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1r BTL CCSS · Real numbers Exercises 16 Sep 2026

4. Homework solutions (16 Sep)

Solutions to the rational-and-irrational exercises and to the first intervals and neighbourhoods.

Rational and irrational numbers

1 Rational or irrational
Say whether these numbers are rational or irrational: $\sqrt{6}$; $2{,}234901$; $5{,}0\overline{4}$; $\pi$; $-5$.

Rational, because they can be written as a fraction:

$$2{,}234901 \qquad 5{,}0\overline{4} \qquad -5$$

The first is a terminating decimal, the second is recurring and the third is an integer ($-5 = \tfrac{-5}{1}$).

Irrational, because they have infinitely many decimals with no repeating block:

$$\sqrt{6} \qquad \pi$$

$\sqrt{6}$ is not exact: $6$ is not a perfect square.

2 Placing the numbers in the diagram

Place these numbers in the $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ diagram: $\;0{,}\overline{92}$; $\sqrt[3]{-8}$; $\sqrt{2}$; $\tfrac{8}{5}$; $-7$; $\tfrac{12}{25}$; $-1{,}\overline{9}$; $\pi$; $3$; $\dfrac{2-\sqrt{2}}{3}$; $\sqrt[4]{16}$; $11{,}3$; $0{,}1010010001\ldots$; $2{,}3\overline{5}$.

See the completed diagram
$\mathbb{N}$$3$$\sqrt[4]{16}$
$\mathbb{Z}$$\sqrt[3]{-8}$$-7$$-1{,}\overline{9}$
$\mathbb{Q}$$0{,}\overline{92}$$\tfrac{8}{5}$$\tfrac{12}{25}$$11{,}3$$2{,}3\overline{5}$
$\mathbb{R}$$\sqrt{2}$$\pi$$\dfrac{2-\sqrt{2}}{3}$$0{,}1010010001\ldots$

Every number goes into the smallest set that contains it. The three tricky ones:

→ $\sqrt[4]{16} = 2$, which is natural, not irrational.

→ $\sqrt[3]{-8} = -2$: with an odd index the root of a negative number exists, and here it comes out an integer.

→ $-1{,}\overline{9} = -2$. A recurring $9$ does not approach $-2$: it equals $-2$, so the number is an integer.

Only the irrationals are left in the $\mathbb{R}$ band: $\sqrt{2}$, $\pi$, $\dfrac{2-\sqrt{2}}{3}$ (an irrational added to and divided by rationals stays irrational) and $0{,}1010010001\ldots$, whose infinitely many decimals never repeat.

Intervals and neighbourhoods

Neighbourhood

The neighbourhood with centre $a$ and radius $r$ is the open interval running from $a - r$ to $a + r$:

$$E(a,\, r) = (a - r,\; a + r)$$
1 Represent the intervals

For each interval, the set notation and the picture on the real line.

a) $[-3,\, 4)$
$$\{-3 \le x < 4 \mid x \in \mathbb{R}\}$$
−34
b) $(-10,\, -5)$
$$\{-10 < x < -5 \mid x \in \mathbb{R}\}$$
−10−5
c) $[-100,\, 200]$
$$\{-100 \le x \le 200 \mid x \in \mathbb{R}\}$$
−100200
e) $E(0,\, 2)$

Centre $0$ and radius $2$, that is, from $0-2$ to $0+2$:

$$E(0,\, 2) = (-2,\, 2) = \{-2 < x < 2 \mid x \in \mathbb{R}\}$$
−22

Parts d, f and g of the exercise are still to be done.