2.4

Rational Numbers, Irrational Numbers and Real Numbers

ARational Numbers

We can express the following numbers in the form a/b, where a and b are integers and b ≠ 0, by using an equivalent form.

Zero: 0

Positive integers: 1, 2, 3, …

Negative integers: −1, −2, −3, …

Fractions and mixed numbers: ½, 4⁄3, 5 2⁄7, …

Decimal numbers with finite decimal places: 0.4, 7.75, …

For example,

0 = 0/7, 2 = 2/1, −3 = −3/1, 5 2⁄7 = 37/7, 0.4 = 2/5 and 7.75 = 7 3⁄4 = 31/4.

We classify these numbers as rational numbers.

Rational numbers are numbers that can be expressed in the form a/b, where a and b are integers and b ≠ 0.

Let us look at the relationship between the different types of numbers below:

Rational Numbers

e.g., −9, −5, 0, 0.4, 3, 11, −2 1⁄7, 3⁄8, −9⁄4

Integers

e.g., −9, −5, 0, 3, 11

Whole Numbers

(Positive integers and 0)
e.g., 0, 3, 11

Negative Integers

e.g., −9, −5

Fractions

e.g., −2 1⁄7, 3⁄8, 0.4 (or 2⁄5), −9⁄4

Every rational number, a/b, can be expressed as a decimal by dividing its numerator by its denominator. Let us examine the decimal representation of rational numbers.

📖 WORKED EXAMPLE 12

Convert the following rational numbers into decimals.

(a) 3/8 (b) 1/9 (c) 53/99 (d) 47/999

Solution

(a) 3/8 = 3 ÷ 8 = 0.375

(b) 1/9 = 0.111 111 111… = 0.1 (with a dot above '1')

(d) 47/999 = 0.047 047 047… = 0.047 (dots above '0' and '7')

In (a), the decimal 0.375 is a terminating decimal as it has a finite number of digits.

In (b), (c) and (d), the decimals do not terminate and the digits repeat in a pattern. We call each of these decimals a recurring decimal (or repeating decimal).

In (b), 1/9 = 0.111 111 111…. Hence there is only one repeating digit, '1', and we write it as 1/9 = 0.1.
In (c), 53/99 = 0.535 353 535… has a repeating block '53'. We write 53/99 = 0.53.
In (d), 47/999 = 0.047 047 047… has a repeating block '047'. We write 47/999 = 0.047.

From the above, we observe that rational numbers, when expressed as decimals, are either terminating or recurring decimals.

✏️ TRY IT YOURSELF 12

Convert each fraction to a decimal. State whether the decimal is a terminating decimal or a recurring decimal.

(a) 9/16 = ()

(b) 1/6 = ()

(d) 7/22 = ()

BIrrational Numbers and Real Numbers

There are many other numbers that cannot be expressed in the form a/b, where a and b are integers and b ≠ 0. For such numbers, when expressed as decimals, they are neither terminating nor recurring decimals. We call these numbers irrational numbers. For example, π and √2.

The number π has been calculated using computers to thousands of decimal places with no repeating pattern. In fact, we can prove that √2 has no repeating decimal representation and so √2 is also an irrational number.

Rational numbers and irrational numbers make up real numbers. We can summarise their relationship as follows:

Real Numbers

Rational Numbers

e.g., −2.4, −1⁄2, −1 3⁄8, 5⁄11, 2 2⁄5

Irrational Numbers

e.g., √2, π

A real number can be represented by a point on the number line.

THINK²

Can you write down an irrational number in decimal form using (a) only the digit '1', and (b) only the digits '0' and '1'?

If yes, write down the number. If not, state your reasons.

自评：

📖 WORKED EXAMPLE 13

(a) Determine whether each of the following numbers is rational or irrational.

(i) −∛1331 (ii) −5.234 (read as "−5.234 with 34 recurring") (iii) −π

(b) Arrange all the numbers in descending order.

Solution

(a) (i) −∛1331 = −∛(11³) = −11. −∛1331 is a negative integer. So, it is rational.

(ii) −5.234 = −5.234 343 4… −5.234 is a recurring decimal with the repeating block of digits, '34'. It is a rational number.

(iii) −π = −3.141 592 6… Since π is non-terminating and non-recurring, it is an irrational number. Hence −π is also an irrational number.

(b) We can draw a number line and represent the numbers using points on the line.

The descending order of the numbers is −π, −5.234 and −∛1331.

✏️ TRY IT YOURSELF 13

(a) Determine whether each of the following numbers is rational or irrational.

(i) −8/3 = (ii) ∛(−3375) =

(iii) π + 1 = (iv) 0.034/2 =

(b) Arrange all the numbers in ascending order.

📝 PRACTICE EXERCISE 2.4

BASIC MASTERY

Express each fraction as a decimal. State whether the decimal is a terminating or a recurring decimal.

(a) 3/4 = ()

(b) −1 2/5 = ()

(c) 2/9 = ()

(d) −13/11 = ()

(e) −7/12 = ()

(f) −17/20 = ()
Express each decimal as a fraction or a mixed number in its simplest form.

(a) 0.32 = (b) 0.666 =

(c) 3.45 = (d) −8.25 =
Given the numbers: −2/3, 0.666 66 …, 9.32, π − 3, 3.141 592 6, ∛(−216).

