𝑾𝒐𝒓𝒍𝒅 𝒐𝒇 𝑴𝒂𝒕𝒉𝒆𝒎𝒂𝒕𝒊𝒄𝒔

Happy Fibonacci Day ❤️ 😍
Fibonacci Sequence: A Beautiful Pattern in Mathematics
The Fibonacci sequence is one of the most famous and intriguing patterns in mathematics. It begins with the numbers 0 and 1, and every term after that is the sum of the two preceding numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

This simple rule leads to a sequence that appears throughout mathematics, nature, art, and technology. The sequence was introduced to the Western world by the Italian mathematician Leonardo of Pisa, commonly known as Fibonacci, in the 13th century.

𝐎𝐮𝐫 𝐧𝐞𝐱𝐭 𝐜𝐚𝐥𝐞𝐧𝐝𝐚𝐫 𝐲𝐞𝐚𝐫 𝐢𝐬 𝐚 𝐦𝐚𝐭𝐡𝐞𝐦𝐚𝐭𝐢𝐜𝐚𝐥 𝐰𝐨𝐧𝐝𝐞𝐫

👉🏻𝗜𝗻𝘁𝗲𝗿𝗲𝘀𝘁𝗶𝗻𝗴 𝟮𝟬𝟮𝟱

1) 2025, itself is a square [(45)²= 2025].

2) It's a product of two squares,
Viz. 9² x 5² = 2025

3) It is the sum of 3-squares,
viz. 40²+ 20²+5²= 2025

4) It's the first square after 1936.

5) It's the sum of cubes, of all the single digits, from 1 to 9,
Exp: 1³+2³+3³+4³+5³+6³+7³+8³+9³=2025.

👉🏻👍🏻𝗪𝗼𝗿𝗹𝗱 𝗼𝗳 𝗺𝗮𝘁𝗵𝗲𝗺𝗮𝘁𝗶𝗰𝘀

A mathematics teacher entered the classroom and found the chair he was going to sit on, hung to the ceiling. He looked at the students and smiled without saying a word. He proceeded to the board and wrote:
Test-15min, 30marks.
Q1. Calculate the distance between the chair and the floor in centimetres (1 Mark).
Q2. Calculate the angle of inclination of the chair to the ceiling, show your workings (1 Mark).
Q3. Write the name of the student who hung the chair to the ceiling and the friends who helped him. (28 Marks) 🤣😂

,

𝑻𝙮𝒑𝙚𝒔 𝒐𝙛 𝙁𝒖𝙣𝒄𝙩𝒊𝙤𝒏👇

𝙒𝙝𝙖𝙩 𝙞𝙨 𝙋𝙞?? 𝘼𝙣𝙙 𝙒𝙝𝙮 𝙞𝙩 𝙐𝙨𝙚 𝙞𝙣 𝙀𝙫𝙚𝙧𝙮𝙬𝙝𝙚𝙧𝙚??

پائی ( π ) کیا ہے ۔۔۔؟
کہاں سے آئی ہے ۔۔۔؟
اور ہر جگہ استعمال کیوں ہوتی ہے۔۔۔ ؟
پائی کی مستقل مقدار یا کانسٹینٹ ویلیو 22/7 یا constant Value " 3.14 " کیوں ہے؟
چلیں ہم ایک سرکل لیتے ہیں جسکا نیم قطری ( Radius ) مثلا 3 سینٹی میٹر ہے تو اسکا قطر ( diameter ) 6 سینٹی میٹر ہی ہوگا ...
اور اس سرکل کا پیرامیٹر یعنی سرکمفیرینس ناپنے سے 18.85cm آجائے گا ( سرکل کے پیرامیٹر کو یار لوگ سرکمفیرینس کہتے ہیں )۔۔۔۔
اب اس سرکل کے سرکمفیرینس کو diameter پر تقسیم کرتے ہے تو ویلیو آئے گی ....3.14166

