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Contents

## Number Sets

number sets, natural numbers, integers, rational numbers, irrational numbers, real numbers, complex numbers, relation between number sets

## Natural Numbers

N={0,1,2,3,4,5,6⋅⋅⋅}ℕ=0,1,2,3,4,5,6···

## Integers

Z={⋅⋅⋅,−3,−2,−1,0,1,2,3,⋅⋅⋅}ℤ=···,-3,-2,-1,0,1,2,3,···

## Rational Numbers

Q={pq∣∣∣p,q∈Z, q≠0}ℚ=pq|p,q∈ℤ, q≠0

A rational number is any number that can be expressed as fraction of two integers.

## Irrational Numbers

Q∗={⋅⋅⋅,−√3,√2,π,e,⋅⋅⋅}ℚ*=···,-3,2,π,e,···

Irrational number is any number that cannot be expressed as a ratio between two integers.

## Real Numbers

R=Q∪Q∗ℝ=ℚ∪ℚ*

All Rational and Irrational numbers.

## Complex Numbers

C={a+ib ∣∣ a,b∈R, i=√−1}ℂ=a+ib | a,b∈ℝ, i=-1

N⊂Z⊂Q⊂R⊂C

## Prime Numbers – Prime Factorization

Prime number is a whole number greater than 1, whose only two whole-number factors are 1 and itself.P={ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 …}ℙ={ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 …}

## Prime Factorization

Any positive integer m can be written as a unique product of prime numbers:m=p1α1⋅p2α2⋅… ⋅pkαk p1,p2,…,pk∈P α1,αk,…,αk∈Nm=p1α1·p2α2·… ·pkαk p1,p2,…,pk∈ℙ α1,αk,…,αk∈ℕ

## Example:

60|230|56|23|31|60|230|56|23|31|60=22⋅3⋅5

## Divisibility

If a and m are two integers, than is a divisor of m, or m is divisible by a when:a|m ⇔∃n:n∈N∧n⋅a=ma|m ⇔∃n:n∈ℕ∧n·a=m

## Divisibility Rules

(a|m)∧(a|n)⇒(a|m⋅n)∧(a|(m+n))∧(a||m−n|)a|m∧a|n⇒a|m·n∧a|m+n∧a||m-n|(a|b) ∧(b|c)⇒a|ca|b ∧b|c⇒a|c(p∈P)∧(p|m⋅n)⇒(p|m)∨(p|n)p∈ℙ∧p|m·n⇒p|m∨p|n

## Greatest Common Divisor (factor) – Least Common Multiple

Greatest Common Divisor (factor) – Least Common Multiple

## Greatest Common Factor GCF

Greatest common factor of two integers m and n is:GCF(m;n)=(m;n)=l

Eeuclidean algorithm for computing the greates common factor GCF

Example: GCF (246;132)=(246;132)=6
246=132·1+114132=114·1+18114=18·6+66=6·1+0

## Least Common Multiple LCM

Least common multiple of two integers m and n is:{formula_2329}

## Connection between the greatest common divisor (GCD) and the least common multiple (LCM)

(m;n)·[m;n]=m·n

Example: LCM (246;132)=[246;132]=5412
[246;132]=246·132(246;132)=246·1326=5412

## Relative Primes

Two integers m and n are relatively prime if they share no common positive factors (divisors) except 1. GCF(m;n)=[m;n]=1

## Permutation without Repetition

A permutation is an arrangement, or listing, of n distinct objects in which the order is important.

Total number of permutations in case of n elements:

Pn=n⋅(n−1)⋅(n−2)⋅…⋅2⋅1=n!

Example:

In case of 4 elemts: {a,b,c,d}:n=4,P4=4!=4⋅3⋅2⋅1=24n=4,P4=4!=4·3·2·1=24

## Permutation with Repetition

A permutation is an arrangement, or listing, of n objects in which the order is important. The elements are repeated. Number of repetations:

k1,k2,k3,…,kr;(k1+k2+k3+…+kr≤n)

## Rectangle

A quadrilateral whose opposite sides are equal and the diagonals with each angle at right angles (90º) are also equal.

• Area of ​​rectangle = length (l) × breadth (b)
• Perimeter of rectangle = 2 (length width)
• Area of ​​the four walls of the room = 2 (length and width) × height

## Square

A quadrilateral is called a square whose all sides are equal and every angle is right angle.

