Rational numbers

Among negative numbers, the one closer to zero is greater. ( \frac{-1}{6} ) is closer to zero.

To change the denominator from \(8\) to \(16\), multiply it by \(2\). To keep the fraction equivalent, the numerator must also be multiplied by \(2\): \(5\times2=10\). Therefore, the correct answer is \(10\). Exam tip: In equivalent fractions, multiply or divide both the numerator and denominator by the same non-zero number.

The middle value is ( \frac{\frac{2}{3}+\frac{4}{5}}{2}=\frac{11}{15} ). You can take the average to find a number between two numbers.

The greatest common divisor of 10 and 25 is 5. Dividing both the numerator and denominator by 5 gives \(\frac{10\div5}{25\div5}=\frac{2}{5}\). Option A, \(\frac{1}{5}\), is incorrect because the numerator and denominator must both be divided by the same common factor. Exam tip: To reduce a fraction to its simplest form, divide its numerator and denominator by their greatest common divisor.

Since \(16\) is a perfect square, \(\sqrt{16}=4\). The number \(4\) can be written as \(\frac{4}{1}\), so it is rational. Therefore, option B is correct. It is not irrational because its value is an exact integer, and it is certainly a real, defined number. Exam tip: the square root of a positive perfect square is an integer and hence rational.

A repeating decimal is always rational because it can be expressed as the ratio of two integers. Here, \(2.\overline{3}=2.333\ldots=\frac{7}{3}\), so it is a rational real number. Irrational decimals are non-terminating and non-repeating. Exam tip: identify a repeating decimal as rational and a non-terminating, non-repeating decimal as irrational.

Option B is correct because −5 is an integer, and every integer can be written as \(\frac{p}{q}\) with \(q\neq 0\), so it is rational. Also, 0.4 is a terminating decimal: \(0.4=\frac{4}{10}=\frac{2}{5}\), so it is rational. In contrast, \(\sqrt{3}\), \(\sqrt{7}\), and \(\sqrt{11}\) are irrational, and \(\pi\) is also irrational. Exam tip: Every integer and every terminating or recurring decimal is a rational number.

In \(1.232323\ldots\), the block \(23\) repeats indefinitely, so it is a recurring decimal. Every recurring decimal can be expressed as a fraction; in fact, \(1.232323\ldots=\frac{122}{99}\), so it is rational. It is not irrational because an infinite decimal is irrational only when it is non-terminating and non-recurring. Exam tip: terminating and recurring decimals are always rational.

The block 45 repeats after the decimal point, so this is a recurring decimal. Every recurring decimal is rational; in fact, \(3.454545\ldots=3+\frac{45}{99}=\frac{38}{11}\). It is not an integer because \(\frac{38}{11}\) is not a whole number. Exam tip: Both terminating and recurring decimals are rational numbers.

In 0.272727\ldots, the block 27 repeats continuously, so it is a non-terminating recurring decimal and therefore a rational number. In fact, \(0.272727\ldots=\frac{27}{99}=\frac{3}{11}\). It is not a terminating decimal and it is not an integer. Exam tip: A decimal that terminates or repeats a fixed block of digits is rational.

\(\frac{5}{8}\) is a ratio of two integers, and \(-7\) can be written as \(\frac{-7}{1}\); therefore, both are rational numbers. \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{11}\), and \(\pi\) are irrational, while \(0.101001\ldots\) is a non-terminating, non-repeating decimal. Exam tip: A number is rational if it can be written as \(\frac{p}{q}\), where \(p,q\) are integers and \(q\ne0\).

-12 is an integer. It can be written as \(\frac{-12}{1}\), so it is also a rational number. \(\frac{3}{7}\) is rational but not an integer, while \(\sqrt{15}\) and \(\pi\) are irrational. Exam tip: To check whether an integer is rational, write it as a fraction with denominator 1.

\(\frac{2}{7}\) is a ratio of integers, and \(-4\) can be written as \(\frac{-4}{1}\); therefore, both numbers in option B are rational. In option D, \(\sqrt{49}=7\) is rational, but \(\sqrt{2}\) is irrational. Exam tip: Write an integer with denominator 1 to check whether it is rational.

\(\sqrt{81}=9\), and 9 can be written as \(\frac{9}{1}\); therefore, it is a rational number. \(\sqrt{21}\) and \(\sqrt{11}\) are not square roots of perfect squares, so they are irrational; \(\pi\) is also irrational. Exam tip: The square root of a perfect square is always an integer and hence rational.

\(\sqrt{5}\times\sqrt{5}=(\sqrt{5})^2=5\), and 5 is a rational number. Option A gives \(\sqrt{6}\), while option C gives \(2\sqrt{7}\); both are irrational. Option D remains \(\pi\), which is irrational. Exam tip: the product of two identical square roots equals the number inside the radical.

\( -\frac{1}{3} = -0.3333\ldots \). On the number line, \( -0.3333\ldots \) lies to the right of \( -0.34 \), so \( -\frac{1}{3} > -0.34 \), making option B correct. Option A is a common mistake caused by comparing the decimal digits without considering that, for negative numbers, the number closer to zero is greater. Exam tip: convert the fraction to a decimal or write both numbers to the same number of decimal places before comparing them.

\(22/7\) is the ratio of two integers with a non-zero denominator, so it is a rational number. Its value is very close to π, but it is not exactly equal to π. Therefore, option A is correct; option B is wrong because \(22/7\) is rational, and option D is wrong because \(22/7 = 3\frac{1}{7}\), which is not an integer. Exam tip: Before calling a fraction an integer, check whether its numerator is exactly divisible by its denominator.

In option C, \((\sqrt{13})^2=13\), and 13 is an integer, so it is rational. In the other options, an irrational square root is added to or subtracted from a rational number, so the result remains irrational. Exam tip: squaring the square root of a number removes the radical and gives the number itself.

The governing concept is the definition of coprime, or relatively prime, numbers. Two integers are called coprime when their only positive common factor is 1. Therefore their highest common factor, also called greatest common divisor, is 1. Option C is correct. The numbers themselves need not be prime: for example, 8 and 15 are both composite in the broad sense of ordinary integers, yet their common factors include only 1, so HCF(8,15) = 1. A value of 2 or 3 would mean that both numbers share that factor, which would contradict coprimality. Zero is not the HCF of ordinary nonzero coprime numbers. This property is important in rational-number proofs because a fraction in lowest form has a numerator and denominator whose HCF is 1.

\(\sqrt{2}\times\sqrt{8}=\sqrt{2\times8}=\sqrt{16}=4\). Since \(4=\frac{4}{1}\), it is a rational number. Although \(\sqrt{2}\) and \(\sqrt{8}\) are individually irrational, the product of two irrational numbers is not always irrational. Also, \(4\) is neither negative nor non-real. Exam tip: When multiplying square roots, multiply the radicands first and look for a perfect square.