https://youtu.be/EU1qspe8lwE

Dear students please like share and subscribe to my channel for class 9th 10th maths and Science and for class 11th maths

Number Systems:

Natural Numbers:
(i) The counting numbers are called Natural numbers.
(ii) The symbol for natural numbers is N.
(iii) N = {1,2,3,4,5....}
(iv) The smallest natural number is 1.

Whole Numbers:
(i) If we include 0 (zero) with the natural numbers, we get Whole numbers.
(ii) The symbol for whole numbers is W.
(iii) W = {0,1,2,3,4,5....}
(iv) The smallest whole number is 0 (zero)

Integers:
(i) If we include opposite of natural numbers with whole numbers, we get Integers.
(ii) The symbol for Integers is Z (or) I
(iii) Z (or) I = {Minus Infinity to Plus Infinity}
(iv) Z (or) I = {....-5,-4,-3,-2,-1,0,1,2,3,4,5....}

Rational Numbers:
(i) If we can write any number in the form a/b (b is not equal to 0), is a Rational number.
(ii) The Symbol for Rational numbers is Q.
(iii) Every Natural numbers, Whole numbers and Integers are also Rational Numbers.
(iv) Every Rational numbers are not Natural numbers, Whole numbers and Integers.
(v) We have two types of Rational numbers.
(a) Terminating Decimal places.
Example: 5/2 = 2.5
(b) Non-Terminating but Repeating Decimal places.
Example: 10/3 = 3.3333333....... and 22/7 = 3.142857 142857 142857........

Irrational Numbers:
(i) Irrational numbers, we can't write in a/b form.
(ii) The Symbol for Irrational numbers is R-Q.
(iii) Irrational numbers has Non-Terminating and Non_Repeating Decimal places.
(iv) Square root p is an irrational number, where p is a prime number.
Example:
square root 2 = 1.414213562373095
square root 3 = 1.732050807568877
Pi = 3.14159265358979

Remember: 22/7 is not equal to Pi.

Real Numbers:
(i) The set of Rational numbers and Irrational numbers are called Real Numbers.
(ii) The Symbol for Real Numbers is R.
(iii) R = {Rational Numbers + Irrational Numbers}
Dear All, Thanks. Be safe. Stay Home. Stay Healthy.

