Question

One way to do multiplication is by repeated addition. For example, \(47 \times 25\) can be evaluated as \(47+47+47+\ldots+47\) (25 times). Sketch out an algorithm for multiplying two positive numbers \(a\) and \(b\) using this technique.

Short answer

Multiply \(a\) by \(b\) by adding \(a\), \(b\) times using a loop.

Step-by-step solution

Initialization

Start by defining your two numbers. Let's assume that we want to multiply two positive numbers, \(a\) and \(b\). Also, initialize a variable \(sum\) to zero. It will hold the result of the repeated addition.

Repeat Addition

Perform repeated addition by adding the number \(a\) to the \(sum\) variable, \(b\) times. This is done in a loop that iterates \(b\) times. In each iteration, \(sum = sum + a\). This simulates the multiplication operation by adding \(a\) repeatedly.

Return Result

Once the loop completes, the \(sum\) variable will hold the product of \(a\) and \(b\). Return \(sum\) as the output of the algorithm.

Key concepts

Multiplication

Multiplication is a fundamental mathematical operation used in various areas of math and science. It involves scaling one number by another, resulting in a product. Consider the multiplication of two numbers: if you multiply 3 by 4, you are essentially taking the number 3 and adding it to itself 4 times. This operation reflects the core relationship between multiplication and addition.

Traditional multiplication involves an algorithm that involves breaking down numbers into smaller parts, often using a standard multiplication table as a guide. However, another approach, as seen in this exercise, is to treat multiplication as repeated addition.

• For instance, multiplying 6 by 3 can be viewed as starting with zero and adding the number 6, three times: 6 + 6 + 6 = 18.
• This method is particularly useful for understanding the concept of multiplication without relying on memorization.

Through repeated addition, the basic principle of multiplication becomes intuitive, allowing learners to comprehend how one number increases by being added repeatedly.

Repeated Addition

Repeated addition is a straightforward way to understand multiplication, especially for new learners. This method involves iteratively adding one number a specific number of times. In essence, if you want to find the product of two numbers, you start with zero and repeatedly add one of the numbers the other number of times.

For example, if you want to multiply 5 by 4 using repeated addition, you'd follow these steps:

• Begin with 0.
• Add 5 to 0, resulting in 5.
• Add 5 again, which totals 10.
• Add 5 again, reaching 15.
• One last addition of 5 gives you 20.

The final result, 20, is equivalent to the product of 5 and 4.

This approach, albeit not the most efficient, emphasizes the relationship between addition and multiplication. It illustrates how multiplication is essentially a shortcut to adding the same number repeatedly. Teachers often use this method to simplify the concept for students who find multiplication intimidating initially.

Loop Iteration

Loop iteration is a crucial concept in programming and algorithm design, especially when implementing multiplication through repeated addition. In programming, a loop allows you to execute a block of code multiple times, which is handy when you need to perform repetitive tasks.

In the context of using repeated addition for multiplication, the loop iterates as many times as specified by one of the numbers in the multiplication. This ensures the second number is added to the total the correct number of times. Here's a simple illustration:

• Initialize a sum variable to zero.
• Set up a loop that runs a specific number of times (equal to the second multiplier).
• During each iteration, add the first multiplier to the sum.

For instance, to solve the equation 7 multiplied by 3, your loop will run three times, adding 7 to the sum each time. After the third iteration, the sum will be 21.

This process not only mirrors the manual repetition involved in addition but also introduces students to fundamental programming structures, laying the groundwork for more advanced algorithmic thinking.