NUMBER SYSTEMS(LECTURE 6)

Good morning boys!!!!!!

Dear Students

In the previous class we covered the following learning outcomes:

·  Recall that any real number has a decimal expansion.

· How to  Visualize the decimal expansion real numbers on the number line.

Let us go through the guidelines for  the  blog once again:

·         The text in Red, is to be written in your register

·         The text in blue is to be viewed by clicking on it

·         The text in green is to be practiced for home work

1.      take your SET-A Mathematics Register

2.      Our handwriting reflects a lot about us. It will be awesome if you use good presentation and cursive hand writing

3.      Make a column on the right hand side, if you need to do any rough work

4.      Leave two lines where you finished yesterday’s work and draw a horizontal line

5.      Write today's date on the line after that.

6.      Pending Queries from yesterday (if any).......

7.      Please write the learning outcomes as mentioned below

I will be able to  :

· perform operations on real numbers and find the desired outcome.

The rational and irrational numbers together form the real numbers.Here, we will discuss operations on real numbers – both rational and irrational.

Please click on 👇 link.

https://www.youtube.com/watch?v=UJZyx12Odh8

Operations on Real Numbers Rules

The following pointers are to be kept in mind when you deal with real numbers and mathematical operations on them:

• When the addition or subtraction operation is done on a rational and irrational number, the result is an irrational number.
• When the multiplication or division operation is done on a rational number with an irrational number, the result is an irrational number.
• When two irrational numbers are added, subtracted, multiplied or divided, the result may be a rational or an irrational number.

If a and b are positive real numbers, then we have,

• √ab = √a √b 
• (√a + √b)(√a – √b) = a – b
• (a + √b)(a – √b) = a² – b
• (√a + √b)(√c + √d) = √ac + √ad + √bc + √bd
• (√a + √b)² = a + 2√ab + b
• √(a/b) = √a /√b.

Operations on Real Numbers Examples (Do all the following examples in your register)

Example 1:

Solve (2√2 + 7√7) + (13√2 – 4√7).

Solution:

(2√2 + 7√7) + (13√2 – 4√7)

= (2√2 + 13√2) + (7√7 – 4√7)

= (2 + 13)√2 + (7-4)√7

= 15√2 + 3√7

Example 2:

Solve (7√7) x (- 4√7)

Solution:

(7√7) x (- 4√7)

= 7 x -4 x √7 x √7

= -28 x 7 = -196

Example 3:

Solve (8√21 / 4√7)

Solution:

(8√21 / 4√7)

= (8√7 x √3) / (4√7)

= 2 x √7 x √3 / √7 = 2√3

Example 4:

Solve (2√2 + 7√7)(2√2 – 7√7).

Solution:

(2√2 + 7√7) (2√2 – 7√7)

= (2√2)² – (7√7)²

= 4 x 2 – 49 x 7

= 8 – 343 = -335

Example 5:

Solve (√2 + √7)(√3 – √11).

Solution:

(√2 + √7)(√3 – √11)

= √2√3 – √2√11 + √7√3 – √7√11

= √6 – √22 + √21 – √77

Operations on Real Numbers Examples

Example 1:

Solve (2√2 + 7√7) + (13√2 – 4√7).

Solution:

(2√2 + 7√7) + (13√2 – 4√7)

= (2√2 + 13√2) + (7√7 – 4√7)

= (2 + 13)√2 + (7-4)√7

= 15√2 + 3√7

Example 2:

Solve (7√7) x (- 4√7)

Solution:

(7√7) x (- 4√7)

= 7 x -4 x √7 x √7

= -28 x 7 = -196

Example 3:

Solve (8√21 / 4√7)

Solution:

(8√21 / 4√7)

= (8√7 x √3 / 4√7)

= 2 x √7 x √3 / √7 = 2√3

Example 4:

Solve (2√2 + 7√7)(2√2 – 7√7).

Solution:

(2√2 + 7√7) (2√2 – 7√7)

= (2√2)² – (7√7)²

= 4 x 2 – 49 x 7

= 8 – 343 = -335

Example 5:

Solve (√2 + √7)(√3 – √11).

Solution:

(√2 + √7)(√3 – √11)

= √2√3 – √2√11 + √7√3 – √7√11

= √6 – √22 + √21 – √77

CLASS WORK 

 (Do all the following examples in your register)

Q1. Add (3√2 +7√3) and (√2 - 5√3)

 = (3√2 +7√3) +(√2 - 5√3)

 = (3 +1)√2 + (7 – 5)√3

 = (4√2 + 2√3).

Q2. Multiply 5√11 by 3√11

 = 5 x 3 x √11 x √11

 = 5 x 3 x 11

 = 165

Q3. Divide 15√15 by 3√5

 =15√15 ÷ 3√5

 = 5 x √5 x √3

        √5

 = 5√3

Q4. Simplify:  (i) (5 +√5) (5 -√5)

                             =(5)²  - (√5)²

                              =25 -5

                              = 20

                          (ii) (√3 + √2)²

                           = (√3)² + (√2)² + 2 x √3 x √2

                          = 3 +2 + 2√6

                          = 5 + 2√6

Now you are able to solve Exercise questions.

Q3. Recall, p is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, p = c/d. This seems to contradict the fact that p is irrational. How will you resolve this contradiction?

Solution:

When we measure a length with a scale or any other device, we only get an approximate rational value. 

Therefore we may not realize that c or d is irrational.

Circumference c or the perimeter of a circle is given by 2πr,
where r is the radius of the circle
π is approximated as 3.14 or 722​ 
Also diameter(longest chord of circle) of the circle is equal to 2r.

Hence, c=(2πr),d=2r⇒dc​=π

This is analogous to the approximated value of 722​ which though looks like a rational number of the form q p​ (q!=0)

But when computed corresponds to a real value of ~3.14.

And real numbers consists of irrational numbers.

Hence, there is no contradiction in the equation =d c​.

INTERESTING FACTS
In Measurement of a Circle written circa 250 BCE, Archimedes showed that this ratio (C/d, since he did not use the name π) was greater than 310/71 but less than 31/7 by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides.^[5] This method for approximating π was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by Christoph Grienberger who used polygons with 10⁴⁰ sides.

HOME WORK

Exercise 1.5 ; Q 1 and 2.

That is all for today!!!!!!!!!!!!!
Good morning and Thank you boys.

See you all after Easter break