Sri Lankan Math Geek

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A Rose from the Realm of Mathematics

Mathematics isn't just about numbers; it's a profound way to explore and represent patterns that resonate with beauty. Recently, I stumbled upon a fascinating way to blend mathematical equations with graphical representation using Python. Let me introduce you to two gorgeous 3D visualizations, both rooted in a simple yet intricate equation. import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D # Parameters n = 800 A = 1.995653 B = 1.27689 C = 8 petalNum = 3.6 # Create grid r = np.linspace ( 0 , 1 , n ) theta = np.linspace ( -2 , 20 * np.pi , n ) R , THETA = np.meshgrid ( r , theta ) # Calculate coordinates x = 1 - ( 1 / 2 ) * (( 5 / 4 ) * ( 1 - np.mod ( petalNum * THETA , 2 * np.pi ) / np.pi ) ** 2 - 1 / 4 ) ** 2 phi = ( np.pi/ 2 ) * np.exp ( -THETA / ( C * np.pi )) y = A * ( R** 2 ) * ( B * R - 1 ) ** 2 * np.sin ( phi ) R2 = x * ( R * np.sin ( phi ) + y * np.cos ( phi )) X = R2 * np.sin ( THETA ) Y = R2 * np.cos (...

The Hilarious "i at Home" Meme: A Mathematical Twist

Have you ever come across a meme that perfectly blends humor with a touch of mathematics? One such meme that caught my eye recently goes like this: At first glance, it might seem like just another funny meme, but there's a clever twist that makes it stand out. Let's break it down and appreciate the humor behind it. Explanation:

$i$ in mathematics usually refers to the imaginary unit, where

$i = − 1 i = \sqrt{-1} In$ \mathbb{Z}_17

$,$ 16=-1

$Then$ sqrt{16}=4=\sqrt{-1}=i` Have you come across any other math-related memes that made you laugh?

Exploring Trigonometric Identities

Trigonometric identities often present fascinating challenges that require a blend of algebraic manipulation and understanding of trigonometric properties. They allow us to transform and simplify expressions, making it easier to solve equations and understand the relationships between different trigonometric functions. If

$sin(x)+cos(x)=\sqrt{3}-1$ , then what is

$sin(2x)$ ? . . . . . . . . . . . . . . . . . . . Show Answer Answer:

$3-2\sqrt{3}$ Explanation:

$(sin(x)+cos(x))^2=(\sqrt{3}-1)^2$

$\underbrace{\sin^2(x)+\cos^2(x)}_1+\underbrace{2\sin(x)\cos(x)}_\sin(2x)=3-2\sqrt{3}+1$

$1+\sin(2x)=3-2\sqrt{3}+1$

$\sin(2x)=3-2\sqrt{3}$

Exploring Digit Properties: Solving for Two-Digit Numbers

Mathematics often presents intriguing problems that involve the properties of digits within numbers. In this blog post, we'll explore a problem where we need to find all two-digit numbers $ N $ that satisfy a specific equation involving the product and sum of their digits. Let's dive into the details and solve this problem step-by-step. Let

$P(n)$ and

$S(n)$ denote the product and sum, respectively, of the digits of a positive integer

$n$ . Determine all two-digit numbers

$N$ that satisfy the equation:

$P(N) + 2S(N) = N$ For example,

$P(23)=2\times 6$ and

$S(23)=2+3=5$ . . . . . . . . . . . . . . . . Show Answer Answer:

$14, 36, 77$ Explanation: Let

$N = 10a + b$ , where

$a$ and

$b$ are the tens and units digits of

$N$ , respectively. Here,

$a$ and

$b$ are integers such that

$1 \leq a \leq 9$ and

$0 \leq b \leq 9$ . Then , The product of the digits

$P(N)$ is

$P(N) = a \cdot b$ The sum of the digits

$S(N)$ is: ...

Exploring Maximum Values: Finding the Peak of a Rational Function

In mathematics, finding the maximum value of a function is a common and intriguing problem. In this blog post, we'll explore how to determine the maximum value of the function

$E=\frac{48}{p^2-2p+3}$ . where

$p$ is a real number. Let's dive into the details and solve this problem step-by-step. What is the maximum value of

$E$ if

$p$ is real number?

