Random facts, just for fun and curiosity

[Sarah B]

Hi Everyone

If you thought Pi was okay, you are wrong.  Let me show you something:

Introduction
Pi (π) and Euler's number (e) are two of the most fundamental mathematical constants, appearing in a wide range of mathematical and scientific disciplines.  While they originate from different areas π from geometry and e from calculus and exponential growth they are deeply connected in various ways. Here are some key relationships between them:

1. Euler's Identity (The Most Famous Relationship)
One of the most beautiful and profound equations in mathematics is Euler's identity:

e^iπ + 1 = 0

This equation connects five of the most important mathematical constants: e, π, i (the imaginary unit), 1, and 0. It demonstrates the deep relationship between exponential growth, trigonometry, and complex numbers.

2. Euler's Formula (General Form)
Euler's identity is a special case of a more general formula:

e^iπ = cos(𝑥) + isin(𝑥)

When 𝑥 = 𝜋, this reduces to Euler's identity.

3. Connection in Calculus

• π appears in trigonometry (e.g., sine and cosine functions, Fourier series).
• e appears in exponential and logarithmic functions (e.g., growth/decay, compound interest).

Both are connected through the Taylor series expansions of sine, cosine, and the exponential function.
For example, the Taylor series for 𝑒^𝑥

𝑒^𝑥 = ^n=∞ ∑ _𝑛=0 (𝑥^𝑛/𝑛!)
 
The Taylor series for sine and cosine involve π and have a structure similar to e^𝑥.

4. Gaussian Integral (Probability and Statistics)
The normal distribution in probability theory is described by the function:

𝑓(𝑥) = 1/(Sqrt(2𝜋)) x 𝑒^-(𝑥2/2)

Here, both π and e appear naturally, demonstrating their fundamental role in statistics and probability.

5. Wallis Product (Connecting π and e)
The Wallis product for π can be derived using an integral that also involves e:

𝜋/2 = ^∞ ∏ _𝑛=1 (2𝑛)²/((2n − 1)(2n + 1))

A similar infinite product for e can be found in number theory and combinatorics.

6. Complex Analysis and the Gamma Function
The Gamma function, which generalizes the factorial to complex numbers, has deep connections to both π and e. One key relationship is:

Γ(1/2) = sqrt(𝜋)
 
Which is derived from an integral involving e.

Conclusion
Pi (π) and Euler's number (e) may seem to come from different fields, but they are fundamentally intertwined through complex numbers, calculus, probability, and special functions. Euler's identity, e^iπ + 1 = 0 is one of the most famous and elegant demonstrations of their deep connection.

Now back to your regular programming: "Random facts, just for fun and curiosity."

Best Wishes Always
Sarah B
Global Moderator
PS Never let a Math Geek loose.

[ChrissyRyan]

Sarah,

Fascinating.


Chrissy

[ChrissyRyan]

Tanks do not get good fuel mileage.

I wonder how many tanks are lost in battle because they ran out of fuel.

Chrissy

[ChrissyRyan]

Yes!  You can ride in real tanks at some "tank driving experience" locations in the USA, such as Minnesota and Florida.  You can do a Web search and find some.

[ChrissyRyan]

"2S" in some of the LGBTQIA+ or similar labels stands for "Two Spirit" or "Two Spirited."

Such as 2SLGBTQ++

[ChrissyRyan]

Circles have two (2) sides:  The inside and the outside.

Actually, some mathematicians argue that a circle has no side or an infinite number of sides.



P. S.  I do not care.  Seems obvious to me:  an inside and an outside.

[ChrissyRyan]

Lori's Age: One-half of Lori's age two years from now plus one-third of her age three years ago is twenty years. 

To find her current age, you can set up the equation (setting up equations is fascinating and sometimes harder than it initially seems):


1/2(x+2) + 1/3(x−3) = 20, where x is her current age. 

Easy Peasy after you setup the basic algebra equations, right?  What is Lori's current age?

[ChrissyRyan]

Nancy's and Sarah's Ages:  Sarah is three times Nancy's age.  In ten years, Sarah will be twelve years older than Nancy. 

