1.3 Rational and irrational numbers

First simplify the fraction: \(-\frac{12}{3} = -4\)

\(-\text{4}\) is an integer, so it falls into the set \(\mathbb{Z}\).

Consider each option carefully:

1. There are integers which do not fall into the natural numbers (all negative numbers), so this is false.
2. The natural numbers are \(\left\{1; 2; 3; \ldots\right\}\) and whole numbers are \(\left\{0; 1; 2; 3; \ldots\right\}\) (the circle \(\mathbb{N}\) is inside \(\mathbb{N}_{0}\)) so if a number is a natural number it must be a whole number. This is true.
3. Whole numbers \(\left\{0; 1; 2; 3; \ldots\right\}\) only go up in steps of 1, so there cannot be any decimal numbers in the whole numbers, making this false.

So only (ii) is true.

\(- \frac{1}{2}\) is in its simplest form, therefore it is not in \(\mathbb{N}\), \(\mathbb{N}_0\) or \(\mathbb{Z}\). It is in the space between the rectangle and \(\mathbb{Z}\).

Consider each option carefully:

1. There are integers which do not fall into the natural numbers (all negative numbers), so this is false.
2. The integers are \(\left\{\ldots; -3; -2; -1; 0; 1; 2; 3; \ldots\right\}\) and the whole numbers are \(\left\{0; 1; 2; 3; \ldots\right\}\) (the circle \(\mathbb{Z}\) is inside \(\mathbb{N}_{0}\)) so if a number is an integer it must be a whole number. This is true.
3. Whole numbers \(\left\{0; 1; 2; 3; 4; \ldots \right\}\) only go up in steps of 1, so there cannot be any decimal numbers in the whole numbers, making this false.

So only (ii) is true.

\(-\sqrt{3}\) has no minus sign under the square root (the minus is outside the root) and is not divided by zero, so it is real.

\(\dfrac{0}{\sqrt{2}}\) has no minus sign under the square root (the minus is outside the root) and is not divided by zero, so it is real.

\(\sqrt{-9}\) has a minus sign under the square root so it is non-real.

\(\dfrac{-\sqrt{7}}{0}\) has division by zero so it is undefined.

\(-\sqrt{-16}\) has a negative number under the square root so it is non-real.

\(\sqrt{2}\) has no minus under the square root (the minus is outside the root), is not divided by zero, so it is real.

\(-\frac{1}{3}\) is rational. A fraction of integers is a rational number.

\(\text{0,651268962154862...}\) is irrational. It cannot be simplified to a fraction of integers.

\(\dfrac{\sqrt{9}}{3}\) is rational, an integer, a whole number and a natural number. An integer is a rational number.

\(\pi^{2}\) is irrational. It cannot be simplified to a fraction of integers.

\(\pi^4\) is irrational. It cannot be simplified to a fraction of integers.

\(\sqrt[3]{19}\) is irrational. It cannot be simplified to a fraction of integers.

\(\left( \sqrt[3]{1} \right)^7\) is rational, an integer, a whole number and a natural number. It can be written as an integer.

\(\pi\) is irrational. \(\text{3}\) is rational (it is an integer). Any rational number added to any irrational number is irrational.

Therefore \(\pi + 3\) is irrational.

\(\pi\) is irrational. \(\text{0,858408346}\) is rational (it is a terminating decimal). Any rational number added to any irrational number is irrational.

Therefore \(\pi + \text{0,858408346}\) is irrational.

\(\frac{5}{6}\) is rational.

Since \(a\) is an integer, \(\frac{a}{3}\) is rational.

Since \(b\) is an integer, \(\frac{-2}{b}\) is rational.

Note that \(b\) cannot be \(\text{0}\) as that makes the fraction undefined.

Since \(c\) is irrational, \(\frac{1}{c}\) is irrational.

\(\frac{a}{14} = \frac{1}{14}\) is rational.

\(\frac{a}{14} = \frac{-10}{14}\) is rational.

\(\frac{a}{14} = \frac{\sqrt{2}}{14}\) is irrational.

\(\frac{a}{14} = \frac{\text{2,1}}{14}\) is rational.

Check which of the numbers are in the set \(\left\{1; 2; 3; 4;\ldots\right\}\). Therefore \(\text{7}\) and \(\text{11}\) are natural numbers.

