Practice identifying rational and irrational numbers with this engaging hidden message puzzle. Show
The puzzle asks students to shade the irrational numbers. The problems are simple to complex. Students will need to simplify many of the expressions before knowing if they are rational or irrational. Once the puzzle is shaded correctly, it says, "3.14 is pi day." A great formative assessment and self-assessment for students. It is included as a printable PDF for in person learning and a Google Slides download for virtual or hybrid learning. You will receive a PDF of the one-page activity, a link to download the drag and drop Google Slides version, and the answer key. Common Core Standards: CCSS.Math.Content.8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. CCSS.Math.Content.HSN.RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
A rational number (\(\mathbb{Q}\)) is any number which can be written as: where \(a\) and \(b\) are integers and \(b \ne 0\). The following numbers are all rational numbers: We see that all numerators and all denominators are integers. This means that all integers are rational
numbers, because they can be written with a denominator of \(\text{1}\). Irrational numbers (\(\mathbb{Q}'\)) are numbers that cannot be written as a fraction with the numerator and denominator as integers. Examples of irrational numbers: These are not rational numbers, because either the numerator or the denominator is not an integer. All integers and fractions with integer numerators and non-zero integer denominators are rational numbers. Remember that when the denominator of a fraction is zero then the fraction is undefined. You can write any rational number as a decimal number but not all decimal numbers are rational numbers. These types of decimal numbers are rational numbers: Decimal numbers that end (or terminate). For example, the
fraction \(\frac{4}{10}\) can be written as \(\text{0,4}\). Decimal numbers that have a repeating single digit. For example, the fraction \(\frac{1}{3}\) can be written as \(\text{0,}\dot{3}\) or \(\text{0,}\overline{3}\). The dot and bar notations are equivalent and both represent recurring \(\text{3}\)'s, i.e. \(\text{0,}\dot{3} = \text{0,}\overline{3} = \text{0,333...}\). Decimal numbers that have a recurring pattern of multiple digits. For example, the fraction
\(\frac{2}{11}\) can also be written as \(\text{0,}\overline{18}\). The bar represents a recurring pattern of \(\text{1}\)'s and \(\text{8}\)'s, i.e. \(\text{0,}\overline{18} = \text{0,181818...}\). You may see a full stop instead of a comma used to indicate a decimal number. So the number \(\text{0,4}\) can also be written as 0.4 Notation: You can use a dot or a bar over the repeated digits to indicate that the decimal is a recurring decimal. If the bar
covers more than one digit, then all numbers beneath the bar are recurring. If you are asked to identify whether a number is rational or irrational, first write the number in decimal form. If the number terminates then it is rational. If it goes on forever, then look for a repeated pattern of digits. If there is no repeated pattern, then the number is irrational. When you write irrational numbers in decimal form, you may continue writing them for many, many decimal places. However,
this is not convenient and it is often necessary to round off. Rounding off an irrational number makes the number a rational number that approximates the irrational number. Which of the following are not rational numbers? \(\pi =\text{3,14159265358979323846264338327950288419716939937510...}\) \(\text{1,4}\) \(\text{1,618033989...}\) \(\text{100}\) \(\text{1,7373737373...}\) \(\text{0,}\overline{02}\)
Converting terminating decimals into rational numbers (EMA6)A decimal number has an integer part and a fractional part. For example, \(\text{10,589}\) has an integer part of \(\text{10}\) and a fractional part of \(\text{0,589}\) because \(10 + \text{0,589} = \text{10,589}\). Each digit after the decimal point is a fraction with a denominator in increasing powers of \(\text{10}\). For example:
