What Are Rational Numbers? Definition, Examples and Effective Ways to Learn Them (Grade 7 Maths) Many Grade 7 students get their calculations right but still go wrong when asked: “Is this a rational number?” The usual reason is that they mix up rational numbers with fractions, or confuse rational numbers with irrational numbers. This is one of the foundational concepts in the Grade 7 Maths curriculum, and it directly affects later lessons on real numbers. This article explains rational numbers in the simplest way possible: the definition, how to identify them, worked examples, how to represent and compare them on a number line, and common mistakes to avoid. What is a rational number? Definition of a rational number A rational number is a number that can be written as a fraction a/b, where a and b are integers and b is not 0. In other words, if a number can be expressed as “an integer divided by a non-zero integer”, it is a rational number. For example: 5, −3, 0.5 and 2/7 are all rational numbers, because each of them can be written as a fraction. The symbol ℚ for the set of rational numbers The set of all rational numbers is denoted by ℚ. An important point students often overlook: many different fractions can represent the same rational number. For example, 1/2, 2/4 and 3/6 are all the same rational number. So “rational number” and “fraction” are not exactly the same thing. A rational number is a value; a fraction is just one way of writing that value. Quick reminder: Rational number = a number that can be written as a/b (a, b ∈ ℤ, b ≠ 0). The set of rational numbers is denoted by ℚ. “Rational numbers”: the term in international curricula In English, these numbers are called rational numbers, and the set ℚ is read as “the set of rational numbers”. In Vietnamese, the term is số hữu tỉ. The word “rational” comes from “ratio”, which matches the very nature of a rational number: “the ratio of two integers”. Students in bilingual programmes will meet this concept with the same definition, just in a different language. Understanding both the Vietnamese and English terms helps them move easily between the two curricula. Which numbers are rational? To check whether a number is rational, we see whether it can be written in the form a/b. Integers and fractions are rational numbers Every integer is a rational number. The reason: an integer n can always be written as n/1. For example: 4 = 4/1; −7 = −7/1; 0 = 0/1. Every fraction whose numerator and denominator are integers (with a non-zero denominator) is also a rational number. This is the standard form of a rational number. Terminating and recurring decimals A terminating decimal is a rational number, because it can always be converted into a fraction. For example: 0.25 = 25/100 = 1/4; and 1.5 = 3/2. A recurring decimal is also a rational number. “Recurring” means there is a group of digits that repeats forever. For example: 0.(3) = 1/3; and 0.(6) = 2/3. Converting 0.(3) into a fraction is quick. Let x = 0.(3). Multiply both sides by 10 to get 10x = 3.(3). Subtract one equation from the other: 10x − x = 3, so 9x = 3, which gives x = 1/3. On the other hand, a non-terminating, non-recurring decimal is not a rational number. It is an irrational number, which we’ll look at later. Is this number rational? Why? Number Rational? Reason -8 Yes Can be written as -8/1 3/5 Yes A fraction with an integer numerator and denominator 0.75 Yes Equal to 3/4 0.(3) Yes A recurring decimal, equal to 1/3 2 No A non-terminating, non-recurring decimal No A non-terminating, non-recurring decimal Rational numbers and other number sets Rational numbers don’t stand alone. They sit within a system of number sets that contain one another. The relationship N ⊂ Z ⊂ Q ⊂ R The four basic number sets are: ℕ: natural numbers (0, 1, 2, 3…) ℤ: integers (…, −2, −1, 0, 1, 2…) ℚ: rational numbers ℝ: real numbers The relationship between them is: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. This means every natural number is an integer, every integer is a rational number, and every rational number is a real number. Diagram of the relationship between natural numbers, integers, rational numbers and real numbers Telling rational and irrational numbers apart Real numbers are made up of two groups: rational numbers and irrational numbers. An irrational number is a non-terminating, non-recurring decimal that cannot be written in the form a/b. Typical examples: √2 ≈ 1.41421356… and π ≈ 3.14159265… Quick tip: if the decimal part stops or repeats in a cycle, the number is rational; if it goes on forever without repeating, the number is irrational. Comparing 0.(3) with √2 makes this clear. 0.