الرياضيات (بالإنجليزية) الصف الثاني الإعدادي، الأسبوع 1: Irrational numbers: roots that are not perfect, and π

Real numbers

الأسبوع 1 · الأعداد الحقيقية: الأعداد غير النسبية وتقدير الجذور على خط الأعداد ومجموعة الأعداد الحقيقية · 3 دروس

Irrational numbers: roots that are not perfect, and π

Mathematics Prep 2 Term 1, Unit 1, Lesson 1

GoalTell rational numbers from irrational ones, and know which roots and which decimals are irrational.

New words

rational number · a number that can be written as a/b with b ≠ 0irrational number · a number that cannot be written as a fraction; its decimal never ends and never repeatsperfect square · a number that is the square of a whole number, such as 25recurring decimal · a decimal in which a digit or a block of digits repeats for ever

Explanation

A rational number can be written as a fraction a/b with b not zero. Whole numbers, fractions, decimals that stop (0.25) and decimals that repeat (3.666…) are all rational. An irrational number cannot be written as a fraction: its decimal goes on for ever with no repeating block. The square root of a number that is not a perfect square is irrational, so √2, √3, √5 and √11 are irrational; so is the cube root of a number that is not a perfect cube, such as ∛9. The number π is irrational too. But √25 = 5 and ∛27 = 3 are rational, because 25 and 27 are perfect powers. Why can’t √2 be a fraction? A fraction in simplest form is still in simplest form when you square it. To equal 2 = 2/1 the squared fraction would need denominator 1, so the fraction would be a whole number whose square is 2, and there is no such whole number. A calculator shows only the first digits of √2 = 1.41421356…; that is an approximation, not the number itself.

Number	Type	Reason
√49	rational	49 = 7²
√50	irrational	50 is not a perfect square
∛27	rational	27 = 3³
∛9	irrational	9 is not a perfect cube
3.666…	rational	the 6 repeats
π	irrational	never ends, never repeats

Worked examples

Decide for each number: √16, √20, ∛8, ∛10 and 0.125. √16 = 4 is rational. 20 lies between the perfect squares 16 and 25, so √20 is irrational. ∛8 = 2 is rational. 10 is not a perfect cube, so ∛10 is irrational. 0.125 = 1/8 is rational because the decimal stops.

Number	Value
100x	12.1212…
x	0.1212…
99x	12

Is 0.1212… rational? The block 12 repeats, so yes. Let x = 0.1212…; then 100x = 12.1212…. Subtract: 100x − x = 12, so 99x = 12 and x = 12/99 = 4/33.

Is 22/7 equal to π? No. 22/7 = 3.142857142857… repeats, so it is rational, but π = 3.14159265… never repeats. 22/7 is only a handy approximation; it is a little too big (3.1429 against 3.1416).

Class activity

Root hunt. Use a calculator to find √1, √2, √3 and so on up to √16. Which of them give whole numbers? The others are irrational. How many of the sixteen roots are rational?

Practice

Which of these are irrational: √36, √37, ∛64, ∛65, 0.3333…?

Write 0.777… as a fraction.

Is √(9/25) rational?

Is √0.09 rational?

Layla says √2 is exactly 1.4142 because her calculator shows 1.4142…. Is she right? Square 1.4142 to check.