How to invent 2 digit multiplication if you are a farmer using 18 x 24 as an example

Question

A farmer might have been the first person to invent 2 digit multiplication

Answer 1

In the context of a farmer needing to multiply two-digit numbers, such as 18 × 24, it's likely that early methods of multiplication relied on visual, practical, and possibly intuitive approaches. A farmer might have come up with an efficient way of calculating this product by breaking it into smaller parts that were easier to handle. This approach resembles a technique often called "partial products" or "lattice multiplication", which involves breaking the problem into smaller, manageable steps. Here's how a farmer might approach 18 × 24:

Step 1: Break the numbers into their place values

For 18 and 24, break them down like this:

1. 18 = 10 + 8
2. 24 = 20 + 4

Step 2: Use the distributive property

Next, use the distributive property to multiply each part of 18 by each part of 24. This is like multiplying using a grid:

(10+8)×(20+4)(10 + 8) \times (20 + 4)This gives us four smaller multiplications to perform:

(10×20)+(10×4)+(8×20)+(8×4)(10 \times 20) + (10 \times 4) + (8 \times 20) + (8 \times 4)Step 3: Perform each multiplication

Now, let's calculate each of these:

1. 10×20=20010 \times 20 = 200
2. 10×4=4010 \times 4 = 40
3. 8×20=1608 \times 20 = 160
4. 8×4=328 \times 4 = 32

Step 4: Add the partial products

Now, add all the results together:

200+40+160+32=432200 + 40 + 160 + 32 = 432Final Answer:

Thus, 18×24=43218 \times 24 = 432.

How this might relate to a farmer:

A farmer in ancient times, without calculators, might have had to rely on practical mental math skills. This method would allow the farmer to break down the complex task of multiplying large numbers into simpler steps that could be managed using everyday knowledge of numbers like tens and ones, much like how they might deal with groups of animals, bundles of hay, or other divisible items.

Using this method, they could use smaller, familiar numbers to handle multiplication without the need for memorizing multiplication tables for all possible two-digit numbers. This is one of the earliest known methods of simplifying multiplication.

Answer 2

A farmer might have invented 2-digit multiplication as a practical way to handle larger numbers in everyday tasks like calculating area, crop yields, or inventory. To understand how they might have approached multiplying 18 by 24, we can break it down into a step-by-step process.

Step-by-Step Breakdown (Using the Distributive Property)

One possible method the farmer could have used is called the distributive property, which makes multiplication of larger numbers more manageable. Here's how it works for 18 × 24:

1. Break down the numbers into tens and ones:
2. 18 = 10 + 8
3. 24 = 20 + 4
4. Distribute the multiplication: Now, multiply each part of the first number by each part of the second number.
5. 18×24=(10+8)×(20+4)18 \times 24 = (10 + 8) \times (20 + 4)18×24=(10+8)×(20+4)
6. We expand this using the distributive property:
7. (10×20)+(10×4)+(8×20)+(8×4)(10 \times 20) + (10 \times 4) + (8 \times 20) + (8 \times 4)(10×20)+(10×4)+(8×20)+(8×4)
8. Which gives:
9. 200+40+160+32200 + 40 + 160 + 32200+40+160+32
10. Add all the results together:
11. 200+40+160+32=432200 + 40 + 160 + 32 = 432200+40+160+32=432

So, 18 × 24 = 432.

Why Would This Be Practical for a Farmer?

For a farmer, using a method like this would allow them to quickly calculate things like:

1. The area of a rectangular field (if the dimensions were 18 meters by 24 meters).
2. The total cost of purchasing multiple items (like bundles of hay, priced at $18 each for 24 bundles).

This technique is much easier than trying to multiply the full numbers directly without breaking them down. By using the distributive property, the farmer can multiply using simpler numbers, which are easier to manage mentally or on paper.

Inventing 2-Digit Multiplication

The farmer would have developed this method to simplify everyday calculations. Farmers typically worked with practical, real-world problems involving measurements, prices, and quantities, which made breaking numbers down into smaller parts a natural and useful approach to complex multiplication.

Answer 3