Circumference/diameter= π

اب ایک دوسرا لیکن تھوڑا سا بڑا سرکل لیتے ہیں جسکا ریڈیس 4cm ہے یعنی ڈایامیٹر 8 cm ہوگا۔۔۔
تو اس سرکل کا سرکمفیرینس 25.1cm ہوگا ۔۔۔
اس سرکمفیرینس (25.1) کو ڈایامیٹر (8) پر تقسیم کرنے سے ویلیو آئے گی ...3.14166

یعنی کسی بھی سائز کے ڈایامیٹر کو سرکل کے سرکمفیرینس پر تقسیم کرنے سے ایک ہی ویلیو آئے گی جوکہ π برابر ہے۔۔۔۔

نتیجہ یہ کہ جب کبھی ایک یونٹ کے ڈایامیٹر کا سرکل ملے اس کا سرکمفیرینس 3.14 یونٹس ہی ہوگا ۔۔۔

Circumference of circul= 2πr

اس فارمولے میں 2r تو ڈبل ریڈیس یعنی ڈایامیٹر ہے جب اسکو دائیں طرف سرکمفیرینس کی طرف لے جاتے ہیں تو
circumference/diameter = π
ہی آتا ہے ۔۔۔
ایک دلچسپ بات یہ کہ 22/7 کی ویلیو بھی 3.14 ہی آتی ہے یعنی
π = 22/7

اب ایک سوال رہتا ہے کہ π ہی کیوں ۔۔۔
دراصل پائی" π " یونانی رسم الخط اور حروف تہجی کا سولہواں حرف یا الفابیٹ ہے، جبکہ یونانی زبان میں circumference کو "περιφέρεια" کا لفظ استعمال ہوتا ہے جبکہ Parameter كو "παράμετρος" کہتے ہیں اور یہ دونوں الفاظ π سے ہی شروع ہوتے ہیں لھذا یہ مستقل مقدار یا کانسٹینٹ ویلیو بھی π ہی کہلائی ۔

بعضے فزیشینز نے اس پر کافی تطویل سے کام لیا ہے ۔۔۔
لوڈولف وین نے اپنی پوری زندگی π کی ویلیو 35 ڈیجٹس کو کلکولیٹ کرنے میں لگا دی ۔۔۔
اسی طرح ڈاکٹر لیبنیز نے بھی ایک سیریز دی جو π کی آپروکسیمیشن کرتی ہے ۔۔۔

😍😊

For metric level.

Pi, denoted by the symbol π, is a mathematical constant with a value of approximately 3.14159. It is an irrational number, which means it cannot be expressed as a fraction and its decimal representation goes on indefinitely without repeating.

In engineering, pi plays a significant role in many different applications, including:

Calculating Circumference and Area of Circles: The most basic application of pi in engineering is to calculate the circumference and area of circles. The circumference of a circle is given by 2πr, where r is the radius of the circle. Similarly, the area of a circle is given by πr^2.

Electrical Engineering: In electrical engineering, pi is used in calculations related to alternating current (AC) circuits. For example, the frequency of an AC signal is given by f = 2πf, where f is the angular frequency and f is the frequency in hertz.

Structural Engineering: Pi is used in structural engineering to calculate the stress and strain in materials. For example, the stress in a beam under bending is given by σ = My/I, where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia. Pi appears in the calculation of I, which is given by I = πr^4/4 for a circular cross-section.

Fluid Mechanics: Pi is used in fluid mechanics to calculate the Reynolds number, which is a dimensionless quantity that characterizes the flow of fluids. The Reynolds number is given by Re = ρvL/μ, where ρ is the density of the fluid, v is the velocity, L is the characteristic length scale, and μ is the viscosity of the fluid. Pi appears in the calculation of L for cylindrical geometries, where L is the diameter of the cylinder.

