Problem 6 - A Gentle Introduction to the Art of Mathematics

Question

Find the first 20 decimal places of \(\pi, 3 / 7, \sqrt{2}, 2 / 5,16 / 17, \sqrt{3}, 1 / 2\) and \(42 / 100 .\) Classify each of these quantity's decimal expansion as: terminating, having a repeating pattern, or showing no discernible pattern.

Step-by-Step Solution

Answer

Terminating: 0.4, 0.5, 0.42. Repeating: 3/7, 16/17. Non-discernible: \(\pi\), \sqrt{2}\), \sqrt{3}\).

Step 1 - Identify decimal expansions

First, let's convert each number to its decimal form. - \(\pi \): \ 3.14159265358979323846 - \(\frac{3}{7} \): \ 0.42857142857142857143 - \sqrt{2} \: \ 1.41421356237309504880 - \( \frac{2}{5} \): \ 0.4 - \( \frac{16}{17} \): \ 0.94117647058823529412 - \sqrt{3} \: \ 1.73205080756887729352 - \( \frac{1}{2} \): \ 0.5 - \( \frac{42}{100} \): \ 0.42

Step 2 - Classify decimal expansions

Next, we classify each decimal expansion: - \(\pi \): shows no discernible pattern (non-repeating, non-terminating)- \(\frac{3}{7} \): has a repeating pattern \( (0.428571) \)- \sqrt{2} \: shows no discernible pattern (non-repeating, non-terminating)- \( \frac{2}{5} \): terminating \( (0.4) \)- \( \frac{16}{17} \): has a repeating pattern \( (0.941176470588235294) \)- \sqrt{3} \: shows no discernible pattern (non-repeating, non-terminating)- \( \frac{1}{2} \): terminating \( (0.5) \)- \( \frac{42}{100} \): terminating \( (0.42) \)

Key Concepts

terminating decimals

A terminating decimal is a decimal expansion that comes to an end after a certain number of digits. This means that after a specific point, no more digits appear. For example, \(\frac{2}{5} = 0.4\) and \(\frac{1}{2} = 0.5\). Here's a quick guide to identifying terminating decimals:

- If you can express the fraction (in simplest form) as \(\frac{a}{b}\) where \text{b}\ has only the factors 2 and/or 5, the decimal will be terminating.

Let's consider a few more examples from the exercise:

• \(\frac{42}{100} = 0.42\)

Notice that these decimals end: \(\frac{42}{100}\) (factors of 100 are 2s and 5s).

Understanding this concept is vital as it helps simplify problems and aids in quickly recognizing terminating decimals in various contexts.

repeating decimals

Repeating decimals are decimals that continue indefinitely with a repeating pattern of digits. Unlike terminating decimals, these do not come to an end, but instead, they recur at regular intervals. For instance, \(\frac{3}{7} = 0.428571428571...\). Here, '428571' is the repeating block.

Here's how to spot a repeating decimal:

• Perform the division: When you divide and the remainder starts repeating, the decimal portion will repeat from that point.
• Expressible fractions: Most fractions where the denominator isn't solely composed of 2's and 5's will yield a repeating decimal.

More examples from our exercise include:

• \(\frac{16}{17} = 0.941176470588235294...\)

This massive block '941176470588235294' is the repeating sequence.

Understanding and identifying repeating decimals is important because it allows you to work more efficiently with complex fractions and prepares you for higher-level math concepts.

non-repeating decimals

Non-repeating decimals are also known as non-terminating, non-repeating decimals. These decimals go on infinitely without any repeating pattern. Numbers like \(ic \) and other irrational numbers belong to this category. In our exercise, examples include \(\pi\) and \(\frac{2}{5}\).

Characteristics of non-repeating decimals:

• Infinite sequence: These decimals continue forever without repeating.
• Non-recurrence: There's no predictable pattern or sequence in their digits.
• Common with irrationals: Often, these decimals are derived from irrational numbers like square roots and specific constants like \(\pi\).

Let's take a couple of examples from the exercise:

• \(\pi = 3.14159265358979323846...\)
• \(\sqrt{2}= 1.41421356237309504880...\)

Notice how neither decimal terminates nor repeats.

Grasping this concept is crucial as it lays the foundation for understanding irrational numbers and dealing with more advanced mathematical theories.