Fix the mistake: Real Numbers and the Number System
Real Numbers ① Classification of Numbers and Properties of Real Numbers
In high school mathematics, we begin by reorganizing the numbers we have studied so far. Let us look at the overall structure of the world of numbers and understand where each type of number belongs.
Classification of Numbers
- Natural numbers: 1, 2, 3, … (positive integers)
- Integers: …, −2, −1, 0, 1, 2, …
- Rational numbers: Numbers that can be expressed as a fraction a/b (a and b are integers, b ≠ 0)
- Irrational numbers: Real numbers that cannot be expressed as fractions. Examples: √2, π, √3
- Real numbers: The set consisting of rational and irrational numbers
Rational and Irrational Numbers
When rational numbers are expressed as decimals, they become terminating decimals or repeating decimals. Irrational numbers are non-repeating infinite decimals. They have no pattern, as in √2 = 1.41421356…
Example ①
For each of the following numbers, state all applicable categories: natural number, integer, rational number, irrational number, or real number.
(1) 5 (2) −3 (3) 2/7 (4) √5 (5) 0
Answer:
(1) Natural number, integer, rational number, real number
(2) Integer, rational number, real number
(3) Rational number, real number
(4) Irrational number, real number
(5) Integer, rational number, real number
For each of the following numbers, state all applicable categories: natural number, integer, rational number, irrational number, or real number.
(1) 5 (2) −3 (3) 2/7 (4) √5 (5) 0
Answer:
(1) Natural number, integer, rational number, real number
(2) Integer, rational number, real number
(3) Rational number, real number
(4) Irrational number, real number
(5) Integer, rational number, real number
Fractional Representation of Repeating Decimals
Let x = 0.333… . Then 10x = 3.333…. Therefore, 10x − x = 3 → 9x = 3 → x = 1/3.
Example ②
Express the repeating decimal 0.272727… as a fraction.
Answer: Let x = 0.272727… . Then 100x = 27.272727…. Thus, 99x = 27 → x = 3/11
Express the repeating decimal 0.272727… as a fraction.
Answer: Let x = 0.272727… . Then 100x = 27.272727…. Thus, 99x = 27 → x = 3/11
Key Point
- Natural numbers ⊂ integers ⊂ rational numbers ⊂ real numbers
- Rational numbers ⇔ terminating decimals or repeating decimals
- Irrational numbers ⇔ non-repeating infinite decimals
- Convert a repeating decimal into a fraction by “multiplying by 10 and subtracting”
Practice Problems
- Choose all the irrational numbers: 3, √4, √7, π, 0.5, 22/7
- Express the repeating decimal 0.6666… as a fraction.
- Express the repeating decimal 0.121212… as a fraction.
Answers and Explanations
- Answer: √7, π
Explanation: √4 = 2 is rational. 22/7 is rational because it is a fraction. - Answer: 2/3
Explanation: 10x − x = 6 → 9x = 6 → x = 2/3. - Answer: 4/33
Explanation: 100x − x = 12 → 99x = 12 → x = 4/33.
Previous and next lessons in this course
- Previous lesson: Warm-up (Math Test Grade 3)
- Next lesson: Fix the mistake: Real Numbers and the Number System