Problem 2 - A History of Mathematics: An Introduction

Question

Use Egyptian techniques to multiply 34 by 18 and to divide 93 by 5

Step-by-Step Solution

Answer

Question: Multiply 34 by 18 and divide 93 by 5 using Egyptian techniques. Answer: Multiplication: 34 * 18 = 646. Division: 93 / 5 = 18.6.

Step 1: Express 18 as the sum of powers of 2

Write 18 as a sum of powers of 2: \(18 = 2^0 + 2^1 + 2^4\)

Step 2: Repeatedly double 34 and align it with powers of 2

Double 34 as many times as necessary and align each value with its corresponding power of 2: \(2^0\): 34 \(2^1\): \(34\times 2 = 68\) \(2^2\): \(68\times 2 = 136\) \(2^3\): \(136\times 2 = 272\) \(2^4\): \(272\times 2 = 544\)

Step 3: Add the corresponding numbers

Add the values corresponding to the powers of 2 from Step 1: \(34 + 68 + 544 = 646\) Thus, \(34\times 18 = 646\). Now, let's divide 93 by 5 using Egyptian techniques.

Step 1: Express 5 as the sum of powers of 2

Write 5 as a sum of powers of 2: \(5 = 2^0 + 2^2\)

Step 2: Repeatedly double 1 and align it with powers of 2

Double 1 as many times as necessary and align each value with its corresponding power of 2: \(2^0\): 1 \(2^1\): \(1\times 2 = 2\) \(2^2\): \(2\times 2 = 4\)

Step 3: Multiply 93 by the numbers from step 2

Multiply 93 by the numbers corresponding to the powers of 2 from Step 1: \(93\times 1 = 93\) \(93\times 4 = 372\)

Step 4: Add the results from step 3

Add the values obtained from Step 3: \(93 + 372 = 465\)

Step 5: Divide the result from step 4 by the original divisor

Divide 465 by 5: \(\frac{465}{5} = 93\) So, \(93\div 5 = 18.6\)

Key Concepts

Mathematical History

The history of mathematics stretches back to ancient times and encompasses a wide range of cultures and civilizations. Mathematics in ancient Egypt holds a special place, being one of the earliest systems to reach a high level of sophistication. The Egyptians used mathematics primarily for practical tasks such as surveying land, constructing pyramids, and managing trade. Their numerical system was based on tens, similar to ours, but they had specific symbols for each power of ten.

However, unlike our present-day system which is positional, Egyptian mathematics was more hierarchical. This means that the value of a symbol depended not only on its inherent value but also on its placement relative to other symbols. As the ancient Egyptian civilization thrived, its mathematicians developed arithmetic techniques, some of which are quite different from those used today, yet their legacies persist in our understanding of mathematics' evolution over time.

Ancient Multiplication Techniques

Ancient Egyptian multiplication is notably different from the methods taught today. The technique is known as 'duplation,' which is a process similar to doubling or 'Russian Peasant Multiplication.' The method involves breaking down one of the numbers to be multiplied into powers of two and then doubling the other number as many times as necessary.

To multiply 34 by 18, the Egyptians would first express 18 in terms of the sum of powers of two. They would then line up these powers of two next to repeated doublings of 34. Finally, they would sum the numbers next to the powers of two that add up to 18. This ingenious method reduces multiplication, which is more complex, into a series of doubling operations that are simpler to perform, especially given the Egyptians' lack of a positional numerical system.

Egyptian Division Methods

The Egyptian division method was somewhat similar to their multiplication technique. It was based on a 'subtract and double' approach where the dividend was successively subtracted by a divisor that was repeatedly doubled.

For dividing 93 by 5, they would express 5 as a sum of powers of two and then double the number 1 (which is easier to work with than 93) to align with these powers. Multiplying these values by 93 and summing the appropriate products would provide a number which, when divided by the original divisor, would result in the quotient. This process effectively transformed division into a form of subtraction and addition—operations that were more straightforward for ancient mathematicians.

Arithmetic Methods

Arithmetic methods of ancient Egyptian math were practical and efficient for their time. They allowed for the calculation of areas, volumes, and the solving of simple algebraic equations known as 'aha' problems—precursors to what we call linear equations. The arithmetic was performed using unit fractions, hieroglyphs, and a system that favored addition and subtraction over multiplication and division.

The Egyptians' approach to arithmetic exploited the properties of numbers and aimed at minimizing the complexity of calculations. By exploring these ancient techniques, we can appreciate how our modern procedures have evolved and also understand that the principles underlying arithmetic have remained mostly uniform —finding ways to simplify and solve problems using the available numerical systems and tools.