Problem 17 - A History of Mathematics: An Introduction

Question

Convert the fractions \(7 / 5,13 / 15,11 / 24\), and \(33 / 50\) to sexagesimal notation. (Do not worry about initial zeros, since the product of a number with its reciprocal can be any power of \(60 .\) ) What is the condition on the integer \(n\) that ensures it is a regular sexagesimal, that is, that its reciprocal is a finite sexagesimal fraction?

Step-by-Step Solution

Answer

Question: Convert the given fractions into sexagesimal notation and find the condition on an integer n to ensure it is a regular sexagesimal. Solution: 1. Convert the following fractions to sexagesimal notation: a. \(7/5 = 1^{\circ}24'\) b. \(13/15 \approx 52'\) c. \(11/24 \approx 27' 30''\) d. \(33/50 = 39' 36''\) 2. An integer n is a regular sexagesimal if it contains only the prime factors 2, 3, and 5 in its reciprocal.

Step 1: Convert fractions to decimals

To do this, you can simply divide the numerator by the denominator for each fraction, as shown below: \(7/5 = 1.4\) \(13/15 \approx 0.8667\) \(11/24 \approx 0.4583\) \(33/50 = 0.66\)

Step 2: Convert decimals to sexagesimal notation

To express a decimal number in sexagesimal notation, we follow these steps: 1. Multiply the decimal part of the number by 60. The result is the number of arcminutes (') in sexagesimal notation. 2. Get the integer part of the above calculation. 3. If there's still a decimal part, multiply it by 60 to get the number of arcseconds (") in sexagesimal notation. Let's apply these steps for each of the four decimals obtained above. \(7/5 = 1.4\) in decimal notation becomes: \(0.4 * 60 = 24\) (arcminutes) So, \(7/5 = 1^{\circ}24'\) in sexagesimal notation. \(13/15 \approx 0.8667\) in decimal notation becomes: \(0.8667 * 60 \approx 52\) (arcminutes) So, \(13/15 \approx 52'\) in sexagesimal notation. \(11/24 \approx 0.4583\) in decimal notation becomes: \(0.4583 * 60 \approx 27.5\) \(0.5 * 60 = 30\) (arcseconds) So, \(11/24 \approx 27' 30''\) in sexagesimal notation. \(33/50 = 0.66\) in decimal notation becomes: \(0.66 * 60 = 39.6\) \(0.6 * 60 = 36\) (arcseconds) So, \(33/50 = 39' 36''\) in sexagesimal notation. Now, let's find the condition on the integer n that ensures it is a regular sexagesimal.

Step 3: Determine the condition for an integer to be a regular sexagesimal

An integer n is a regular sexagesimal if its reciprocal (1/n) can be written as a finite sexagesimal fraction. In terms of prime factors, the only prime factors of 60 are 2, 3, and 5. So, an integer n is a regular sexagesimal if it contains only the prime factors 2, 3, and 5. Any integer containing other prime factors would result in a reciprocal that can't be expressed as a finite sexagesimal fraction.

Key Concepts

Fractions to Decimals

Understanding how to convert fractions to decimals is key when working with sexagesimal notation. A fraction represents a part of a whole and is expressed with a numerator (top number) and a denominator (bottom number), like \( \frac{7}{5} \). To convert this fractional form into a decimal form, you simply divide the numerator by the denominator. Here’s how it works:

• For \( \frac{7}{5} \), divide 7 by 5 to get 1.4.
• For \( \frac{13}{15} \), dividing 13 by 15 gives approximately 0.8667.
• For \( \frac{11}{24} \), division yields approximately 0.4583.
• Finally, \( \frac{33}{50} \) equals 0.66 when divided.

Each result is a decimal number, which represents the same value as the original fraction, just in a different form. This decimal representation is the first step towards expressing it in sexagesimal notation.

Prime Factor

When converting fractions to sexagesimal notation, understanding prime factors is essential for determining regular sexagesimal numbers. Prime factors are the prime numbers that multiply together to give a particular number. For instance, the number 60 is the basis for sexagesimal notation. Its prime factors are 2, 3, and 5, since \( 60 = 2^2 \times 3 \times 5 \).

An integer \( n \) is considered a regular sexagesimal if its reciprocal can be expressed as a finite sexagesimal fraction. This happens only if \( n \) itself contains only the prime factors 2, 3, and 5.

• If \( n \) has any prime factors beyond 2, 3, and 5, the reciprocal \( \frac{1}{n} \) will result in a repeating or infinite sexagesimal fraction, making it irregular.
• Examples of regular sexagesimal numbers include those like 60, 30, or 12, which are composed only of these prime factors.

Thus, ensuring an integer is a regular sexagesimal involves checking its prime factor composition.

Regular Sexagesimal

The concept of regular sexagesimal notation is centered around expressing numbers in a base-60 system, which involves understanding finite sexagesimal fractions. To qualify as a regular sexagesimal notation, a number's reciprocal should be finite when expressed in this form, implying that the number itself must meet specific conditions.

Here’s how you determine if an integer is regular in sexagesimal format:

• The number must be composed entirely of the prime factors 2, 3, and 5.
• This means it could be numbers like 4 (\( 2^2 \)), 9 (\( 3^2 \)), 25 (\( 5^2 \)), or combinations where these factors appear in different powers.
• Integers with any additional prime numbers, such as 7 or 11, will not produce a finite sexagesimal reciprocal, making them irregular.

Therefore, checking whether only 2, 3, and 5 are factors of an integer \( n \) helps in identifying it as a regular sexagesimal number.