the binary numbers given in the exercise like this: 10111 x 1101 ___________

Step 3: Multiply the First Row

Multiply each digit of the first number (10111) with the first digit (1) of the second number (1101). Write each product underneath the corresponding digit: 10111 x 1101 ___________ 10111

Step 4: Multiply the Second Row

Multiply each digit of the first number (10111) with the second digit (0) of the second number (1101). Write each product underneath the corresponding digit, shifted one position to the left: 10111 x 1101 ___________ 10111 00000

Step 5: Multiply the Third Row

Multiply each digit of the first number (10111) with the third digit (1) of the second number (1101). Write each product underneath the corresponding digit, shifted two positions to the left: 10111 x 1101 ___________ 10111 00000 10111

Step 6: Multiply the Fourth Row

Multiply each digit of the first number (10111) with the fourth digit (1) of the second number (1101). Write each product underneath the corresponding digit, shifted three positions to the left: 10111 x 1101 ___________ 10111 00000 10111 10111

Step 7: Add the Partial Products

Now, add the partial products to get the final result: 10111 x 1101 ___________ 10111 00000 10111 10111 ___________ 100011111 The multiplication of the two binary numbers, 10111 and 1101, is 100011111.

Key Concepts

Binary Number System

The binary number system is a simple yet powerful numerical system that uses only two digits: 0 and 1. Unlike the decimal system, which is based on ten digits (0 through 9), binary is based on powers of two. This system is particularly useful in computing and digital electronics because it aligns perfectly with the on-off nature of electronic circuitry.
Binary numbers are read from right to left, with each digit representing a power of two. For example, the binary number 10111 can be translated to decimal by evaluating it as:

• 1 x 2⁴ = 16
• 0 x 2³ = 0
• 1 x 2² = 4
• 1 x 2¹ = 2
• 1 x 2⁰ = 1

Adding all these values gives us the decimal number 23.
Understanding the binary number system is foundational to grasping how binary arithmetic works, including multiplication and addition.

Step-by-Step Solution

When it comes to solving binary multiplication problems, following a structured step-by-step solution can simplify the process. Breaking down binary multiplication into manageable steps ensures that you're systematically handling every part of the operation without missing critical details.
Here's a simple approach to solving a binary multiplication:

• Align the binary numbers correctly, keeping in mind that you write the multiplier below the multiplicand.
• Multiply each digit of the upper binary number by each digit of the lower one separately.
• Shift your results appropriately, based on which digit you are currently multiplying, similar to the decimal system's use of place value.
• Sum up the resulted partial products to find the final product.

By approaching binary multiplication step-by-step, you ensure accuracy and avoid errors that might occur with the more abstract thinking required when considering the entire problem at once.

Partial Products

In binary multiplication, just like in decimal, partial products play a crucial role. Each multiplication of a single digit in the multiplicand by the entire multiplier yields what we call a partial product.
To calculate these, simply:

• Multiply the entire multiplicand by one digit of the multiplier at a time.
• Start from the least significant bit (rightmost bit) of the multiplier, and move leftwards.
• Each successive operation results in a shift of the partial product to the left, increasing the leftward shift by one more position than the previous step.

The alignment and summation of these partial products is an essential part of reaching the final solution in the binary multiplication process. For example, when multiplying 10111 by 1101, each line of the 'product' in the scaffold is a partial product. Understanding how these partial products accumulate to form the complete product exemplifies the logic behind binary multiplication.

Binary Addition

Once you have all the partial products, the next step in binary multiplication is adding them together using binary addition. Binary addition is quite similar to regular addition in the decimal system, except it operates only with the digits 0 and 1.
Key rules for binary addition include:

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

When you're adding the partial products in binary multiplication, always keep in mind the requirements for handling carries. For instance, when adding: - 10111 - 00000 - 101110 and - 10111000, start from the rightmost column and work your way left, applying carry rules as necessary.
The correct implementation of binary addition is vital to achieve the accurate final product of the multiplication.