(3) repeats the digit 3, so it is rational; √2 doesn’t repeat, so it is irrational. Find out more about irrational numbers and real numbers in related articles. How to represent rational numbers on a number line Each rational number corresponds to exactly one point on the number line. Representing numbers this way helps students “see” how big a number is. Steps to represent a rational number Write the rational number as a fraction or decimal. Work out which two integers it lies between. Divide the segment between the two integers into equal parts according to the denominator. Count and mark the point corresponding to the numerator. For negative numbers, do the same but move to the left of 0. Step-by-step example Represent 3/2 on the number line. 3/2 = 1.5, which lies between 1 and 2. Divide the segment from 1 to 2 into 2 equal parts. The point 3/2 is the midpoint, just after the 1 mark. To represent −1/2, do the opposite: the point lies between 0 and −1. Representing the rational numbers 3/2 and −1/2 on a number line How to compare two rational numbers When you need to know which number is greater, there are two common and reliable methods. Comparing by finding a common positive denominator This is the most reliable method and works in every case. Write both rational numbers as fractions with positive denominators. Find a common denominator for the two fractions. Compare the two numerators: the one with the larger numerator is the greater number. Example: compare −2/3 and −3/4. With a common denominator of 12: −2/3 = −8/12 and −3/4 = −9/12. Since −8 > −9, −2/3 > −3/4. Comparing by position on the number line On a number line, a number on the right is always greater than a number on the left. This method is visual and works well when both numbers are easy to place. Example: 1/4 lies to the right of −1/2, so 1/4 > −1/2. A table classifying numbers as rational or irrational Common mistakes when learning rational numbers and how to avoid them When learning about rational numbers, Grade 7 students often repeat the four mistakes below. Spotting them early helps students avoid losing marks unnecessarily. Mistake 1: Confusing rational numbers with fractions. Students think two different ways of writing a number are two different numbers. In fact, 2/4 and 1/2 are the same rational number. Mistake 2: Confusing rational numbers with irrational numbers. Students see a long decimal and rush to call it irrational. They need to check whether it repeats in a cycle. Mistake 3: Forgetting that the denominator must not be 0. When writing a/b, many students overlook the requirement b ≠ 0. This is an essential condition of the definition. Mistake 4: Converting recurring decimals into fractions incorrectly. For example, writing 0.(3) as 3/10 instead of 1/3. Mistake How to fix it Thinking 2/4 is "a different number from 1/2" Simplify before drawing a conclusion. "Long decimal = irrational" Check for a repeating cycle. Forgetting b 0 Always state the condition on the denominator. 0.(3) = 3/10 Use the method of letting x equal the number and multiplying by 10, as shown above. Rational number exercises with worked solutions Practice helps students remember the definition and how to identify rational numbers. Below are four basic question types that closely follow Vietnam’s Grade 7 Maths textbooks. Exercise 1. Which of the following numbers are rational: −5; 0.75; 2/3; √2; 0.(6)? Solution: −5 = −5/1, 0.75 = 3/4, 2/3 and 0.(6) = 2/3 are all rational numbers. Only √2 is irrational. Exercise 2. Write −0.25 as a fraction in its simplest form. Solution: −0.25 = −25/100 = −1/4. Exercise 3. Compare the rational numbers −2/3 and −3/4. Solution: With a common denominator of 12: −2/3 = −8/12; −3/4 = −9/12. Since −8 > −9, −2/3 > −3/4. Exercise 4. Is 0 a rational number? Explain. Solution: Yes. 0 can be written as 0/1, which is in the form a/b with b ≠ 0. For more in-depth practice, see the next articles on adding and subtracting rational numbers and the set of advanced exercises. Learning rational numbers in the bilingual programme at Westlink Rational numbers are the starting point for students to gain a deeper understanding of real numbers and higher-level maths. At Westlink, Maths is taught bilingually with a focus on helping students understand the underlying concepts rather than learning by rote. Students approach concepts through questions and real-life examples, instead of just memorising formulas. Grade 7 students can follow the Bilingual MYP, which combines the framework of Vietnam’s Ministry of Education and Training with a US core curriculum, or the internationally oriented IB Middle Years Programme (MYP). Students in a bilingual Maths lesson at Westlink Learning in both Vietnamese and English helps students master the terminology of both curricula. For students who need extra language support, the programme provides tailored support based on each student’s needs. Parents who want to see how Westlink teaches Maths bilingually can find out more about the Bilingual MYP, or book a tour and join a trial lesson with their child.