In summary, pi is a fundamental mathematical constant that plays a critical role in many different areas of engineering, from basic calculations of circles to more complex applications in electrical engineering, structural engineering, and fluid mechanics.

In addition to its use in engineering, pi plays a significant role in many different fields, including:

Mathematics: Pi is a fundamental constant in mathematics, appearing in many different formulas and equations. It is used in trigonometry, calculus, number theory, and other areas of mathematics.

Physics: Pi is used in many different areas of physics, including thermodynamics, fluid mechanics, quantum mechanics, and relativity. For example, pi appears in the calculation of the period of a pendulum, the frequency of an oscillator, and the wavelength of light.

Statistics: Pi is used in statistics to calculate the probability density function of the normal distribution, which is a commonly used probability distribution in statistics and probability theory.

Computer Science: Pi is used in computer science in a variety of ways, including as a test for randomness, as a constant in algorithms and formulas, and as a way to represent irrational numbers in computer programs.

Music: Pi has been used to create musical compositions, with some composers using the digits of pi to determine the notes and rhythms of their compositions.

The mystery of pi arises from its unique properties and its seemingly random and infinite decimal expansion. Pi is an irrational number, which means it cannot be expressed as a simple fraction, and its decimal expansion goes on indefinitely without repeating. Despite its seemingly random nature, pi has been found to appear in many unexpected places throughout mathematics, physics, and other fields.

One of the most intriguing aspects of pi is its connection to circles. Pi is defined as the ratio of the circumference of a circle to its diameter, and this relationship between pi and circles has fascinated mathematicians for centuries. The fact that pi is an irrational number also means that the exact value of the circumference of a circle or the area of a circle cannot be expressed as a finite decimal or fraction.

The decimal expansion of pi has been studied for centuries, and mathematicians have found a number of interesting patterns and properties within the seemingly random digits. For example, there are many known algorithms for calculating the digits of pi, some of which have been used to calculate pi to billions or even trillions of digits.

Despite its long history of study, there are still many open questions and mysteries surrounding pi. For example, it is not known whether pi is a normal number, which means that each digit occurs with equal frequency in its decimal expansion. There are also many unsolved problems related to the distribution of the digits of pi, and its relationship to other important mathematical constants such as e and the golden ratio.

Basic Mathematics

Zero is neither positive nor negative.
Zero is an even number.
Whole number starts with zero.
Negative numbers are numbers that are smaller than zero, and positive numbers are numbers that are bigger than zero.
Zero is not a natural number.

Natural numbers start with 1.
Natural numbers are called counting numbers and cardinal numbers.
1 is a Natural Number.
1 is the cardinal number.
1 is an odd number.
1 is the whole number.
Number 1 is not a prime number.
1 is not a composite number.
One is the only positive integer (whole number) which is neither prime (exactly two factors: one and itself) nor composite (more than two factors).

The smallest prime Number is 2.
Smallest composite number is 4.

In Maths, integers are the numbers which can be positive, negative or zero, but cannot be a fraction or decimal.

Digits 10 symbols .

Numeral : group of digits.

Place Value : Local value due to place

Face value : Actual value of digit

Types of Numbers

Natural Numbers
Counting numbers , cardinal numbers.
1,2,3,3,4,5,6,7,8,9

Ordinal numbers : in order
First, second ,third, eleventh, twenty first

Whole Numbers : zero , all natural Numbers

0,1,2,3,4,5,6,7,8,9

Integers : natural Number , positive numbers, whole Number, negative numbers. Not decimals and not fractions.

