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Chapter 1: Problem 17

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?

Short Answer

Expert verified
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 by step solution

01

Convert fractions to decimals

To do this, you can simply divide the numerator by the denominator for each fraction, as shown below: \(7/5 = 1.4\) \(13/15 \approx 0.8667\) \(11/24 \approx 0.4583\) \(33/50 = 0.66\)
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 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.

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Most popular questions from this chapter

Problem 72 of the Rhind Mathematical Papyrus reads " 100 loaves of pesu 10 are exchanged for loaves of pesu 45 . How many of these loaves are there? The solution is given as, "Find the excess of 45 over \(10 .\) It is 35 . Divide this 35 by 10. You get \(3 \overline{2}\). Multiply \(3 \overline{2}\) by 100. Result: 350. Add 100 to this 350 . You get 450 . Say then that the exchange is 100 loaves of pesu 10 for 450 loaves of pesu \(45 . "^{18}\) Translate this solution into modern terminology. How does this solution demonstrate proportionality?

Show that the area of the Babylonian "bull's-eye" is given by \(A=(9 / 32) a^{2}\), where \(a\) is the length of the arc (one-third of the circumference). Also show that the length of the long transversal of the bull's-eye is \((7 / 8) a\), whereas the length of the short transversal is \((1 / 2) a\). (Use the Babylonian values of \(C^{2} / 12\) for the area of a circle and \(7 / 4\) for \(\sqrt{3}\).)

Various conjectures have been made for the derivation of the Egyptian formula \(A=\left(\frac{8}{9} d\right)^{2}\) for the area \(A\) of a circle of diameter \(d\). One of these uses circular counters, known to have been used in ancient Egypt. Show by experiment using pennies, for example, whose diameter can be taken as 1, that a circle of diameter 9 can essentially be filled by 64 circles of diameter 1. (Begin with one penny in the center; surround it with a circle of six pennies, and so on.) Use the obvious fact that 64 circles of diameter 1 also fill a square

Solve the Babylonian problem taken from a tablet found at Susa: Let the width of a rectangle measure a quarter less than the length. Let 40 be the length of the diagonal. What are the length and width? Use false position, beginning with the assumption that 1 (or 60 ) is the length of the rectangle.

In the Babylonian system, multiply 25 by 1,04 and 18 by 1,21 . Divide 50 by 18 and 1,21 by 32 (using reciprocals). Use our standard multiplication algorithm modified for base \(60 .\)
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