4. Which set does it belong to?
Say which set each of the following numbers belongs to.
a) $\displaystyle -\dfrac{2}{5}$
It is a fraction with an integer numerator and denominator, and it is not an integer: $-2 : 5 = -0{,}4$.
$-\dfrac{2}{5} \in \mathbb{Q}$, but it is neither an integer nor a natural number.
b) $\displaystyle \sqrt{4}$
First work out the root:
$\sqrt{4} = 2 \in \mathbb{N}$ (so it is also in $\mathbb{Z}$ and $\mathbb{Q}$).
c) $\displaystyle 2{,}0\overline{3}$
It is a mixed recurring decimal: the non-repeating part is $0$ and the period is $3$. Recurring decimals can be written as a fraction:
$2{,}0\overline{3} \in \mathbb{Q}$, but it is not an integer.
d) $\displaystyle -6$
It is a negative integer. It can also be written as a fraction: $-6 = -\dfrac{6}{1}$.
$-6 \in \mathbb{Z}$ (and $\mathbb{Q}$), but it is not a natural number.
e) $\displaystyle \dfrac{4}{9}$
It is a fraction that does not give an integer:
$\dfrac{4}{9} \in \mathbb{Q}$, but it is not an integer.
f) $\displaystyle 0{,}71$
It is a terminating decimal, and it can be written as a fraction:
$0{,}71 \in \mathbb{Q}$, but it is not an integer.
g) $\displaystyle \dfrac{12}{3}$
The division is exact:
$\dfrac{12}{3} = 4 \in \mathbb{N}$ (and $\mathbb{Z}$ and $\mathbb{Q}$).
Summary
Since $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$, a natural number is also an integer and a rational number. For each one we give the smallest set.
| Number | $\mathbb{N}$ | $\mathbb{Z}$ | $\mathbb{Q}$ |
|---|---|---|---|
| $-\dfrac{2}{5}$ | — | — | ✓ |
| $\sqrt{4}$ | ✓ | ✓ | ✓ |
| $2{,}0\overline{3}$ | — | — | ✓ |
| $-6$ | — | ✓ | ✓ |
| $\dfrac{4}{9}$ | — | — | ✓ |
| $0{,}71$ | — | — | ✓ |
| $\dfrac{12}{3}$ | ✓ | ✓ | ✓ |
5. True or false
Say whether the following statements are true or false and, if false, correct them in your exercise book.
a) All integers are rational numbers.
Any integer can be written as a fraction with denominator $1$:
True.
b) A fraction is the quotient of two natural numbers.
The numerator and the denominator can be negative, and $0$ can be the numerator. For example, $-\dfrac{3}{4}$ is a fraction and $-3$ is not a natural number. The only condition is that the denominator is not $0$.
False. Corrected: “A fraction is the quotient of two integers, with a denominator other than $0$”.
c) Recurring decimals are rational numbers.
Every recurring decimal, pure or mixed, can be written as a fraction. For example:
True.
2. A company's profit
A company has fixed monthly costs of 15 000 € and daily costs of 350 € from making its product. If in a given month (30 days) the company sold 250 units of this product at 70 € each, what is its monthly profit?
See the solution
Daily costs for the whole month:
Total costs for the month:
Income from sales:
Profit is what comes in minus what goes out:
The profit is $-8\,000$ €: this month the company loses $8\,000$ €. A negative profit is a loss, which is why we need integers.
Change one of the figures, solve it again and answer the question. What differences do you find? Does the result make sense? Justify your answers.
One example
There are many ways to do it. For example, if the company sells 400 units instead of 250, the costs stay the same:
Now the profit is positive: the company makes money. The result makes sense: the more units it sells, the more it takes in, and the costs do not change.
What is the smallest number of units it must sell so as not to lose money?
With 364 units it still loses money ($364 \cdot 70 = 25\,480$ €); from 365 units onwards it makes a profit.