Question

CRITICAL THINKING One of the major sources of our knowledge of Egyptian mathematics is the Ahmes papyrus, which is a scroll copied in 1650 B.C. by an Egyptian scribe. The following problem is from the Ahmes papyrus. Divide 10 hekats of barley among 10 men so that the common difference is \(\frac{1}{8}\) of a hekat of barley. Use what you know about arithmetic sequences and series to determine what portion of a hekat each man should receive.

Short answer

Using the concept of arithmetic sequence, each man's portion can be calculated by identifying the first term using the sum formula and then determining each term using the term formula and concept of common difference.

Step-by-step solution

Understand the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is commonly known as the 'common difference'. Here, the common difference is given as \( \frac{1}{8} \) hekat of barley.

Define Terms of the Sequence

For an arithmetic sequence, the term can be expressed as \( a_n = a_1 + (n - 1) * d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the position of the term in the sequence. Here, \( d = \frac{1}{8} \) hekat and we have a total of 10 terms (10 men).

Calculate the Sum of the Sequence

The sum \( S \) of an arithmetic series up to \( n \) terms can be calculated using the formula \( S = \frac{n}{2} * (2a + (n - 1)d) \). As per the problem, this sum total is 10 hekats, so we have the equation \( 10 = \frac{10}{2} * (2a + (10 - 1) * \frac{1}{8}) \).

Determine the first term

Solving the equation from Step 3 for \( a \) will give us the first term of the sequence, i.e., the amount of hekat the first man would receive.

Determine the portions for each man

Now, knowing the first term and the common difference, we can determine the portion each man would receive, by plugging in the respective values of \( n \) in the formula for \( a_n \).

Key concepts

Ahmes Papyrus

The Ahmes Papyrus, also known as the Rhind Mathematical Papyrus, is a significant artifact in the study of ancient mathematics. It dates back to around 1650 B.C. and was copied by an Egyptian scribe named Ahmes. This papyrus is crucial because it provides insight into the mathematical practices and concepts of ancient Egypt.

In particular, the Ahmes Papyrus includes a wide variety of mathematical problems, with procedures and solutions documented in hieratic script. One of the problems detailed within involves dividing 10 hekats of barley among 10 men with a common difference of \( \frac{1}{8} \) hekat of barley.

This artifact shows the existence of what we now understand as arithmetic sequences and vividly demonstrates the application of these concepts in day-to-day Egyptian life. It is one of the earliest records of mathematical problem-solving and significantly enhances our understanding of arithmetic concepts like series and sequences in the context of ancient Egyptian culture.

Egyptian Mathematics

Egyptian Mathematics is an intriguing field that highlights the mathematical prowess of ancient Egyptian civilizations. The Egyptians employed a base-10 numeral system and were notable for their practical approach to solving everyday problems. Their mathematical concepts formed the foundation for many modern-day arithmetic theories.

They tackled issues such as building dimensions, land measurements, and provisions distribution, much like the example we have from the Ahmes Papyrus. Egyptian mathematicians used rudimentary tools yet achieved complex calculations by breaking down problems step by step. Typically, they relied heavily on fractions, as seen in their distribution problems, like dividing barley with a set common difference.

Through documents like the Ahmes Papyrus, we observe their intuitive understanding and application of arithmetic sequences and series long before these concepts were formally defined. Egyptian Mathematics remains a testament to the intelligence and ingenuity of the ancient Egyptians in handling intricate mathematical tasks.

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence. In such a sequence, each term differs from the previous one by a constant value, known as the common difference. This concept is vital because it allows us to predict the sum of a set of numbers efficiently.

To find the sum of an arithmetic series, we use the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \( S_n \) is the sum of the series, \( n \) is the number of terms, \( a \) is the first term, and \( d \) is the common difference.

In the case of the problem from the Ahmes Papyrus, we use this formula to distribute 10 hekats of barley among 10 men. This demonstrates how arithmetic series can solve real-world distribution problems by calculating fair and equal portions based on given constraints.

Common Difference

The common difference is a crucial part of arithmetic sequences and series. It is the fixed amount between each term in the sequence and defines the pattern of progression. Understanding this concept helps in formulating and solving problems involving sequential increases or decreases.

In the example from the Ahmes Papyrus, the common difference is \( \frac{1}{8} \) hekat of barley. This indicates that each man's portion increases by \( \frac{1}{8} \) hekat compared to what the previous man received.

This value is key to setting up the sequence and calculating each portion efficiently. Applying the concept of common differences allows one to determine how the portions of barley were distributed, showcasing its importance in solving calculations involving arithmetic sequences.