Chapter 1:Number system-Irrational numbers.pptx

Discussed about irrational number

1. NUMBER SYSTEMS  Irrational Number Presented by Dr.T.Gandhimathi Associate Professor of Mathematics P.A.C.E.T, Pollachi

2. History of Irrational Number The Pythagoreans in Greece, followers of the famous mathematician and philosopher Pythagoras, were the first to discover the numbers which were not rationals , around 400 BC. These numbers are called irrational numbers ( irrationals ), because they cannot be written in the form of a ratio of integers.

3. Definition The number ‘s’ which cannot be written in the form of p/q is called irrational, where p and q are integers and q ≠ 0 or the numbers which are not rational are called  Irrational Numbers The decimal expansion of an irrational number is non-terminating and non-recurring Example : √2, √11,  , 0.10110111011110 ...

4. Irrational means  not Rational Value of π : π  =  3.14 15926535897932384626433832795...  The popular approximation of  22 / 7  = 3.14 28571428571... is close but not accurate .

5. Example 1: Find the value of √ 2 Solution:

6. Pythagoras Theorem In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides

7. Example 2: Locate √2 on the number line Solution: Step I: Draw a number line and mark the centre point as zero Step II: Mark right side of the zero as (1) and the left side as (-1 ). Step III: consider a unit square OABC Oo

8. Step III : Draw a perpendicular of length 1 unit on point A as AB Step IV: Pythagoras Theorem, OB = √ 2 Step V: Take an arc of length OB, and draw it on the number line which meets as E. So, at E, we can represent √2 as shown in the figure

9. Example 3: Locate √3 on the number line Solution: Step I : Draw a √2 on number line Step II : Construct PD of unit length perpendicular to OP Step III : using the Pythagoras theorem, OD = = √ 3 Step IV: Using a compass, with centre O and radius OD, draw an arc which intersects the number line at the point E. Step V: Then E corresponds to √3 . ..\..\Documents\Represent Root 3 on Number line - YouTube (480p).mp4  

10. EXERCISE 1.2 1 . State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form √ m where m is a natural number. (iii) Every real number is an irrational number . Solution: True False False

11. 2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number Solution: The square roots of all positive integers are not irrational. Example: √4 = 2, 2 is a rational. Homework: 3. Show how √5 can be represented on the number line. 4. Locate √10 on the number line

12. Thank you