Exercises

Exercises 1.12 Exercises

1. Integers and Rationals.

Which of the following expressions are integers? Which are rational? Which are both? Which are undefined? Explain.

\(\displaystyle \displaystyle \frac{120}{-12}\)
\(\displaystyle \displaystyle \frac{120}{-12}\)
\(\displaystyle \displaystyle \frac{12\times 0}{-12}\)
\(\displaystyle \displaystyle \frac{12}{-120}\)
\(\displaystyle \displaystyle \frac{10}{5-5}\)
\(\displaystyle \displaystyle \frac{10}{-5-5}\)
\(\displaystyle \displaystyle \sqrt{100}\)
\(\displaystyle \displaystyle \sqrt{10}\)
\(\displaystyle \displaystyle 10.9999\)
\(\displaystyle \displaystyle 10.\overline{9}\)

2. Is it Irrational?

We know that \(\pi\) is irrational. Which of the following are also irrational? Support your reasoning.

\(\displaystyle \displaystyle \frac{\pi}{2\pi}\)
\(\displaystyle \displaystyle \frac{1}{2\pi}\)
\(\displaystyle \displaystyle \frac{3+\pi}{2\pi}\)
\(\displaystyle \displaystyle \frac{3\pi+\pi}{2\pi}\)
\(\displaystyle \displaystyle 3.\overline{1459}\)

3. Order of Operations.

Evaluate each expression. What are your conclusions?

\(\displaystyle \displaystyle (2/3)/4\)
\(\displaystyle \displaystyle 2/(3/4)\)
\(\displaystyle \displaystyle 2/(3 \times 4)\)
\(\displaystyle \displaystyle (2\times 3)/4\)

4. Properties of Real Numbers.

Use the properties of real numbers to expand and simplify the products below. Which properties are you using as you proceed?

\(\displaystyle \displaystyle 48 \times 21\)
\(\displaystyle \displaystyle (a+b)^2\)
\(\displaystyle \displaystyle (a+b)(a-b)\)

Solution.

\begin{align*} 48 \times 21 \amp=\quad (50-2)\times (20+1) \\ \amp=\quad 1000+50-40-2 \\ \amp=\quad 1000 +10-2 \\ \amp=\quad 1008 \end{align*}

\begin{align*} (a+b)^2 \amp=\quad (a+b)(a+b)\\ \amp=\quad a(a+b)+b(a+b)\\ \amp=\quad a^2+ab+ba+b^2\\ \amp=\quad a^2+ab+ab+b^2\\ \amp=\quad a^2+2ab+b^2 \end{align*}

\begin{align*} (a+b)(a-b) \amp=\quad a(a-b)+b(a-b)\\ \amp=\quad a^2-ab+ba-b^2\\ \amp=\quad a^2-ab+ab-b^2\\ \amp=\quad a^2-b^2 \end{align*}

Notice how we use the distributive property and commutativity at key points in each. Associativity is used throughout. Why?

5. Number Sense.

I consider a number to be small if it is close to zero. How would you describe the reciprocal of a small, but non-zero, real number? How would you describe the reciprocal of a big number?

6. Avoiding Common Mistakes.

Are \(\sqrt{4\cdot 9}\) and \(\sqrt{4}\cdot\sqrt{9}\) the same or different real numbers? Are \(\sqrt{9 + 16}\) and \(\sqrt{9}+\sqrt{16}\) the same or different real numbers? Explain your answers.

Solution.

We have \(\sqrt{4\cdot 9} = \sqrt{36} = \sqrt{6^2} = 6\text{,}\) while \(\sqrt{4}\cdot\sqrt{9} = 2 \cdot 3 = 6\) and they agree. In general, \(\sqrt{a\cdot b} = \sqrt{a} \cdot \sqrt{b}\text{,}\) for non-negative \(a\) and \(b\text{.}\)
We have \(\sqrt{9 + 16}=\sqrt{25} = 5\text{,}\) while \(\sqrt{9}+\sqrt{16}= 3+ 4 = 7\) and they do not agree. In general, \(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\)

7. Avoiding Common Mistakes.

Is \(\sqrt{-4}\) a real number? What about \(-\sqrt{4}\text{?}\) What makes them different? What are the solutions to the equation \(x^2 - 4 = 0\text{?}\) What are the solutions to \(x^2 + 4 = 0\text{?}\) Explain your answer.

Solution.

You cannot square a real number and get a negative result. Thus, \(\sqrt{-4}\) is not a real number because nothing may be squared to get \(-4\text{.}\) On the otherhand, \(-\sqrt{4}=- 2\) is a real number.

8. Avoiding Common Mistakes.

Evaluate the expression \(\sqrt{(-2)^2}\) without using a calculator. What should you be careful about when simplifing expressions of the form \(\sqrt{x^2}\text{?}\) Explain your answer.

Hint.

Remember! \(\sqrt{x}\) is defined as the non-negative number whose square is \(x\text{.}\) Emphasis on non-negative!

Answer.

In general, \(\sqrt{x^2} = | x |\text{.}\)

9. Interval Notation.

The speed of light is \(670,616,629\) miles per hour. No object with mass can travel at or faster than the speed of light. What interval represents the allowable speeds an object with mass may acheive?

10. Working with Intervals.

Let \(I = [0,7)\text{,}\) \(J = (-\infty,2]\text{,}\) and \(K = (3,+\infty)\text{.}\) Plot the intervals \(I\text{,}\) \(J\text{,}\) and \(K\) then describe \(I \cap J\text{,}\) \(I \cup J\text{,}\) \(I \cap K\text{,}\) \(I \cup K\text{,}\) \(J \cap K\text{,}\) and \(J \cup K\text{.}\)

Answer.

\(I \cap J = [0,2]\text{,}\) \(I \cup J = (-\infty,7)\text{,}\) \(I \cap K = (3,7)\text{,}\) \(I \cup K = [0,+\infty)\text{,}\) \(J \cap K = \emptyset\text{,}\) and \(J \cup K = (-\infty,2]\cup(3,+\infty)\text{.}\)

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