Testinar Grade 8 math

Rational and Irrational Numbers

Learn to reason about and solve Grade 8 problems involving Rational and Irrational Numbers.

CH01-01 Number Systems and Financial Reasoning

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Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

Based on the number line, which statement is true?
1. \(\sqrt{5}\) is rational
2. \(\sqrt{5}\) is between \(1\) and \(2\)
3. \(\sqrt{5} = 2.5\)
4. \(\sqrt{5} \approx 2.24\)
Which statement is true?
1. \(\sqrt{3} > 2\)
2. \(\sqrt{3} = 1.73\)
3. \(\sqrt{3} < 2\)
4. \(\sqrt{3}\) is rational
Which decimal is rational?
1. \(0.123456\ldots\) (non-repeating)
2. \(\pi\)
3. \(0.454545\ldots\) (repeating)
4. \(e\) (Euler's number)
Which number should be plotted as irrational (height 0)?
1. \(0.5\)
2. \(2\)
3. \(3.\overline{3}\)
4. \(\sqrt{5}\)
Which is true based on the number line?
1. \(\sqrt{6} > \pi\)
2. \(\sqrt{6} + \pi\) is rational
3. \(\sqrt{6} = \pi\)
4. \(\sqrt{6} < \pi\)
The decimal \(1.2\overline{34}\) (where \(34\) repeats) is:
1. Irrational
2. Equal to \(\pi\)
3. Neither rational nor irrational
4. Rational
Which number should be placed in the Rational section?
1. \(\sqrt{13}\)
2. \(2 + \sqrt{5}\)
3. \(-\frac{9}{2}\)
4. \(\sqrt{\pi}\)
Which statement is true?
1. All square roots are irrational
2. All integers are rational
3. Some repeating decimals are irrational
4. Every irrational number can be written as a fraction

Teacher and parent reference

Answer key and explanations

D. \(\sqrt{5} \approx 2.24\)
The number line shows \(\sqrt{5}\) positioned between \(2\) and \(3\), closer to \(2.2\). Since \(2.2^2 = 4.84\) and \(2.24^2 \approx 5.02\), we have \(\sqrt{5} \approx 2.24\). Also, \(\sqrt{5}\) is irrational.
C. \(\sqrt{3} < 2\)
From the number line, \(\sqrt{3} \approx 1.73\) is positioned to the left of \(2\), so \(\sqrt{3} < 2\). We can verify: \(1.73^2 = 2.9929 < 4 = 2^2\). Also, \(\sqrt{3}\) is irrational.
C. \(0.454545\ldots\) (repeating)
The repeating decimal \(0.454545\ldots = 0.\overline{45} = \frac{45}{99} = \frac{5}{11}\), which is rational. Non-repeating non-terminating decimals like \(\pi\) and \(e\) are irrational.
D. \(\sqrt{5}\)
Only \(\sqrt{5}\) is irrational (a square root of a non-perfect square). The others are rational: \(0.5 = \frac{1}{2}\), \(2\) is an integer, and \(3.\overline{3} = \frac{10}{3}\).
D. \(\sqrt{6} < \pi\)
From the number line, \(\sqrt{6} \approx 2.45\) is to the left of \(\pi \approx 3.14\). Therefore \(\sqrt{6} < \pi\). Both are irrational.
D. Rational
Any decimal that eventually repeats is rational. In fact, \(1.2\overline{34} = 1.2343434\ldots = \frac{611}{495}\), so it can be written exactly as a fraction.
C. \(-\frac{9}{2}\)
Only \(-\frac{9}{2}\) is rational (a fraction of two integers). The others are irrational: \(\sqrt{13}\) is a non-perfect-square root, \(2 + \sqrt{5}\) is a sum involving an irrational, and \(\sqrt{\pi}\) is the root of an irrational.
B. All integers are rational
Any integer \(n\) can be written as \(\frac{n}{1}\), so every integer is rational. Choice A is false because square roots of perfect squares, such as \(\sqrt{25}\), are rational. Choice C is false because repeating decimals are always rational. Choice D is false because numbers that can be written as a fraction are rational, not irrational.