$E=\frac{48}{p^2-2p+3}$ . . . . . . . . . . . . . . Show Answer Answer: 24 Explanation: First, observe the numerator is constant. Then, observe that quadric expression, The maximum value of E, we need to minimize the denominator. (E is inversely proportional to denominator Let's say denominator

$\phi(p)=p^2-2p+3$ . It has critical point of'

$\phi'(p)=2p-2=0\implies p=1$ . denominator has minimum when

$p = 1$ .

$E$ has maximum where

$p = 1$ .

Exploring Logarithmic Graphs: Finding the Distance Between Points

Logarithmic functions often present interesting challenges in mathematics. In this blog post, we'll explore a problem involving points on the graph of a logarithmic function and how to find the positive difference between their

$x$ -coordinates. Let's dive into the details and solve this intriguing problem. Points

$A$ and

$B$ lie on the graph

$y=\log_2(x)$ . The mid point of

$AB$ is

$(6,2)$ . What is the positive difference between

$x$ -coordinates of

$A$ and

$B$ ? . . . . . . . . . . . . . . . . . . Show Answer Answer:

$4\sqrt{5}$ Explanation:

Simple Geometry Riddle

Geometry often presents fascinating problems that challenge our understanding of shapes and their relationships. In this blog post, we'll explore a problem involving a circle inscribed in a square and a smaller square inscribed within that circle. Let's dive into the details and find the area of the smaller square. A circle inscribed a in square The radius of the circle is

$r$ . A smaller square is inscribed inside the circle with each of its sides touching the circle. If the area of larger square is

$4r^2$ . What is the area of the smaller square? . . . . . . . . . . . . . . . . . . . . . . Show Answer Answer:

$2r^2$ Explanation:

The Train and Platform Riddle

When it comes to solving real-world problems using mathematics, understanding the relationship between speed, distance, and time is crucial. In this blog post, we'll explore how to calculate the length of a platform given the speed of a train, the length of the train, and the time it takes to cross the platform. A train is

$120m$ long and crosses a platform in

$20$ seconds at

$54km/h$ . What is the platform length? . . . . . . . .. . . . . . . . . . . . . . . . . . . Show Answer Answer:

$180m$ Explanation:

If $x+\frac{1}{x}=3$ , find $x^3+\frac{1}{x^3}$ ?

Mathematics often presents us with intriguing problems that require a blend of algebraic manipulation and logical reasoning. Try to solve the following problem. If

$x+\frac{1}{x}=3$ , find

$x^3+\frac{1}{x^3}$ ? This is very easy one. Even Grade 8 students also can do it. . . . . . . . . . . . . . . Show Answer Answer: 18 Explanation:

$\left( x+\frac{1}{x} \right)^3=x^3+3x+\frac{3}{x}+\frac{1}{x^3}$

$\left( x+\frac{1}{x} \right)^3=x^3+3\left(x+\frac{1}{x}\right)+\frac{1}{x^3}$

$\left(3 \right)^3=x^3+3\left(3\right)+\frac{1}{x^3}$

$27=x^3+9+\frac{1}{x^3}$

$x^3+\frac{1}{x^3}=27-9=18$

Predicting Your Symbol: The Ultimate Number Trick!

Have you ever stumbled upon a math trick so intriguing, it feels like magic? Today, we’re diving into an enchanting number trick that involves simple addition and subtraction. Follow these steps and get ready to be amazed! If you need to see the magic click here Step-by-Step Instructions: 1. Choose a number between 10 and 99. - For example, let's take 74 2. Add the two digits of your number together. - In our example: 7 + 4 = 11. 3. Subtract the sum you just calculated from your original number. - So, 74 - 11 = 63. 4. Find your result in the diagram below. Note the symbol associated with that number. 95 ♣ 84 ☃ 73 ☃ 62 ♣ 51 ☃ 40 ☂ 29 ♦ 18 ♦ 7 ♥ 94 ☂ 83 ☂ 72 ♦ 61 ♠ 50 ♦ 39 ☃ 28 ♠ 17 ★ 6 ♥ 93 ♪ 82 ★ 71 ☀ 60...