TECHNO PEDAGOGIC CONTENT KNOWLEGE ANALYSIS(Practicum)

1. By
SRUTHI K
BNV CTE
1

2. Dear Students,
A basic knowledge of Mathematics is essential to study any science.
Mathematics is both a computational tool and a conceptual framework.
In this book we present the basic lessons of theoretical and Mathematics.
We hope that this will help to enhance your logical faculties and increase your
problem solving skills
By
Sruthi K
2

3. Irrational Number
Men first used numbers to denote the number of members in a group. For this purpose ,natural number
suffice .Later when lengths and area came to be measured it was realized that it could not be done using
natural numbers.
Many ancient mathematicians thought that all measures could be done using natural numbers and
fractions. But even in those times some came to realize that some lengths could not be specified as fraction of
a chosen unit. The new number used to indicate such lengths were called irrational number
In this lesson we study the notion of irrational numbers and some specific irrational numbers. We also
explain the operations with such numbers and how their approximate vales could be found.
We have studied natural numbers, fractions and negative numbers. We also know how to operate with these
numbers. Let's recall these and start to learn irrational numbers.
3

4. Content
1. Natural number
Whole number
Negative number
2. Integers
3. Rational number
4. Irrational number
5. Some irrational Numbers
6. History of Irrational Numbers
7. Square root of 2
8. Addition of irrational number
9. Multiplication of irrational number
10. Division of irrational number
11. Funny Facts
4

5. Natural
numbers
•Numbers used
in counting
• 1,2,3,…
Whole
numbers
•Natural
number with
zero
• 0,1,2,3….
Negative
Numbers
•Numbers less
than zero.
• ….,-3,-2,-1
5

6. Negative
number
Integer
Zero
Positive
number
6

7. Rational Numbers
A rational Number is a number that can be in the form
p/q
Where p and q are integers and q is not equal to zero
Examples
p q p/q =
1 1 1/1 1
1 2 1/2 .5
55 100 55/100 .55
7

8. Irrational Numbers
An irrational number is a real number that cannot be written as a
simple fraction.
Irrational means not Rational
Example:π (Pi) is a famous irrational number.
Π=3.14159265358979323486264338….(and more)
You cannot write down a simple fraction that equals Pi
The Popular approximation of 22/7=3.1428571428571…is
closed but not accurate.
8

9. 9

10. History of Irrational Number
Apparently Hippasus(one of Pythagoras’ students) discovered
irrational numbers when trying to represent the square root of 2 as
fraction. Instead he proved you couldn’t write the square root of 2 as a
fraction and so it was irrational.
However Pythagoras could not accept the existence of irrational
numbers, because he believed that all numbers had perfect values. But
he could not disprove Hippasus “irrational numbers” and so Hippasus
was thrown overboard and drowned! 10

11. Square root of 2
√2 is a irrational number.The
value of 2 is
1.4142135623730950...(etc)
11

12. Addition of irrational number
푆푖푚푝푙푖푓푦 80 + 125 − 45
√80=√16 × 5=4 5
√125=√25 × 5=5 5
√45=√9 × 5=3√5
√80+√125−√45 =4√5+5 5 − 3 5
=(4+5-3) 5
=6√5.
Ans:
12

13. Multiplication of Irrational Number
Let a and b be two positive numbers.
Let x =√푎+√푏

푥2 = (√푎 + √푏) 2
= (√푎)2+( 푏)2
=a×b
=ab
x=√푎푏
For any two positive number a and b,
we have 풂 × 풃 = √풂풃
13

14. Note on Multiplying Irrational Numbers
Have a look at this:
π × π = 휋2is irrational
But √2 × √2 = 2 is rational
So be careful ... multiplying irrational numbers might result in a
rational number
14

15. Division of irrational number
For any two positive numbers a
and b ,we have
풂/풃 = 풂/√풃
15

16. Funny Facts
Pi day is celebrated
on March 14
16

17. 17