This means that \begin{align*} \text{10,589} & = 10 + \frac{5}{10} + \frac{8}{100} + \frac{9}{\text{1 000}}\\ & = \frac{\text{10 000}}{\text{1 000}} + \frac{\text{500}}{\text{1 000}} + \frac{80}{\text{1 000}} + \frac{9}{\text{1 000}}\\ & = \frac{\text{10 589}}{\text{1 000}} \end{align*}The following two videos explain how to convert decimals into rational numbers. Part 1 Video: 2DBJ Part 2 Video: 2DBK Converting recurring decimals into rational numbers (EMA7)When the decimal is a recurring decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction. Worked example 2: Converting decimal numbers to fractionsWrite \(\text{0,}\dot{3}\) in the form \(\frac{a}{b}\) (where \(a\) and \(b\) are integers). Define an equation\[\text{Let } x = \text{0,33333...}\] Multiply by \(\text{10}\) on both sides\[10x = \text{3,33333...}\] Subtract the first equation from the second equation\[9x = 3\] Simplify\[x = \frac{3}{9} = \frac{1}{3}\] Worked example 3: Converting decimal numbers to fractionsWrite \(\text{5,}\dot{4}\dot{3}\dot{2}\) as a rational fraction. Define an equation\[x=\text{5,432432432...}\] Multiply by \(\text{1 000}\) on both sides\[\text{1 000}x=\text{5 432,432432432...}\] Subtract the first equation from the second equation\[\text{999}x = \text{5 427}\] Simplify\[x = \frac{\text{5 427}}{\text{999}} = \frac{\text{201}}{\text{37}} = \text{5}\frac{\text{16}}{\text{37}}\] In the first example, the decimal was multiplied by \(\text{10}\) and in the second example, the decimal was multiplied by \(\text{1 000}\). This is because there was only one digit recurring (i.e. \(\text{3}\)) in the first example, while there were three digits recurring (i.e. \(\text{432}\)) in the second example. In general, if you have one digit recurring, then multiply by \(\text{10}\). If you have two digits recurring, then multiply by \(\text{100}\). If you have three digits recurring, then multiply by \(\text{1 000}\) and so on. Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like \(\sqrt{2}=\text{1,4142135...}\) cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits and they do not terminate. Textbook Exercise 1.1 Where does the number \(-\frac{12}{3}\) belong in the diagram? First simplify the fraction: \(-\frac{12}{3} = -4\) \(-\text{4}\) is an integer, so it falls into the set \(\mathbb{Z}\). In the following list, there are two false statements and one true statement. Which of the statements is true?
Consider each option carefully:
So only (ii) is true. Where does the number \(- \frac{1}{2}\) belong in the diagram? \(- \frac{1}{2}\) 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 \(\mathbb{Z}\). In the following list, there are two false statements and one true statement. Which of the statements is true?
Consider each option carefully:
So only (ii) is true. \(-\sqrt{3}\) \(-\sqrt{3}\) has no minus sign under the square root (the minus is outside the root) and is not divided by zero, so it is real. \(\dfrac{0}{\sqrt{2}}\) \(\dfrac{0}{\sqrt{2}}\) has no minus sign under the square root (the minus is outside the root) and is not divided by zero, so it is real. \(\sqrt{-9}\) \(\sqrt{-9}\) has a minus sign under the square root so it is non-real. \(\dfrac{-\sqrt{7}}{0}\) \(\dfrac{-\sqrt{7}}{0}\) has division by zero so it is undefined. \(-\sqrt{-16}\) \(-\sqrt{-16}\) has a negative number under the square root so it is non-real. \(\sqrt{2}\) \(\sqrt{2}\) has no minus under the square root (the minus is outside the root), is not divided by zero, so it is real. \(-\frac{1}{3}\) \(-\frac{1}{3}\) is rational. A fraction of integers is a rational number. \(\text{0,651268962154862...}\) \(\text{0,651268962154862...}\) is irrational. It cannot be simplified to a fraction of integers. \(\dfrac{\sqrt{9}}{3}\) \(\dfrac{\sqrt{9}}{3}\) is rational, an integer, a whole number and a natural number. An integer is a rational number. \(\pi^{2}\) \(\pi^{2}\) is irrational. It cannot be simplified to a fraction of integers. \(\pi^4\) \(\pi^4\) is irrational. It cannot be simplified to a fraction of integers. \(\sqrt[3]{19}\) \(\sqrt[3]{19}\) is irrational. It cannot be simplified to a fraction of integers. \(\left( \sqrt[3]{1} \right)^7\) \(\left( \sqrt[3]{1} \right)^7\) is rational, an integer, a whole number and a natural number. It can be written as an integer. \(\pi + 3\) \(\pi\) 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. \(\pi + \text{0,858408346}\) \(\pi\) 