Classify the numbers above into

(a) rational numbers:

(b) irrational numbers:

EN
Rational: −2/3, 0.666 66 … (= 2/3, recurring), 9.32 (= 923/99, recurring), 3.141 592 6 (terminating, ≠ π), ∛(−216) = −6

Irrational: π − 3 (π is irrational, so π minus a rational is still irrational)

注意 3.141 592 6 看上去像 π，但它只有 7 位小数且终止——它是有理数。真正的 π 是无穷不循环的。

⏱ 30 秒后查看中文版

中文
有理数： −2/3、0.66666…（= 2/3，循环）、9.32（= 923/99，循环）、3.141 592 6（终止小数，≠ π）、∛(−216) = −6。

无理数： π − 3（π 是无理数，无理数减去有理数仍是无理数）。

自评：

INTERMEDIATE

Arrange the following in ascending order:

23/70, 0.302 ('02' repeats), 0.302 ('302' repeats), 0.302, 1/3

💡 写答案前可以先看四个数的小数展开（不告诉顺序）

23/70 ≈ 0.328 571…；0.302 = 0.302 020 2…；0.302 = 0.302 302 302…；0.302 = 0.302 00；1/3 = 0.333 33…。
请自己比较第三、四、五位小数，得出从小到大的顺序。
(a) Arrange the following in ascending order: 1/14, 1/5, 1/11, 1/8.

(b) Arrange the following in descending order: 13/14, 4/5, 10/11, 7/8.

ADVANCED

What is the maximum possible number of digits in the repeating block when dividing an integer by

(a) 3 → (b) 7 →

Explain your answer:

ENWhen dividing by n, the possible non-zero remainders form a set of size at most n−1 (each remainder is in {1, 2, …, n−1}). Once a remainder repeats, the digits of the quotient also repeat. Hence the maximum length of the repeating block is n−1.

For n = 3: remainders ∈ {1, 2} → max period = 2; but in practice 1/3 = 0.3 with period 1 only. So 1.
For n = 7: remainders ∈ {1, 2, 3, 4, 5, 6} → max period = 6. Verify with 1/7 = 0.142857 (period 6).

⏱ 30 秒后查看中文版

中文除以 n 时，非零余数只能取 {1, 2, …, n−1}，最多 n−1 个。一旦余数重复，商的数字也重复。所以循环节最长 = n−1。
n=3：余数 ∈ {1, 2}，最长 2；实际 1/3 循环节是 1 位。所以答 1。
n=7：最长 6；验证 1/7 = 0.142857 恰好 6 位。

自评：
Determine whether each statement below is TRUE or FALSE. Give a counterexample if the statement is false.

(a) The sum of two irrational numbers is always irrational.

Counterexample (if FALSE):

ENFALSE. Counterexample: π + (−π) = 0 (rational). Two irrationals can sum to a rational.

⏱ 15 秒后查看中文版

中文错误。反例：π + (−π) = 0，0 是有理数。两个无理数之和可以是有理数。

自评：

(b) The sum of a rational number and an irrational number is irrational.

（这条是 TRUE — 不需要反例。如果想验证，请先自己想一个论证，然后展开下面对照。）

💡 思考后再展开

反证法：设 r 是有理数、i 是无理数。如果 r + i = q 是有理数，那么 i = q − r 是两个有理数之差 → 有理数。但 i 是无理数，矛盾。所以 r + i 必须是无理数。

(c) The product of two irrational numbers is always irrational.

Counterexample (if FALSE):

ENFALSE. Counterexample: √2 × √2 = 2 (rational). Two irrationals can multiply to a rational.

⏱ 15 秒后查看中文版

中文错误。反例：√2 × √2 = 2，是有理数。两个无理数之积可以是有理数。

自评：

(d) The product of a non-zero rational number and an irrational number is irrational.

💡 思考后再展开（这条是 TRUE 的理由）

反证：设非零有理数 r，无理数 i。若 r·i = q 是有理数，则 i = q/r 是两个有理数之商（r ≠ 0）→ 有理数。但 i 是无理数，矛盾。所以 r·i 必为无理数。
[OPEN] Find a number a, where −1/2 < a < −1/3, and

(a) a is a terminating decimal: a =

(b) a is a recurring decimal: a =

(c) a is an irrational number:
ENExamples (any one of these works):
- a = −π/7 ≈ −0.4488 (π is irrational and 7 is rational non-zero, so π/7 is irrational; range: −0.5 < −0.449 < −0.333 ✓)
- a = −√0.18 ≈ −0.4243 (0.18 is not a perfect square of a rational, so √0.18 is irrational; in range ✓)
Strategy: take any known irrational (like π or √2) and combine it with rationals (subtract, divide, …) so the result lands inside the open interval (−1/2, −1/3) = (−0.5, −0.333…). Always verify numerically that your value falls in the range.
⏱ 30 秒后查看中文版
中文示例（任意一个即可）：
- a = −π/7 ≈ −0.449（π 无理，7 非零有理，所以 π/7 无理；在 (−0.5, −0.333…) 内 ✓）
- a = −√0.18 ≈ −0.424（0.18 不是有理数的平方，所以 √0.18 无理；在范围内 ✓）
思路：拿一个已知无理数（如 π、√2），与有理数运算后落到 (−1/2, −1/3) 内即可。
自评：

章末概念检查 · Concept Checkpoints

5 道封闭题，自动判分。关键：有理数 = 可写成 a/b 形式 / 小数终止或循环；无理数 = 小数非终止非循环；实数 = 有理 + 无理。