1. (α+в)²= α²+2αв+в²
2. (α+в)²= (α-в)²+4αв
3. (α-в)²= α²-2αв+в²
4. (α-в)²= (α+в)²-4αв
5. α² + в²= (α+в)² - 2αв.
6. α² + в²= (α-в)² + 2αв.
7. α²-в² =(α + в)(α - в)
8. 2(α² + в²) = (α+ в)² + (α - в)²
9. 4αв = (α + в)² -(α-в)²
10. αв =1. (α + в + ¢)² = α² + в² + ¢² + 2(αв + в¢ + ¢α)
12. (α + в)³ = α³ + 3α²в + 3αв² + в³
13. (α + в)³ = α³ + в³ + 3αв(α + в)
14. (α-в)³=α³-3α²в+3αв²-в³
15. α³ + в³ = (α + в) (α² -αв + в²)
16. α³ + в³ = (α+ в)³ -3αв(α+ в)
17. α³ -в³ = (α -в) (α² + αв + в²)
18. α³ -в³ = (α-в)³ + 3αв(α-в)
ѕιη0° =0
ѕιη30° = 1/2
ѕιη45° = 1/√2
ѕιη60° = √3/2
ѕιη90° = 1
¢σѕ ιѕ σρρσѕιтє σƒ ѕιη
тαη0° = 0
тαη30° = 1/√3
тαη45° = 1
тαη60° = √3
тαη90° = ∞
¢σт ιѕ σρρσѕιтє σƒ тαη
ѕє¢0° = 1
ѕє¢30° = 2/√3
ѕє¢45° = √2
ѕє¢60° = 2
ѕє¢90° = ∞
¢σѕє¢ ιѕ σρρσѕιтє σƒ ѕє¢
2ѕιηα¢σѕв=ѕιη(α+в)+ѕιη(α-в)
2¢σѕαѕιηв=ѕιη(α+в)-ѕιη(α-в)
2¢σѕα¢σѕв=¢σѕ(α+в)+¢σѕ(α-в)
2ѕιηαѕιηв=¢σѕ(α-в)-¢σѕ(α+в)
ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв - ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
» ѕιη(α+в)=ѕιηα ¢σѕв+ ¢σѕα ѕιηв.
» ¢σѕ(α+в)=¢σѕα ¢σѕв +ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв+ѕιηαѕιηв.
» тαη(α+в)= (тαηα + тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1+ тαηαтαηв)
» ¢σт(α+в)= (¢σтα¢σтв −1) / (¢σтα + ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв + 1) / (¢σтв− ¢σтα)
α/ѕιηα = в/ѕιηв = ¢/ѕιη¢ = 2я
» α = в ¢σѕ¢ + ¢ ¢σѕв
» в = α ¢σѕ¢ + ¢ ¢σѕα
» ¢ = α ¢σѕв + в ¢σѕα
» ¢σѕα = (в² + ¢²− α²) / 2в¢
» ¢σѕв = (¢² + α²− в²) / 2¢α
» ¢σѕ¢ = (α² + в²− ¢²) / 2¢α
» Δ = αв¢/4я
» ѕιηΘ = 0 тнєη,Θ = ηΠ
» ѕιηΘ = 1 тнєη,Θ = (4η + 1)Π/2
» ѕιηΘ =−1 тнєη,Θ = (4η− 1)Π/2
» ѕιηΘ = ѕιηα тнєη,Θ = ηΠ (−1)^ηα

1. ѕιη2α = 2ѕιηα¢σѕα
2. ¢σѕ2α = ¢σѕ²α − ѕιη²α
3. ¢σѕ2α = 2¢σѕ²α − 1
4. ¢σѕ2α = 1 − ѕιη²α
5. 2ѕιη²α = 1 − ¢σѕ2α
6. 1 + ѕιη2α = (ѕιηα + ¢σѕα)²
7. 1 − ѕιη2α = (ѕιηα − ¢σѕα)²
8. тαη2α = 2тαηα / (1 − тαη²α)
9. ѕιη2α = 2тαηα / (1 + тαη²α)
10. ¢σѕ2α = (1 − тαη²α) / (1 + тαη²α)
11. 4ѕιη³α = 3ѕιηα − ѕιη3α
12. 4¢σѕ³α = 3¢σѕα + ¢σѕ3α

» ѕιη²Θ+¢σѕ²Θ=1
» ѕє¢²Θ-тαη²Θ=1
» ¢σѕє¢²Θ-¢σт²Θ=1
» ѕιηΘ=1/¢σѕє¢Θ
» ¢σѕє¢Θ=1/ѕιηΘ
» ¢σѕΘ=1/ѕє¢Θ
» ѕє¢Θ=1/¢σѕΘ
» тαηΘ=1/¢σтΘ
» ¢σтΘ=1/тαηΘ
» тαηΘ=ѕιηΘ/¢σѕΘ

Arithmetic Progression Class 10 | Arithmetic Progression Class 10 NCERT Solution | IIT JEE FOUNDATION COURSE FOR CLASS 10: https://www.youtube.com/playlist?list=PL-hYriZY1lNypCI36-Dsz5l8cG49N-xMM

Q. What is the role of acid in our stomach?
Ans.  in our stomach? Hydrochloric acid plays a major role in the process of digestion. It creates a medium of gastric juice acidic so that the enzyme pepsin digests the protein and kills the bacteria present in them. ... The acid activates the pepsinogen enzyme required to digest proteins.

Introduction to Number SystemsNumbers

Number: Arithmetical value representing a particular quantity. The various types of numbers are Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers etc.