Remember that rational numbers can be written as \(\frac{a}{b}\) where \(a\) and \(b\) are integers. Also remember that rational numbers include terminating decimal numbers. Therefore \(-\sqrt{8} \; ; \; \text{3,3231089...} \; ; \; 3+\sqrt{2} \; ; \; \pi\) are all irrational.

Any number that is a square root of a negative number is non-real. Therefore only \(\sqrt{-1}\) is non-real.

Remember that rational numbers can be written as \(\frac{a}{b}\) where \(a\) and \(b\) are integers. Also remember that rational numbers include terminating decimal numbers. Therefore \(-3 \; ; \; 0 \; ; \; -8\frac{4}{5} \; ; \; \frac{22}{7} \; ; \; 7 \; ; \; \text{1,}\overline{34} \; ; \; 9\frac{7}{10} \; ; \; 11\) are all rational numbers.

Check which of the numbers are in the set \(\left\{\ldots; -3; -2; -1; 0; 1; 2; 3;\ldots\right\}\). Therefore \(-3 \; ; \; 7 \; ; \; 11\) are integers.

Any fraction divided by \(\text{0}\) is undefined. Therefore only \(\frac{14}{0}\) is undefined.

• Since there is a dot over the 5 we know that the 5 repeats. The next three digits are: \(555\)

• Rational, there is a repeating pattern of digits.

• The number does not terminate (this is shown by the \(\ldots\)). There is also no indication of a repeating pattern of digits since there is not dot or bar over any of the numbers. The next three digits could be any numbers.

  Note that while it looks like there is a pattern in the digits we do not know if this pattern continues on.

• Irrational, there is no repeating pattern.

• The number does not terminate (this is shown by the \(\ldots\)). There is also no indication of a repeating pattern of digits since there is not dot or bar over any of the numbers. The next three digits could be any numbers.

  Note that while it looks like there is a pattern in the digits we do not know if this pattern continues on.

• Irrational, there is no repeating pattern.

• The number does not terminate (this is shown by the \(\ldots\)). There is also no indication of a repeating pattern of digits since there is not dot or bar over any of the numbers. The next three digits could be any numbers.

  Note that while it looks like there is a pattern in the digits we do not know if this pattern continues on.

• Irrational, there is no repeating pattern.

• Since there is a dot over both the 5 and the 4 we know that the pattern 54 repeats. The next three digits are: \(545\)

• Rational, there is a repeating pattern.

\(\text{0,1}=\frac{1}{10}\)

\begin{align*} \text{0,12} & = \frac{1}{10} + \frac{2}{100} \\ & = \frac{10}{100} + \frac{2}{100} \\ & = \frac{12}{100} \\ & = \frac{3}{25} \end{align*}

\begin{align*} \text{0,58} & = \frac{5}{10} + \frac{8}{100} \\ & = \frac{50}{100} + \frac{8}{100} \\ & = \frac{58}{100} \\ & = \frac{29}{50} \end{align*}

\begin{align*} \text{0,2589} & = \frac{2}{10} + \frac{5}{100} + \frac{8}{\text{1 000}} + \frac{9}{\text{10 000}} \\ & = \frac{\text{2 000}}{\text{10 000}} + \frac{500}{\text{10 000}} + \frac{80}{\text{10 000}} + \frac{9}{\text{10 000}} \\ & = \frac{\text{2 589}}{\text{10 000}} \end{align*}

We see that only the digit \(\text{1}\) is repeated and so we can write this as: \(\text{0,}\dot{1}\).

There is a repeating pattern of \(\text{12}\) and so we can write this number as: \(\text{0,}\overline{12}\)

There is a repeating pattern of \(\text{123}\) and so we can write this number as: \(\text{0,}\overline{123}\)

The pattern 4145 repeats and so we can write this number as: \(\text{0,11}\overline{4145}\).

