If I could catch a rainbow

If i could catch a rainbow,

I would just do it for you

And share with you it’s beauty,

On the days you are feeling Blue.

Rainbows are nature’s optical illusion.

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It’s not possible unfortunately to catch a rainbow. They are not objects and are not located at specific distance from the observer that one can physically approach.

Rainbows stems from an optical illusion caused by any water droplets viewed from a
certain angle relative to a light source.

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They are user-specific and everyone sees a different rainbow.

The monochrome rainbow

Not all rainbows that occur in nature are multicolored. Under specific
atmospheric conditions, one can spot the Mono-chrome rainbow i.e It has
only one color.

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                                   PC : rodjonesphotography


Moonbows

A Moonbow / Lunar rainbow /White rainbow  is a
rainbow produced by light reflected off the surface of the moon (as
opposed to direct sunlight) refracting off of moisture-laden clouds in
the atmosphere / from waterfalls.

image

                                            PC:

GanMed64


Each of your eyes sees a different rainbow.

Just as no two people see exactly the same rainbow, even if they’re
standing next to each other, the few inches between your eyes make a
difference in what you are viewing. 

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There is no color- indigo ( sort of )

One can distinguish almost all colors in a rainbow but Indigo.

Legend has it that Newton
included indigo because he felt that there should be seven rather than six colors in a rainbow due to his strong religious beliefs.

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Origins of ‘Iris’

The Greeks and Romans thought a rainbow was the path made
by Iris, the goddess of the rainbow, between heaven and earth, linking
gods with humans. “Rainbow” in Latin is arcus iris or arcus pluvius, a “rainy arch”.

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The iris of the eye is named after her, because of its
colour. 

The
Greeks used the word “iris” to describe any coloured circle, such as
the “eye” of a peacock’s tail. The flower called iris gets its name from
the Greek, as does the chemical iridium (Ir), compounds of which are
highly coloured. Iris is also the root of “iridescent”.

Pulsating Rainbows

Place a linear polarizer over the camera whilst capturing a rainbow and you get pulsating rainbows.

image

                                                   Source


Double Rainbows/ Multiple Rainbows

A double rainbow is a phenomenon in which two rainbows appear. They are caused by a double reflection of sunlight inside the raindrops. Similarly multiple rainbows are a possibility as well.

Observe that the colors in the second rainbow are inverted because the light is reflected twice inside the water droplet

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                                             PC: Janbazian


Viral Double Rainbow Video

This video of a man witnessing a double rainbow for the first time went viral,  featuring on numerous popular talk shows. Pure ecstasy!

The full rainbow

Whilst standing on earth, we see rainbows as magical arcs across the sky, but rainbows are full circles. The bottom part of the full circle is usually blocked by the horizon.

Pilots however do not face this difficulty. Under the right sky conditions, pilots are spectators to one of nature’s most beautiful spectacles – The full rainbow.

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                                          PC :
Steve Kaufman

Everything has beauty, but not everyone sees it.

Have a great day!

Proof without words – Pi

This was intended to be posted on Pi-day earlier this month, but somehow that didn’t happen.

Hope this beautiful pi gif on this sizzling Saturday puts a smile on your face and guides you through the day.

Have a good one!


Photo credit: Lucas V. Barbosa via Wikimedia Commons

** FYP’s Pi-day post ( if you are interested )

An Introduction to Calculus: Part 0

fuckyeahphysica:

image

                                             Source

A lot of people have written in and asked for a down-to-earth, introductory tutorial on calculus. 

This is no small task because of the vastness of the subject, so I’ve decided to write a series of posts to explain the motivation, ideas, and techniques used in calculus!

image

I’m calling this post Part 0 because here, I just want to tell you guys about the big idea! So the question we want to address:

Whats the big idea?!

And unlike most other things in mathematics, the answer here is both succinct and somewhat straightforward, conceptually:

The big idea is: Limits!

image

Calculus is an enormous discipline whose reach spans nearly every quantitative discipline beyond basic algebra, but at the heart of it all is the notion of a limit.

Okay…so…what is a limit?

Intuitively, a limit is a tool which allows you to analyze the behavior of a thing (a sequence of numbers, a function, a collection of spaces…) as it gets near a certain thing (an index, a domain value, a fixed other space…).