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. \(\dfrac{5}{6}\) \(\frac{5}{6}\) is rational. \(\dfrac{a}{3}\) Since \(a\) is an integer, \(\frac{a}{3}\) is rational. \(\dfrac{-2}{b}\) Since \(b\) is an integer, \(\frac{-2}{b}\) is rational. Note that \(b\) cannot be \(\text{0}\) as that makes the fraction undefined. \(\dfrac{1}{c}\) Since \(c\) is irrational, \(\frac{1}{c}\) is irrational. \(\text{1}\) \(\frac{a}{14} = \frac{1}{14}\) is rational. \(-\text{10}\) \(\frac{a}{14} = \frac{-10}{14}\) is rational. \(\sqrt{2}\) \(\frac{a}{14} = \frac{\sqrt{2}}{14}\) is irrational. \(\text{2,1}\) \(\frac{a}{14} = \frac{\text{2,1}}{14}\) is rational. natural numbers 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. irrational 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. non-real numbers Any number that is a square root of a negative number is non-real. Therefore only \(\sqrt{-1}\) is non-real. rational 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 \(-3 \; ; \; 0 \; ; \; -8\frac{4}{5} \; ; \; \frac{22}{7} \; ; \; 7 \; ; \; \text{1,}\overline{34} \; ; \; 9\frac{7}{10} \; ; \; 11\) are all rational numbers. integers 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. undefined Any fraction divided by \(\text{0}\) is undefined. Therefore only \(\frac{14}{0}\) is undefined. \(\text{1,1}\dot{5}\)
\(\text{2,121314...}\)
\(\text{1,242244246...}\)
\(\text{3,324354...}\)
\(\text{3,3243}\dot{5}\dot{4}\)
\(\text{0,1}\) \(\text{0,1}=\frac{1}{10}\) \(\text{0,12}\) \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*} \(\text{0,58}\) \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*} \(\text{0,2589}\) \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*} \(\text{0,1111111...}\) We see that only the digit \(\text{1}\) is repeated and so we can write this as: \(\text{0,}\dot{1}\). \(\text{0,1212121212...}\) There is a repeating pattern of \(\text{12}\) and so we can write this number as: \(\text{0,}\overline{12}\) \(\text{0,123123123123...}\) There is a repeating pattern of \(\text{123}\) and so we can write this number as: \(\text{0,}\overline{123}\) \(\text{0,11414541454145...}\) The pattern 4145 repeats and so we can write this number as: \(\text{0,11}\overline{4145}\). \(\dfrac{\text{25}}{\text{45}}\) \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*} \(\dfrac{\text{10}}{\text{18}}\) \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*} \(\dfrac{\text{7}}{\text{33}}\) \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*} \(\dfrac{2}{3}\) \begin{align*} \frac{2}{3} &= 2\left(\frac{1}{3}\right) \\ &= 2(\text{0,333333...}) \\ &= \text{0,666666...} \\ &= \text{0,}\dot{6} \end{align*} \(1\dfrac{3}{11}\) \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*} \(4\dfrac{5}{6}\) \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*} \(2\dfrac{1}{9}\) \begin{align*} 2\frac{1}{9} &= 2 + \text{0,1111111...} \\ & = \text{2,}\dot{1} \end{align*} \(\text{0,}\dot{5}\) \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*} \(\text{0,6}\dot{3}\) \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*} \(\text{0,}\dot{4}\) \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*} \(\text{5,}\overline{31}\) \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*} \(\text{4,}\overline{\text{93}}\) \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*} \(\text{3,}\overline{\text{93}}\) \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*} What are rational and irrational numbers Class 7?A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. But an irrational number cannot be written in the form of simple fractions. ⅔ is an example of a rational number whereas √2 is an irrational number.
Why is the number 22 7 irrational?Here, the given number, 22⁄7 is a fraction of two integers and has recurring decimal value (3.142857). Hence, it is a rational number.
Is 5'1 25 rational or irrational and why?As a ratio of 2 real numbers that are going to give us a decimal that is ending so a couple of ways to look at 5 and one twenty fifth to show this 5 and one twenty fifth would be 5 points and whatever are decimal from 1. Divided by 25 is, and that would be 5.04, so this is a rational number and again.
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