Non negative integers : 0,1,2,2
Non Positive integers 0,-1,-2,-3,

Prime Number : factor 1 and itself
Smallest prime : 2
2,3,5,7,11

Composite Numbers : More than two factors ,not prime

Smallest composite : 4

4,6,8,9,10,12,14

Even number : divisible by 2
Odd number : not divisible by 2

Co Prime : Highest Common factor 1

Angle

Straight line 180
Straight angle 180
Triangle all angles 180
Circle 360
Square 360
Rectangle 360
Complete angle 360
Acute angle smaller than 90
Right Angle equal to 90
Obtuse angle bigger than 90 smaller than 180
Reflex bigger than 180 smaller than 360
Triangle all angles 180

Triangle

All angles equal 180

Types on Side based:

Equilateral all sides equal size
Isosceles two sides equal
Scalene all sides and angles unequal

Types of Angle based :

Acute angle triangle : every angle smaller than 90
Right angle triangle = one angle 90
Obtuse angle triangle = angles bigger than 90

Complementary angles are pair angles with the sum of 90 degrees.

Two angles are called supplementary when their measures add up to 180 degrees.

Thus two angles are said to be adjacent angles, if they have a vertex, a common arm.

The side opposite the right angle is called the hypotenuse.

What is the meaning of a right angle triangle?

A triangle in which one of the interior angles is 90° is called a right triangle. The longest side of the right triangle, which is also the side opposite the right angle, is the hypotenuse and the two arms of the right angle are the height and the base.

Line
Set of infinite points.

Line segment
a part of a line which has two endpoints called the line segment.

Vertex (of an angle) The vertex of an angle is the common endpoint of two rays that form the angle.

The circumference is the distance measured around a circle.
380 degrees

Diameter is the distance from one side to the other crossing center.

Diameter x pi = Circumference

Pi is the ratio of circumference and diameter.

Circumference divided by diameter= pi

Diameter = 2r (Radius)

Circumference = Pi x D

= Pi x 2r

= Pi 2r

If the radius of the circle is 4cm then find its circumference.

Given: Radius = 4cm

Circumference = 2πr

= 2 x 3.14 x 4

= 25.12 cm

What is a quadrilateral shape?
A quadrilateral is a polygon that has exactly four sides. (This also means that a quadrilateral has exactly four vertices, and exactly four angles.)

A quadrilateral should be a closed shape with 4 sides. All the internal angles of a quadrilateral sum up to 360°.

Properties of a Quadrilateral:
A quadrilateral has 4 sides, 4 angles and 4 vertices.
A quadrilateral can be regular or irregular.
The sum of all the interior angles of a quadrilateral is 360°.

Square.

All the four sides are equal .
Opposite sides parallel.
Each angle is of 90.

Rectangle

All the four sides are equal .
Opposite sides parallel.
Each angle is of 90.

Parallelogram
Opposite sides equal
Opposite sides are parallel
Opposite angles are equal.
None of the angles measure 90

Rhombus
Four equal sides
Opposite sides are parallel.
Opposite angles are equal.
None of the angles measure 90.

Trapezium
Only one pair of opposite parallel sides.

Kite

Two pairs of adjacent equal sides.
Here one pair of equal angles.

Perimeter of a circle is called circumference.

Perimeter is the boundary distance of shapes, square, rectangle etc.

Perimeter
Sum all sides

If area is given
2 (L + B )

Area

Consist of square units.

Area of Rectangle

Area = length x width

Area of circle

A = π r ²

Find perimeter and area

L 5.3 cm
W 5.3

P = 2 (l+b)
= 2 ( 5.3 + 5.3)
= 21.2 cm

Area = l x w
5.3 x 5.3
28.09 cm²

Perimeter and area of rectangle shape

Perimeter
2 (l + b )

Area = l x width

Perimeter and area of Square shape

Perimeter = 4 x side
Or
a + a + a + a
Solved Problems Using Perimeter of Square Concept
Question 1: Find the perimeter of a square whose side is 5 cm.

Solution:

Given:

Side, s = 5 cm

The formula to find the perimeter of a square is given by:

The perimeter of Square = 4s units

Substitute the value of ‘s’ in the perimeter formula,

P= 4 × 5 cm

P = 20 cm

Therefore, the perimeter of square = 20 cm

Question 2: Calculate the perimeter of a square having a side of 16 cm.

