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... Read more

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 $1000$.

2. Confirming the Decimal Expression

To verify, divide:

$$ 1 \div 1000 = 0.001 $$

This confirms that the fraction represents the decimal correctly.

3. 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, $0.9999... = 1$.

4. 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.

5. 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.

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

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

Dividing:

$$ 1 \div 17 $$

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

7. 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.

8. Finding Irrational Numbers

Examples include:

• $\sqrt{2}$
• $\sqrt{3}$
• $\pi$

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

9. Classifying the Following

Classify the provided numbers:

• ( \sqrt{2} ): Irrational
• ( \sqrt{3} ): Irrational
• ( 7.478478... ): Rational (because it has repeating decimals)