• Area of ​​square = (side)2 (diagonal)2
• Diagonal of Square = Side
• Perimeter of square = 4 × (side)2

Note: If area of ​​a square = area of ​​a rectangle, then the perimeter of the rectangle will always be greater than the perimeter of the square.

## Parallelogram

A quadrilateral whose opposite sides are parallel and equal is called a parallelogram. The diagonals of a parallelogram bisect each other. A diagonal divides the parallelogram into two similar triangles.

• Area of ​​parallelogram = base × height
• The perimeter of parallelogram = 2 × sum of adjacent sides

## Rhombus

A parallelogram is called a rhombus whose all sides are equal and the diagonals bisect each other at right angles, but no angle is at right angles.

• Area of ​​rhombus = product of diagonals
• The perimeter of rhombus = 4 × one side

## Trapezium

A quadrilateral in which one pair is parallel, the other pair of sides are parallel, then it is a trapezoid.

## Mathematical Formula

• Area of ​​trapezium = sum of parallel sides × height

Rhombus : A quadrilateral whose four sides are equal and the opposite sides are parallel to each other is called a rhombus.

• The perimeter of rhombus = 4 × side
• Area of ​​rhombus = base × height

In this quadrilateral the opposite angles are equal and its diagonals bisect each other at right angles.

Circle: – A circle is a locus of points in which the distance between each other point revolving from a fixed point is the same, the fixed point is called the center of the circle.

Radius: The straight line joining the circumference of a circle to the center is called the radius.

Diameter:- The line passing through the circumference of the circle and touching the vertices of the other circumference of the circle, which passes through the center of the circle, is called the diameter.

Chord: – A line segment joining any two points on the circumference of a circle is called a chord of a circle.

Sector: The figure made up of two radii of a circle and an arc under it is called a radius.

Segment:- The area enclosed by the chord and arc of a circle is called a segment. Here the shaded part is the segment.

Concentric Circle:- If two or more circles have the same center, then those circles are called concentric circles.

## maths formula sheet : Maths formula

• Area of ​​circle = r2
• Circumference of circle = 2πr
• Area of ​​the radius (arc AB) × r (where = central angle)
• Area of ​​the rings of concentric circles = (r2 – r2)
• The perimeter of the semicircle = (π 2) r

## Mathematical Formula : maths formula sheet

All the formulas of mathematics, important formulas of mathematics,Mathematical Formula,maths formula sheet, major formulas of mathematics, ganit ke sutra pdf, read the complete information related to ganit ke sutra through the points given below.

Important Points:-

• In questions related to running/laying wires around a rectangular/square/circular field, it is necessary to find their perimeter.
• The ratio between the area of ​​a square and another square is drawn on the diagonal of the same square will be 1:2.
• The length of the square/rectangular wire is equal to the perimeter of that square or rectangle.
• The length of a circular wire is equal to the perimeter or circumference of that circle.
• The distance covered by a wheel in one revolution will be equal to the circumference of the circular wheel.

Triangle : The region bounded by three sides is called a triangle.

• Area of ​​Triangle Base × Height
• Perimeter of triangle = sum of all sides

Right-angle Triangle : A triangle whose one angle is right angle i.e. 90º. In this triangle, the side opposite to the right angle is called the hypotenuse.

(hypotenuse)2 = (perpendicular)2 (base)2
Area of ​​right angled triangle = base × perpendicular

Equilateral Triangle: A triangle in which all the sides are equal and each angle is 60º.

• Area of ​​equilateral triangle =(side)2
• Perimeter of an equilateral triangle = 3 × one side

Isosceles Triangle : A triangle whose only two sides are equal is called an isosceles triangle.

• Perimeter of isosceles triangle = 2a b

Scalene Triangle : A triangle whose all sides are unequal. Area Formulas In Maths

## H.C.F. And L.C.M Formula

• No.-1. Greatest Common Factor – The ‘highest common factor is that maximum number, which exactly divides the given numbers. For example, the greatest common factor of the numbers 10, 20, 30 is 10.
• No.-2. Common Factor – A number that divides each of two or more numbers completely, such as the common factor of 10, 20, 30 is 2, 5, 10.
• No.-3. Least Common Factor – The ‘least common factor’ of two or more numbers is the least number which is greater than each of those given numbers.
• No.-4. Common Multiple – A number that is in two or more numbers. If the number is divided completely by each of the numbers, then that number is called the commonality of those numbers, such as the commonality of 3, 5, 6 is 30, 60, 90, etc.
• No.-5. Factor and Multiple – If a number m intersects another number n completely, then m is called a factor of n, and n is called multiple of m.

