Problem 74 - Beginning and Intermediate Algebra

Question

The Greek astronomer and mathematician Geminus died in 60 A.D. at the age of \(70 .\) When was he born?

Step-by-Step Solution

Answer

Geminus was born in 10 B.C.

Step 1: Understand the Information

Geminus died in 60 A.D., and he was 70 years old at that time. We need to determine the year of his birth.

Step 2: Set Up the Equation

To find the birth year, subtract his age when he died from the year he died. This can be expressed as:\[\text{Birth Year} = 60 - 70\]

Step 3: Perform the Calculation

Subtract 70 from 60:\[60 - 70 = -10\]The resulting number, -10, represents 10 B.C., as negative years correspond to years before the common era (B.C.).

Step 4: Interpret the Result

Since the calculation gives us -10, Geminus was born in 10 B.C. A year of -10 in historical terms refers to 10 B.C.

Key Concepts

Equation Solving

Equation solving is a fundamental aspect of algebra, often helping us find unknown values based on the given data. In the problem about Geminus, we needed to solve for his birth year. The given data tells us he died in 60 A.D. at the age of 70. With these facts, we set up a simple subtraction equation:

• The year of death - The age at death = Year of birth
• This translates into: 60 - 70 = Birth Year

This approach is a straightforward application of solving equations where subtraction is used to identify the unknown variable, which in this case is the year of birth. It's important to perform the arithmetic accurately while also considering the historical context, which brings us to the understanding that a negative result here represents a year B.C.

Historical Mathematics

In historical mathematics, context is key to understanding numerical results. For ancient dates, our modern number line integrates both A.D. (Anno Domini) and B.C. (Before Christ) systems. When dealing with ancient figures like Geminus, calculations often cross the threshold from A.D. to B.C. As seen in the exercise, solving the equation gives us -10:

• A negative result indicates a year B.C.
• Therefore, Geminus was born in 10 B.C.

Being aware of historico-mathematical conventions helps students correctly interpret answers involving dates and timeframes, especially when transitioning between B.C. and A.D. It’s crucial when solving similar problems in historical mathematics to always remember that negative numbers often point towards years in B.C.

Negative Numbers

Negative numbers play a unique role in mathematics, particularly when working with historical timelines. In this problem, a negative number resulted from subtracting a larger number (the age of Geminus: 70) from a smaller number (the year he died: 60). Thus, we have:

• 60 - 70 = -10
• In historical terms, this "-10" represents 10 B.C.

For students, understanding negative numbers in non-standard contexts, like historical timelines, is a great way to apply algebraic concepts creatively. It illustrates how numbers can not only indicate less than zero but also signify years prior to the start of the common era. This understanding deepens one's grasp of mathematical principles as they apply to real-world chronologies.