Solution:

Given,
Side of the square = a = 16 cm

Perimeter of a Square = 4a
= 4 × 16
= 64 cmSolved Problems Using Perimeter of Square Concept
Question 1: Find the perimeter of a square whose side is 5 cm.

Solution:

Given:

Side, s = 5 cm

The formula to find the perimeter of a square is given by:

The perimeter of Square = 4s units

Substitute the value of ‘s’ in the perimeter formula,

P= 4 × 5 cm

P = 20 cm

Therefore, the perimeter of square = 20 cm

Question 2: Calculate the perimeter of a square having a side of 16 cm.

Solution:

Given,
Side of the square = a = 16 cm

Perimeter of a Square = 4a
= 4 × 16
= 64 cm

Solved Problems Using Perimeter of Square Concept

Question 1: Find the perimeter of a square whose side is 5 cm.

Solution:

Given:

Side, s = 5 cm

The formula to find the perimeter of a square is given by:

The perimeter of Square = 4s units

Substitute the value of ‘s’ in the perimeter formula,

P= 4 × 5 cm

P = 20 cm

Therefore, the perimeter of square = 20 cm

Question 2: Calculate the perimeter of a square having a side of 16 cm.

Solution:

Given,
Side of the square = a = 16 cm

Perimeter of a Square = 4a
= 4 × 16
= 64 cm

Area of square

Side x side = Side ²

Area = a²

Perimeter of circle
Circumference of circle

π x d
Or
31.4 x d

Area of circle

A = π r²

How to Calculate the Perimeter of a Triangle?
To calculate the perimeter of a triangle, add the length of its sides. For example, if a triangle has sides a, b, and c, then the perimeter of that triangle will be P = a + b + c

Let us consider some of the examples on the perimeter of a triangle:

Example 1: Find the perimeter of a polygon whose sides are 5 cm, 4 cm and 2 cm.

Solution: Let,

a = 5 cm

b = 4 cm

c = 2 cm

Perimeter = Sum of all sides = a + b + c = 5 + 4 + 2 = 11

Therefore, the answer is 11 cm.

Example 2: Find the perimeter of a triangle whose each side is 10 cm.

Solution: Since all three sides are equal in length, the triangle is an equilateral triangle.

i.e. a = b = c = 10 cm

Perimeter = a + b + c

= 10 + 10 + 10

= 30

Perimeter = 30 cm.

Area of triangle

Base and height given

1/2bh

3 sides given Heron's formula

Rational numbers

All integers are Rational numbers.
Means that can be written in ratio or fraction form or decimal.
P/Q
P and Q integers.
Q, Denominator is not equal to zero.

Rational numbers can be written in fractions and decimals.

Irrational numbers can't be written in fractions but can be written in decimals.

All rational numbers can be written in decimals.

All irrational numbers can be written in decimals.

Irrational numbers can't be written in fractions.

All Real numbers can be written in decimals.

Real numbers can be written in fractions except irrational numbers.

Real numbers

are simply the combination of rational and irrational numbers, in the number system

Real numbers =
Whole Numbers
Natural Numbers
Integers
Rational numbers
Irrational numbers.

Area of right triangle

A = b x h/2
Base x height both divided by 2

Fahrenheit to Celsius

1 Minus 32
2 Multiply 5/9

Celsius to Fahrenheit

C x 9/5 + 32

Part / whole percentage

Part / whole = part given / whole

Cross multiply

16/100 = 32/whole

16/100 = 32/x

16x= 3200
= 3200/16
= 200

Rational numbers = Perfect squares + Terminating decimals + Repeating decimals

👉𝓜𝓪𝓽𝓻𝓲𝓬𝓮𝓼 𝓪𝓷𝓭 𝓲𝓽𝓼 𝓣𝔂𝓹𝓮𝓼 ✨

Determinate & Indeterminate forms👇