Natural Numbers

Natural numbers(N) are positive numbers i.e. 1, 2, 3 ..and so on.

Whole Numbers

Whole numbers (W) are 0, 1, 2,..and so on. Whole numbers are all Natural Numbers including ‘0’. Whole numbers do not include any fractions, negative numbers or decimals.

Integers

Integers are the numbers that includes whole numbers along with the negative numbers.

Rational Numbers

A number ‘r’ is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0.

Irrational Numbers

Any number that cannot be expressed in the form of p/q, where p and q are integers and q≠0, is an irrational number. Examples: √2, 1.010024563…, e, π

Real Numbers

Any number which can be represented on the number line is a Real Number(R). It includes both rational and irrational numbers. Every point on the number line represents a unique real number.

Irrational NumbersRepresentation of Irrational numbers on the Number line

Let √x be an irrational number. To represent it on the number line we will follow the following steps:

Take any point A. Draw a line AB = x units.

Extend AB to point C such that BC = 1 unit.

Find out the mid-point of AC and name it ‘O’. With ‘O’ as the centre draw a semi-circle with radius OC.

Draw a straight line from B which is perpendicular to AC, such that it intersects the semi-circle at point D.

Length of BD=√x.

Constructions to Find the root of x.

With BD as the radius and origin as the centre, cut the positive side of the number line to get √x.

Identities for Irrational Numbers

Arithmetic operations between:

rational and irrational will give an irrational number.

irrational and irrational will give a rational or irrational number.

Example : 2 × √3 = 2√3 i.e. irrational. √3 × √3 = 3 which is rational.

Identities for irrational numbers

If a and b are real numbers then:

√ab = √a√b

√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)(√c−√d) = √ac−√ad+√bc−√bd

(√a+√b)2 = a+2√(ab)+b

Rationalisation

Rationalisation is converting an irrational number into a rational number. Suppose if we have to rationalise 1/√a.

1/√a × 1/√a = 1/a

Rationalisation of 1/√a+b:

(1/√a+b) × (1/√a−b) = (1/a−b²)

Laws of Exponents for Real Numbers

If a, b, m and n are real numbers then:

am × an= am+n

(am) n = amn

am/an = am−n

ambm=(ab)m

Here, a and b are the bases and m and n are exponents.

Exponential representation of irrational numbers

If a > 0 and n is a positive integer, then: n√a=a1n Let a > 0 be a real number and p and q be rational numbers, then:

ap × aq = ap + q

(ap)q = apq

ap/ aq= ap−q

apbp = (ab)p

Decimal Representation of Rational NumbersDecimal expansion of Rational and Irrational Numbers

The decimal expansion of a rational number is either terminating or non- terminating and recurring.

Example: 1/2 = 0.5 , 1/3 = 3.33…….

The decimal expansion of an irrational number is non terminating and non-recurring.

Examples: √2 = 1.41421356..

Expressing Decimals as rational numbers

Case 1 – Terminating Decimals

Example – 0.625

Let x=0.625

If the number of digits after the decimal point is y, then multiply and divide the number by 10y.

So, x = 0.625 × 1000/1000 = 625/1000 Then, reduce the obtained fraction to its simplest form.

Hence, x = 5/8

Case 2: Recurring Decimals

If the number is non-terminating and recurring, then we will follow the following steps to convert it into a rational number:

Example –

Step 1. Let x =   (1)

Step 2. Multiply the first equation with 10y, where y is the number of digits that are recurring.

Thus, 100x = (2)Steps 3. Subtract equation 1 from equation 2.On subtracting equation 1 from 2, we get99x = 103.2x=103.2/99 = 1032/990

Which is the required rational number.

Reduce the obtained rational number to its simplest form Thus,

x=172/165

French revolution👍

Class 10 @Garg tutorials
Please subscribe to my YouTube channel

https://www.youtube.com/gargtutorials

Join your friends
https://t.me/garg_tutorials