this is a short ppt but it will give you basic information about real numbers and enhance you mathematics skills.

1. REAL NUMBERS

2. Euclid’s Division Lemma And Algorithm
𝐺𝑖𝑣𝑒𝑛 𝑡𝑤𝑜 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎 𝑎𝑛𝑑 𝑏 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑢𝑛𝑖𝑞𝑢𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑞 𝑎𝑛𝑑 𝑟 𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔 𝑎
= 𝑏𝑞 + 𝑟, 0 ≤ 𝑟 < 𝑏
Here ,
𝑎 = 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑏 = 𝑑𝑖𝑣𝑖𝑠𝑜𝑟 𝑞 = 𝑞𝑢𝑜𝑡𝑒𝑖𝑛𝑡 𝑟 =
𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
Example
13 = 2 × 6 + 1

3. Euclid’s division Algorithm
 Euclid’s division Algorithm –
𝑇𝑜 𝑜𝑏𝑡𝑎𝑖𝑛 𝑡ℎ𝑒 𝐻𝐶𝐹 𝑜𝑓 𝑡𝑤𝑜 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑠𝑎𝑦 𝑎 𝑎𝑛𝑑 𝑏 𝑤𝑖𝑡ℎ 𝑎
> 𝑏, 𝑓𝑜𝑙𝑙𝑜𝑤 𝑡ℎ𝑒 𝑠𝑡𝑒𝑝𝑠 𝑏𝑒𝑙𝑜𝑤 ∶
1. Apply Euclid’s division lemma to 𝑎 and 𝑏 . So , we find whole numbers , 𝑞 𝑎𝑛𝑑 𝑟
such that 𝑎 = 𝑏𝑞 + 𝑟, 0 ≤ 𝑟 < 𝑏.
2. If 𝑟 = 0 , d is the HCF of 𝑎 and 𝑏 . If 𝑟 ≠ 0 apply the division lemma to 𝑏 and 𝑟 .
3. Continue the process till the remainder is zero . The divisor at this stage will be the
required HCF .

4. Example :- Using Euclid’s division algorithm find the HCF of 12576 and 4052
.
Since 12576 > 4052 we apply the division lemma to 12576 and 4052 to get
12576 = 4052 × 3 + 420
Since the remainder 420 ≠ 0 , we apply the division lemma to 4052 and 420 to get
4052 = 420 × 9 + 272
We consider the new divisor 420 and new remainder 272 apply the division lemma to get
420 = 272 × 1 + 148
Now we continue this process till remainder is zero .
272 = 148 × 1 + 124
148 = 124 × 1 + 24
124 = 24 × 5 + 4
24 = 4 × 6 + 0
The remainder has now become 0 , so our procedure stops . Since the divisor at this stage is 4 ,
the HCF of 12576 and 4052 is 4 .

5. Fundamental Theorem of Arithmetic
𝐸𝑣𝑒𝑟𝑦 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑐𝑎𝑛 𝑏𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑎𝑠 𝑎 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑝𝑟𝑖𝑚𝑒𝑠, 𝑎𝑛𝑑
𝑡ℎ𝑖𝑠 𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑢𝑛𝑖𝑞𝑢𝑒 , 𝑎𝑝𝑎𝑟𝑡 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑜𝑟𝑑𝑒𝑟 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒𝑦 𝑜𝑐𝑐𝑢𝑟.
Now factorise a large number say 32760
2 32760
2 16380
2 8190
3 4095
3 1365
5 455
7 91
13 13

6. Revisiting Irrational Numbers
𝐿𝑒𝑡 𝑝 𝑏𝑒 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑓 𝑝 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎2, 𝑡ℎ𝑒𝑛 𝑝 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 , 𝑎 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
Theorem - 2 𝑖𝑠 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙.
proof
Let us assume on contrary that 2 is rational where a and b are co-prime .
→ 2 =
𝑎
𝑏
(𝑏 ≠ 0)
squaring on both sides
2
2
=
𝑎
𝑏
2
𝑏2
=
𝑎2
2
Here 2 divides 𝑎2
, so it also divides 𝑎 .
so we can write a=2c for some integer c .

7. Substituting for 𝑎
we get
2𝑏2
= 4c2
𝑏2
= 2c2
𝑐2
=
𝑏2
2
Here 2 divides 𝑏2 , so it also divides 𝑏 .
This creates a contradiction that a and b have no common factors other than 1 .
This contradiction has arisen because of our wrong assumption .
So we conclude that 2 is a irrational number .

8. Revisiting Rational numbers and their decimal expansions
Theorem
𝐿𝑒𝑡 𝑥 𝑏𝑒 𝑎 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑤ℎ𝑜𝑠𝑒 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑒𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒𝑠 .
𝑇ℎ𝑒𝑛 𝑥 𝑐𝑎𝑛 𝑏𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚
𝑝
𝑞
, 𝑤ℎ𝑒𝑟𝑒 𝑝 𝑎𝑛𝑑 𝑞 𝑎𝑟𝑒 𝑐𝑜𝑝𝑟𝑖𝑚𝑒,
𝑎𝑛𝑑 𝑡ℎ𝑒 𝑝𝑟𝑖𝑚𝑒 𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑠𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑞 𝑖𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 2 𝑛
5 𝑛
, where n and m
𝑎𝑟𝑒 𝑛𝑜𝑛 − 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠
Example
1.
3
8
=
3
23
2.
13
125
=
13
53

9. Thank You
Made by :- Amit Choube
Class :- 10th B