Express the following in the form p/q where p and q are integers and q ≠ 0. (i) 0.001 (ii) 0.99999... (in the form p/q) (iii) 0.6 (iv) 0.4. Are you surprised by your answer? With y... Express the following in the form p/q where p and q are integers and q ≠ 0. (i) 0.001 (ii) 0.99999... (in the form p/q) (iii) 0.6 (iv) 0.4. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense. What can be the maximum number of digits in the decimal expansion of 1/7? Perform the division to check your answer. Look at several examples of rational numbers in the form p/q, where q ≠ 0, and having terminating decimal representations. Can you guess what property q must satisfy? Write three numbers whose decimal expansions are non-terminating non-repeating. Find three different irrational numbers between the rational numbers 5/9 and 7/11. Classify the following numbers as rational or irrational. (i) √225 (ii) √3 (iii) 1.10610000100001 (iv) 7.478478...

Question image

Understand the Problem

The question presents a series of mathematical problems focused on expressing numbers as fractions, understanding decimal expansions, and classifying numbers. It appears to require solutions to mathematical expressions and discussions on properties of numbers.

Answer

$0.001 = \frac{1}{1000}$, $0.9999... = 1$, $\text{Terminating fraction} \Rightarrow q \text{ has factors } 2 \text{ or } 5$ only.
Answer for screen readers
  1. $0.001 = \frac{1}{1000}$
  2. $0.99999... = 1$
  3. The condition for $\frac{p}{q}$ to be terminating: $q$ has only factors $2$ and $5$.
  4. Rational numbers between $5$ and $11$: {$6$, $7$, $8$, $9$, $10$, $11$}
  5. Decimal expansion of $\frac{1}{17}$: $0.058823...$
  6. Examples with no common factors: $\frac{2}{5}, \frac{3}{7}, \frac{4}{9}$
  7. Irrational numbers: {$\sqrt{2}, \sqrt{3}, \pi$}
  8. Classifications: $\sqrt{2}$ - Irrational, $\sqrt{3}$ - Irrational, $7.478478...$ - Rational

Steps to Solve

  1. Expressing 0.001 as a Fraction

To express the decimal $0.001$ as a fraction, recognize that it can be written as:

$$ 0.001 = \frac{1}{1000} $$

This is because there are three decimal places, placing it over $1000$.

  1. Confirming the Decimal Expression

To verify, divide:

$$ 1 \div 1000 = 0.001 $$

This confirms that the fraction represents the decimal correctly.

  1. Analyzing 0.9999...

For the next part, express $0.9999...$ as a fraction. This repeating decimal can be expressed as:

Let $x = 0.9999...$

Then, multiplying by $10$:

$$ 10x = 9.9999... $$

Subtracting the first equation from the second:

$$ 10x - x = 9.999... - 0.999... $$

This simplifies to:

$$ 9x = 9 $$

Dividing both sides by $9$:

$$ x = 1 $$

Thus, $0.9999... = 1$.

  1. Determining Properties of Fractions

For a fraction $\frac{p}{q}$ (where $q \neq 0$) to have a terminating decimal:

The denominator $q$ must have only the prime factors $2$ and/or $5$. If any other prime factor exists, the decimal representation will be non-terminating.

  1. Identifying Rational and Irrational Numbers

For rational numbers between $5$, $9$, $7$, and $11$, we can choose:

  • $\frac{5+9}{2} = 7$
  • $\frac{7+11}{2} = 9$

Rational numbers such as $6$, $8$, and $10$ are also included.

  1. Writing Decimal Expansion of $\frac{1}{17}$

Using long division to express $\frac{1}{17}$:

Dividing:

$$ 1 \div 17 $$

This gives a repeating decimal: $0.058823...$.

  1. Writing Examples with No Common Factors

Take rational numbers like:

  • $\frac{2}{5}$
  • $\frac{3}{7}$
  • $\frac{4}{9}$

These can be shown to have no common factors.

  1. Finding Irrational Numbers

Examples include:

  • $\sqrt{2}$
  • $\sqrt{3}$
  • $\pi$

Each of these cannot be expressed as a ratio of integers.

  1. Classifying the Following

Classify the provided numbers:

  • ( \sqrt{2} ): Irrational
  • ( \sqrt{3} ): Irrational
  • ( 7.478478... ): Rational (because it has repeating decimals)
  1. $0.001 = \frac{1}{1000}$
  2. $0.99999... = 1$
  3. The condition for $\frac{p}{q}$ to be terminating: $q$ has only factors $2$ and $5$.
  4. Rational numbers between $5$ and $11$: {$6$, $7$, $8$, $9$, $10$, $11$}
  5. Decimal expansion of $\frac{1}{17}$: $0.058823...$
  6. Examples with no common factors: $\frac{2}{5}, \frac{3}{7}, \frac{4}{9}$
  7. Irrational numbers: {$\sqrt{2}, \sqrt{3}, \pi$}
  8. Classifications: $\sqrt{2}$ - Irrational, $\sqrt{3}$ - Irrational, $7.478478...$ - Rational

More Information

The fraction $0.001$ illustrates how simple decimals can be represented. The expression $0.9999... = 1$ surprises many, highlighting interesting properties of infinity in mathematics. Understanding conditions for decimal expansions aids in distinguishing rational from irrational numbers.

Tips

  • Confusing terminating decimals with non-terminating ones. It’s essential to check the denominator's prime factors.
  • Miscalculating decimal expansions through approaches like long division. Practicing accurate long division steps can enhance clarity.
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