## Number System

• No.-1. Natural Numbers: The numbers used to count things are called counting numbers or ‘natural numbers’.
• Like- 1, 2, 3, 4, 5, 6, 7, . , , ,
• No.-2. Whole Numbers: The numbers obtained by adding zero to the natural numbers are called ‘whole numbers’.
• Like- 0, 1, 2, 3, 4, 5, 6, 7, . , , ,
• No.-6. Prime Numbers: Those numbers which are not divisible by any number other than themselves and 1 are called ‘prime numbers’.
• Eg- 2, 3, 7, 11, 13, 17 ……….
• Note – ‘1’ is neither a prime number nor a composite number
• No.-7. Composite Numbers: Those numbers which are completely divisible by any number other than itself and 1, are called ‘composite numbers’.
• Like- 4, 6, 8, 9, 10, …………

## Relationship In Trigonometry Formula

• No.-1. Sin θ = 1 / cosec θ
• No.-2. cosec θ = 1 / Sin θ
• No.-3. cos θ = 1 / sec θ
• No.-4. sec θ = 1/ cos θ
• No.-5. sin θ.cosec θ = 1
• No.-6. cos θ.sec θ = 1
• No.-7. tan θ.cot θ = 1
• No.-8. tan θ = sin θ / cos θ
• No.-9. cot θ = cos θ / sin θ
• No.-10. tan θ = 1 / cot θ
• No.-11. cot θ= 1 / tan θ

### 1. Squaring Of A Number Whose Unit Digit Is 5

For example Find (55) ² =?

Step 1. 55 x 55 = . . 25 (end terms)

Step 2. 5x (5+1) = 30

### 2. Multiply a Number By 5

Even Number:

2464 x 5 =?

Step 1. 2464 / 2 = 1232

The answer will be 2464 x 5 = 12320

Odd Number:

3775 x 5

Step 1. Odd number; so ( 3775 – 1) / 2 = 1887

Step 2. As it is an odd number, so instead of 0 we will put 5

The answer will be 3775 x 5 = 18875

Now try —- 1234 x 5, 123 x 5

So our answer will be 3025.

### 3. Subtraction From 1000, 10000, 100000

For example:

1000 – 573 =? (Subtraction from 1000)

We simply subtract each figure in 573 from 9 and then subtract the last figure from 10.

Step 1. 9 – 5 = 4

Step 2. 9 – 7 = 2

Step 3. 10 – 3 = 7

So, the answer is: (1000 – 573) = 427

### 4. Multiplication Of Any 2-digit Numbers (11 – 19)

There are 4 steps to get the result:

Step 1. Add the unit digit of the smaller number to the larger number.
Step 2. Next, multiply the result by 10.
Step 3. Now, multiply the unit digits of both the 2-digit numbers.
Step 4. Then add both the numbers.

For example: Let’s take two numbers 13 & 15.
Step 1. 15 + 3 =18.
Step 2. 18*10 = 180.
Step 3. 3*5 = 15
Step 4. Add the two numbers, 180+15 and the answer is 195.

## Trigonometry Math Formulas

» 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

## Algebra Math Formula

(Natural Numbers) – an – bn = (a – b)(an-1 + an-2 +…+ bn-2a + bn-1)

(Even) –  (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)

(Odd no.) – (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +…- bn-2a + bn-1)

(a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….

(Low Of Formula Exponents)

(am)(an) = am+n
(ab)m = ambm
(am)n = amn

a2 – b2 = (a – b)(a + b)

(a+b)2 = a2 + 2ab + b2

a2 + b2 = (a – b)2 + 2ab

(a – b)2 = a2 – 2ab + b2

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc

(a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc

(a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)

(a – b)3 = a3 – 3a2b + 3ab2 – b3

a3 – b3 = (a – b)(a2 + ab + b2)

a3 + b3 = (a + b)(a2 – ab + b2)

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a – b)3 = a3 – 3a2b + 3ab2 – b3

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4)

(a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4)

a4 – b4 = (a – b)(a + b)(a2 + b2)

a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)