You can set up the equations:
S=3N and
S+10 = N+10+12, where
S is Sarah's age and
N is Nancy's age.

What are each of their ages?  Easy Peasy after you setup the algebra equations.

[davina61]

Over my head Chrissy-----------
 The Malvern hills were formed from volcanos bordering a shallow lagoon, Malvern water is filtered through the granite and gives it a distinct taste. Sand and gravel from the lagoon are dug up around here and there is fossil shells found in it.

[MaryT]

Lori's Age: One-half of Lori's age two years from now plus one-third of her age three years ago is twenty years. 

To find her current age, you can set up the equation (setting up equations is fascinating and sometimes harder than it initially seems):


1/2(x+2) + 1/3(x−3) = 20, where x is her current age. 

Easy Peasy after you setup the basic algebra equations, right?  What is Lori's current age?

I got 24 which is about as old as she looks.

[Lori Dee]

I got 24 which is about as old as she looks.

 

You are too kind. I turn 68 in September. Your equation is missing a 3x in there somewhere. 

[ChrissyRyan]

I got 24 which is about as old as she looks.

Mary,


You are correct on both counts for Lori.  For Lori Dee too.  Could be the same person.


Chrissy

[ChrissyRyan]

I will forgo the trains going in opposite directions on parallel tracks problems, where you may be asked when the trains will initially pass each other.

So much fun it was in junior high math class, right? 


Chrissy

[ChrissyRyan]

The fear of long words is called Hippopotomonstrosesquippedaliophobia.

[Sarah B]

Hi All

Nancy's and Sarah's Ages:  Sarah is three times Nancy's age.  In ten years, Sarah will be twelve years older than Nancy. 

You can set up the equations:
S=3N and
S+10 = N+10+12, where
S is Sarah's age and
N is Nancy's age.

What are each of their ages?  Easy Peasy after you setup the algebra equations.

Sarah is 18 years old and Nancy is 6 years old

Hugs
Sarah B

[Sarah B]

Hi All

Since Chrissy is reluctant to do a train problem then I shall.

Problem

Two trains start at the same time from two different cities that are 300 km apart and travel toward each other on parallel tracks.

Train A travels at 80 km/h
Train B travels at 100 km/h
At what time will the two trains meet?

Hugs
Sarah B

[ChrissyRyan]

Mount Everest is not the tallest mountain on Earth.
Mauna Kea and Mauna Loa in Hawaii, twin volcanoes, are taller than Mount Everest because 4.2 km of their heights are submerged.

These twin volcanoes measure 10.2 km in total height compared to Everest's 8.8 km.

[ChrissyRyan]

Hi All

Sarah is 18 years old and Nancy is 6 years old

Hugs
Sarah B

Sarah,

Those ages are correct.

Chrissy

[Sarah B]

Hi All

Solution

Step 1: Define Variables
Let t be the time (in hours) it takes for the trains to meet.

Distance covered by Train A in time t = 80t km
Distance covered by Train B in time t = 100t km

The total distance between them is 300 km

Since they are moving toward each other, their combined distance covered is equal to the total distance:

80𝑡 + 100𝑡 = 300

Step 2: Solve for t

80𝑡 + 100𝑡 = 300

180𝑡 = 300

𝑡 = 300/180

𝑡 = 1.67 hours

Since 0.67 is about 40 minues we conclude:

𝑡 = 1 hour 40  minutes approximately

Final Answer:

The trains will meet 1 hour and 40 minutes after they start traveling toward each other.

Hugs
Sarah B

[ChrissyRyan]

Hi All

Since Chrissy is reluctant to do a train problem then I shall.

Problem

Two trains start at the same time from two different cities that are 300 km apart and travel toward each other on parallel tracks.

Train A travels at 80 km/h
Train B travels at 100 km/h
At what time will the two trains meet?

Hugs
Sarah B

First, calculate the relative speed at which the two trains are closing the distance between them. The combined speed of Train A and Train B is 80km/h+100km/h=180km/h

Second, because the distance between the two cities is 300 km, the time it takes for the trains to meet is 300 km / 180 km/h, which is 5/3 hours, or 1 hour and 40 minutes, or 100 minutes.