AJAY PATHAK (MATH)

Solution: Triangle Inequality Statement: For all real numbers x and y, |x + y| ≤ |x| + |y| (a) Verification Case (i): x = 2, y = 3 |x + y| = |2 + 3| = |5| = 5 |x| + |y| = |2| + |3| = 2 + 3 = 5 ∴ |x + y| = |x| + |y| Case (ii): x = −2, y = −3 |x + y| = |−2 − 3| = |−5| = 5 |x| + |y| = |−2| + |−3| = 2 + 3 = 5 ∴ |x + y| = |x| + |y| Case (iii): x = −2, y = 3 |x + y| = |−2 + 3| = |1| = 1 |x| + |y| = |−2| + |3| = 2 + 3 = 5 ∴ |x + y| ≤ |x| + |y| (b) Proof for all real numbers We prove the inequality by considering different cases. Case 1: x ≥ 0, y ≥ 0 |x + y| = x + y = |x| + |y| Hence, |x + y| ≤ |x| + |y|. Case 2: x ≤ 0, y ≤ 0 |x + y| = −(x + y) = −x − y = |x| + |y| Hence, |x + y| ≤ |x| + |y|. Case 3: x and y have opposite signs In this case, the sum x + y is reduced in magnitude. Therefore, |x + y| < |x| + |y| Hence, the inequality holds. Therefore, the Triangle Inequality holds for all real numbers x and ...

Max and Min of Two Numbers Using Absolute Value Maximum and Minimum of Two Numbers Using Absolute Value The maximum and minimum of two real numbers a and b can be expressed using the absolute value function. These formulas are very useful in mathematical analysis and proofs. 1. Formula for Maximum max(a, b) = (a + b + |a − b|) / 2 Proof Case 1: a ≥ b a − b ≥ 0 ⇒ |a − b| = a − b (a + b + |a − b|) / 2 = (a + b + (a − b)) / 2 = a Hence, max(a, b) = a . Case 2: b > a a − b < 0 ⇒ |a − b| = b − a (a + b + |a − b|) / 2 = (a + b + (b − a)) / 2 = b Hence, max(a, b) = b . 2. Formula for Minimum min(a, b) = (a + b − |a − b|) / 2 Proof Case 1: a ≤ b a − b ≤ 0 ⇒ |a − b| = b − a (a + b − |a − b|) / 2 = (a + b − (b − a)) / 2 = a Hence, min(a, b) = a . Case 2: b < a a − b > 0 ⇒ |a − b| = a − b (a + b − |a − b|) / 2 = (a + b − (a − b)) / 2 = b Hence, min(a, b) = b . Importa...

Probability and Statistics — Formula Book (GTU: BE03000251) Probability and Statistics — Formula Book GTU: BE03000251 — Semester 3 Contents Unit 1: Basic Probability Unit 2: Special Probability Distributions Unit 3: Basic Statistics (Grouped and Ungrouped Data) Correlation and Regression Unit 4: Hypothesis Testing (Applied) Unit 5: Curve Fitting by Least Squares Appendices Unit 1: Basic Probability Basic concepts Experiment , Outcome , Sample space \(S\). Event : subset \(A\subseteq S\). Probability measure \(P\) satisfies: \(0\le P(A)\le1\), \(P(S)=1\), countable additivity. Axioms and simple results \(P(\varnothing)=0,\qquad P(S)=1.\) \(P(A^c)=1-P(A).\) Addition rule \(P(A\cup B)=P(A)+P(B)-P(A\cap B).\) For mutually exclusive events (\(A\cap B=\varnothing\)): \(P(A\cup B)=P(A)+P(B)\). Conditional probabili...

: South Korea's Declining Population South Korea is experiencing a significant demographic shift characterized by a declining population. This trend is driven by several socio-economic and cultural factors, posing challenges for the country's future. Current Population Trend As of 2024, South Korea's population has slightly declined by 0.02% from the previous year, bringing the total population to approximately 51.7 million. Projections indicate that this trend will accelerate, with the population potentially dropping to around 34.4 million by 2070. Key Causes of Population Decline Low Fertility Rate: South Korea's fertility rate is one of the lowest globally, at 0.78 children per woman, far below the replacement level of 2.1. High Costs of Living: Child-rearing costs are prohibitively expensive, discouraging families from having children. Cultural Shifts: Many young adults prior...

The Golden Ratio in Art, Architecture, and Nature The Golden Ratio in Art, Architecture, and Nature The Golden Ratio , also known as the divine proportion, has fascinated artists, architects, and mathematicians for centuries. This special number, approximately equal to 1.618 , appears in many forms in art, architecture, and nature. Let's explore how it shapes our world. What is the Golden Ratio? The Golden Ratio is derived from the Fibonacci sequence, where each number is the sum of the two preceding ones. If you divide a number in the sequence by its previous number, the result converges to 1.618 as the numbers grow larger. "Mathematics is the language of the universe, and the Golden Ratio is its poetry." Golden Ratio in Art Many artists have used the Golden Ratio to create aesthetically pleasing compositions. Some notable examples include: Leonardo da Vinci's "Vitruvian Man,...

History of the October 1582 Calendar The October 1582 Calendar: A Historic Transition The year 1582 marked a pivotal moment in the history of timekeeping. This year witnessed the adoption of the Gregorian calendar , which replaced the Julian calendar in several Catholic countries. This transition was introduced by Pope Gregory XIII to correct inaccuracies in the Julian calendar, particularly in the calculation of the spring equinox and Easter. Why the Change Was Necessary The Julian calendar, introduced by Julius Caesar in 45 BCE, was based on a solar year of 365.25 days. However, the true solar year is approximately 365.2422 days long. This slight discrepancy of about 11 minutes per year caused the calendar to drift gradually over centuries, misaligning with astronomical events like equinoxes. By the 16th century, the spring equinox had shifted by approximately 10 days, affecting the date of Easter . This was a significant concer...

First Invention of Science: Wheel First Invention of Science: Wheel Introduction The wheel is one of the most significant inventions in human history, revolutionizing transportation, industry, and technology. Its invention marks a major milestone in the development of civilization, enabling advancements in almost every aspect of human life. The Origin of the Wheel The wheel is believed to have been invented around 3500 BCE in ancient Mesopotamia, which is part of Iraq in present-day. The first use of the wheel was not for transport, as most people assume, but as a component of a potter’s wheel. As a result of this early use of the wheel, craftsmen were able to produce more rounded and more effective vessels that were important for the pottery trade and everyday needs. Evolution of the Wheel The wheel began as a simple, solid disc, but over time, its design evolved....