Revisiting Irrational Numbers
Chapter Name : Real Numbers |
Sub Topic Code : 104_10_01_04_01 |
Topic Name : Revisiting Irrational Numbers |
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Sub Topic Name : Revisiting Irrational Numbers |
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Introduction
What is an irrational number?
Pre-Requisites:
Basic knowledge on rational numbers.
Activity:
Take scale, pencil, protractor to draw rt angled triangles with two its lengths being (1,1), ( ?2,1),(1,2) and measure the lengths of the hypotenuse. You will observe the hypotenuse lengths being ?2, ?3 , ?5.
Real Life Question:
Ravi was a farmer; he wanted to have two triangular plots from a square plot, so that he grows different crops on the either side. Dimensions of square plot are 100m*100m.His triangular fields are to be fenced. The fencing length is it rational or irrational?
Key Words / FlashCards
| Key Words | Definitions (pref. in our own words) |
|---|---|
| irrational number | Any number which cannot be written in the form of p/q, where q?0 is called as irrational number |
Learning aids / Gadgets
| Gadgets | How it can be used |
|---|---|
| Right angled triangles to know about irrational numbers | Draw 1*1 rt angled triangle to get ?2.Similarly take ?2 and 1 to get the length ?3, and do for ?5 and ?7. |
Real life uses :
Length of the hypotenuse in an isosceles rt angled triangle is always an irrational number.
Practical examples around us
| Examples | Explainations |
|---|---|
| Prove that 3?3 is an irrational number? | Start with a simple proof of “If a3 is divisible by a prime number p, then a is divisible by p”. Let a/b = 3?3 where a and b are co primes. Therefore a3 = b3*3, so a3 is divisible by 3, so a is divisible by 3. Therefore 3*3*c3=b3 (a=3*c), so b3 is divisible by 3 implies b is divisible by 3.Contradiction that a and b are coprimes. Therefore 3?3 is irrational. |
What you learn in Theory:
Irrational numbers, representation of irrational numbers.
What you learn in Practice:
How to prove whether a number is rational or irrational.