Question

Multiply \(7 \overline{2} \overline{4} \overline{8}\) by \(12 \overline{3}\) using the Egyptian multiplication technique. Note that it is necessary to multiply each term of the multiplicand by \(\overline{\overline{3}}\) separately.

Short answer

Answer: \(58\overline{6}\overline{4}\overline{4}\)

Step-by-step solution

Write the numbers in Egyptian form

We have two multiplicands: \(7\overline{2}\overline{4}\overline{8}\) and \(12\overline{3}\). We must first write these numbers in Egyptian form. \(7\overline{2}\overline{4}\overline{8} = 7 + (\frac{1}{10})(2+ (\frac{1}{10})(4+(\frac{1}{10})(8)))\) And, \(12\overline{3} = 12 + \frac{1}{10}(3)\). Now, let's perform the Egyptian multiplication technique.

Perform the Egyptian multiplication technique on each number separately

First, let's multiply \(7\overline{2}\overline{4}\overline{8}\) by \(\overline{\overline{3}}\). To multiply \(7\overline{2}\overline{4}\overline{8}\) by \(\overline{\overline{3}}\), we apply the Egyptian multiplication technique: \(7\overline{2}\overline{4}\overline{8}\cdot\overline{\overline{3}}\) 1 | 7\overline{2}\overline{4}\overline{8} 2 | 14\overline{4}\overline{9}\overline{6} 4 | 28\overline{9}\overline{9}\overline{2} Now, let's multiply \(12\overline{3}\) by \(\overline{\overline{3}}\). \(12\overline{3}\cdot\overline{\overline{3}}\) 1 | 12\overline{3} 2 | 24\overline{6} 4 | 49\overline{2}

Add the corresponding rows

Finally, we add the corresponding rows (as given in the Egyptian multiplication technique): For \(7\overline{2}\overline{4}\overline{8}\cdot\overline{\overline{3}}\): 1 | 7\overline{2}\overline{4}\overline{8} + 2 | 14\overline{4}\overline{9}\overline{6} = 21\overline{7}\overline{4}\overline{4} For \(12\overline{3}\cdot\overline{\overline{3}}\): 1 | 12\overline{3} + 2 | 24\overline{6} = 36\overline{9}

Add the final results for each multiplication

Now, we have to add these results to obtain the final answer: \(7\overline{2}\overline{4}\overline{8}\cdot\overline{2}\) + \(12\overline{3}\cdot\overline{2}\) = \(21\overline{7}\overline{4}\overline{4}\) + \(36\overline{9}\) So, the final answer is \(58\overline{6}\overline{4}\overline{4}\).

Key concepts

Ancient Mathematics

Ancient mathematics is a fascinating subject that reveals how different cultures approached complex problems before the advent of modern numerical systems. In particular, Egyptian mathematics stands out for its ingenuity and practicality. The Egyptians, who built one of the greatest civilizations of the ancient world, developed mathematical concepts and techniques essential for constructing their monumental pyramids and temples, as well as for managing the annual flooding of the Nile.

Egyptian mathematicians used a system based on the decimal system we use today, but it was fundamentally different in that it was based on a hieroglyphic script with specific symbols representing multiples of 10. The Egyptians excelled in geometry and arithmetic, evidenced by their ability to calculate areas and volumes, as well as their proficiency in handling fractions and multiplication.

As trades and other needs necessitated more intricate computations, Egyptians devised methods such as the Egyptian multiplication technique, capable of breaking down complex multiplication into simpler addition tasks. Their achievements in ancient mathematics are an early testament to the human capacity for innovation and adaptation in the pursuit of knowledge and functionality.

Multiplication Methods

When it comes to multiplication methods, history has seen a variety of approaches tailored to the cultural and mathematical understanding of each era. One of the most interesting of these is the ancient Egyptian method of multiplication, also known as duplation and mediation.

This method relies on doubling numbers and then adding them to reach the desired product. It is a powerful technique because it reduces the process of multiplication to a series of simpler additions and can be performed without a full numerical writing system, making it well-suited to ancient Egyptian mathematical notation with its limited symbols.

For example, to multiply two numbers using Egyptian multiplication, one would list multiples of one number by doubling, and then select the rows that would add up to the other number being multiplied. This process elegantly exemplifies the transition from additive to multiplicative thinking, a leap that defines a significant development in the history of mathematics.

Mathematical Notation

Mathematical notation provides a universal language for expressing mathematical concepts succinctly and precisely. Over time, notation has evolved to be more efficient and comprehensive, enabling mathematicians to communicate complex ideas clearly.

In ancient systems, such as Egyptian numerals, the notation was often additive: hieroglyphs denoted units, tens, hundreds, and so on, and were repeated as necessary to represent a number. This could be cumbersome for large numbers or when performing complex calculations, so the Egyptians distilled operations into more manageable forms, exemplified by their multiplication techniques which relied heavily on notation that could be easily manipulated through addition.

Egyptians also used a unique method called 'unit fractions' to express other numbers. Each fraction was represented as the sum of different unit fractions, with a notation involving an 'overline' to denote division by a number. Understanding ancient notation like this not only informs us about the numerical system they used but also about the cognitive processes and the pragmatism of ancient civilizations when confronting mathematical challenges.