\begin{align*} \text{45}|\overline{\text{25,}\text{0 000}} &= \text{0} \text{ remainder } \text{25} \\ \text{45}|\overline{\text{25,}^{25}\text{0 000}} &= \text{5} \text{ remainder } \text{25} \\ \text{45}|\overline{\text{25,}^{25}\text{0}^{25}\text{000}} &= \text{5} \text{ remainder } \text{25} \\ \text{45}|\overline{\text{25,}^{25}\text{0}^{25}\text{0}^{25}\text{00}} &= \text{5} \text{ remainder } \text{25} \\ \frac{\text{25}}{\text{45}} &= \text{0,}\text{5 555} \ldots \\ &= \text{0,}\dot{\text{5}} \end{align*}

\begin{align*} \text{18}|\overline{\text{10,0000}} &= \text{0} \text{ remainder } \text{10} \\ \text{18}|\overline{\text{10,}^{10}\text{0 000}} &= \text{5} \text{ remainder } \text{10} \\ \text{18}|\overline{\text{10,}^{10}\text{0}^{10}\text{000}} &= \text{5} \text{ remainder } \text{10} \\ \text{18}|\overline{\text{10,}^{10}\text{0}^{10}\text{0}^{10}\text{00}} &= \text{5} \text{ remainder } \text{10} \\ \frac{\text{10}}{\text{18}} &= \text{0,}\text{5 555} \ldots \\ &= \text{0,}\dot{\text{5}} \end{align*}

\begin{align*} \text{33}|\overline{\text{7,0000}} &= \text{0} \text{ remainder } \text{7} \\ \text{33}|\overline{\text{7,}^7\text{0 000}} &= \text{2} \text{ remainder } \text{4} \\ \text{33}|\overline{\text{7,}^4\text{0}^4\text{000}} &= \text{1} \text{ remainder } \text{7} \\ \text{33}|\overline{\text{7,}^7\text{0}^4\text{0}^7\text{00}} &= \text{2} \text{ remainder } \text{4} \\ \frac{\text{7}}{\text{33}} &= \text{0,}\text{2 121} \ldots \\ &= \text{0,}\dot{\text{2}}\dot{\text{1}} \end{align*}

\begin{align*} \frac{2}{3} &= 2\left(\frac{1}{3}\right) \\ &= 2(\text{0,333333...}) \\ &= \text{0,666666...} \\ &= \text{0,}\dot{6} \end{align*}

\begin{align*} 1\frac{3}{11} &= 1 + 3\left(\frac{1}{11}\right) \\ &= 1 + 3(\text{0,090909...})\\ &= 1 + \text{0,27272727...} \\ &= \text{1,}\overline{27} \end{align*}

\begin{align*} 4\frac{5}{6} &= 4 + 5\left(\frac{1}{6}\right) \\ &= 4+ 5(\text{0,1666666...}) \\ &= 4 + \text{0,833333...} \\ & = \text{4,8}\dot{3} \end{align*}

\begin{align*} 2\frac{1}{9} &= 2 + \text{0,1111111...} \\ & = \text{2,}\dot{1} \end{align*}

\begin{align*} x&=\text{0,55555...} \text{ and} \\ 10x &= \text{5,55555...} \\ 10x - x &= (\text{5,55555...}) - (\text{0,55555...}) \\ \text{9}x &= \text{5} \\ \therefore x&=\frac{5}{9} \end{align*}

\begin{align*} 10x &= \text{6,3333...} \text{ and}\\ 100x &= \text{63,3333...} \\ 100x - 10x &= (\text{63,3333...}) - (\text{6,3333...}) \\ \text{99}x &= \text{57} \\ \therefore x&=\frac{57}{90} \end{align*}

\begin{align*} x & = \text{0,4444...} \text{ and} \\ \text{10}x & = \text{4,4444...} \\ 10x - x &= (\text{4,4444...}) - (\text{0,4444...}) \\ \text{9}x &= \text{4} \\ \therefore x & = \frac{\text{4}}{\text{9}} \end{align*}

\begin{align*} x &= \text{5,313131...} \text{ and} \\ 100x & = \text{531,313131...}\\ 100x - x &= (\text{531,313131...}) - (\text{5,313131...}) \\ \text{99}x &= \text{526} \\ \therefore x&=\frac{526}{99} \end{align*}

\begin{align*} x &= \text{4,939393...} \text{ and} \\ 100x &= \text{493,939393...} \\ 100x - x &= (\text{493,939393...}) - (\text{4,939393...}) \\ \text{99}x &= \text{489} \\ \therefore x &= \frac{\text{163}}{\text{33}} \end{align*}

\begin{align*} x &= \text{3,939393...} \text{ and} \\ 100x &= \text{393,939393...} \\ 100x - x &= (\text{393,939393...}) - (\text{3,939393...}) \\ \text{99}x &= \text{390} \\ \therefore x &= \frac{\text{130}}{\text{33}} \end{align*}