For instance: What is the behavior of the function near the point where Gandalf’s staff is at? That’s an example of what limits measure!

image

Algebra on the other hand is language through which we describe patterns.

image

This, in a nutshell, is what separates calculus from other areas of math:

In much the same way that the shift from numbers to variables marks the jump from arithmetic to algebra, the introduction of the limit marks the jump from algebra to calculus.

image

Said differently, variables allow quantities to vary from one situation to another; limits allow quantities to vary from one situation to another and to change within a single context.

Conceptually, it’s a little strange, but with some practice, it becomes pretty natural.

So, like…is there an example…?

There are tons! 

If you have a sequence of numbers like 1, ½ , ¼ , 1/16 , 1/64..  and you want to know what happens to this sequence as we go farther and farther out?

image

In particular, what do the numbers in the sequence get close to (whatever that means) as you write down more and more numbers which follow the same pattern

image

What you’re really asking is what the limit of the sequence is, and since the numbers are getting smaller while never becoming negative, it seems likely that their limit is 0! Voila!

As another example:

Let’s say we have a function which is defined to be the parabola y=  x² everywhere except at x=2, where it has a hole.

image

Graphically, this looks like the parabola with a hole at ( 2,4 )

image

                                                     Source

In this example, it very clearly doesn’t make sense to talk about what the function does at x=2 since it’s undefined there.

However, it does make sense to ask what the function does near x=2  and even though a picture isn’t a proof, we can probably all be convinced that as the x gets closer to 2, the function approaches 4.

image

So, the obvious next question is:

How do we find a limit?

And there are two answers:

The short answer: Very carefully. Seriously: Don’t muck something up because of carelessness!

The long answer: …

The long answer is far too long to address here, so it’ll be the topic of the next post in the introduction to calculus series! So, you know…stay tuned!

Have a great day everyone!

By the way: I’m Chris and I’m a guest poster here on FYP! I’m a doctoral student in mathematics (I study a branch called topology) and I have a main blog (source) and a side math-type blog (source); the latter needs some freshening up, though. ^_^

An Introduction to Calculus: Part 0

image

                                             Source

A lot of people have written in and asked for a down-to-earth, introductory tutorial on calculus. 

This is no small task because of the vastness of the subject, so I’ve decided to write a series of posts to explain the motivation, ideas, and techniques used in calculus!

image

I’m calling this post Part 0 because here, I just want to tell you guys about the big idea! So the question we want to address:

Whats the big idea?!

And unlike most other things in mathematics, the answer here is both succinct and somewhat straightforward, conceptually:

The big idea is: Limits!

image

Calculus is an enormous discipline whose reach spans nearly every quantitative discipline beyond basic algebra, but at the heart of it all is the notion of a limit.

Okay…so…what is a limit?

Intuitively, a limit is a tool which allows you to analyze the behavior of a thing (a sequence of numbers, a function, a collection of spaces…) as it gets near a certain thing (an index, a domain value, a fixed other space…).

For instance: What is the behavior of the function near the point where Gandalf’s staff is at? That’s an example of what limits measure!

image

Algebra on the other hand is language through which we describe patterns.

image

This, in a nutshell, is what separates calculus from other areas of math:

In much the same way that the shift from numbers to variables marks the jump from arithmetic to algebra, the introduction of the limit marks the jump from algebra to calculus.

image

Said differently, variables allow quantities to vary from one situation to another; limits allow quantities to vary from one situation to another and to change within a single context.

Conceptually, it’s a little strange, but with some practice, it becomes pretty natural.

So, like…is there an example…?

There are tons! 

If you have a sequence of numbers like 1, ½ , ¼ , 1/16 , 1/64..  and you want to know what happens to this sequence as we go farther and farther out?

image

In particular, what do the numbers in the sequence get close to (whatever that means) as you write down more and more numbers which follow the same pattern

image

What you’re really asking is what the limit of the sequence is, and since the numbers are getting smaller while never becoming negative, it seems likely that their limit is 0! Voila!

As another example:

Let’s say we have a function which is defined to be the parabola y=  x² everywhere except at x=2, where it has a hole.

image

Graphically, this looks like the parabola with a hole at ( 2,4 )

image

                                                     Source

In this example, it very clearly doesn’t make sense to talk about what the function does at x=2 since it’s undefined there.

However, it does make sense to ask what the function does near x=2  and even though a picture isn’t a proof, we can probably all be convinced that as the x gets closer to 2, the function approaches 4.

image

So, the obvious next question is:

How do we find a limit?

And there are two answers:

The short answer: Very carefully. Seriously: Don’t muck something up because of carelessness!

The long answer: …

The long answer is far too long to address here, so it’ll be the topic of the next post in the introduction to calculus series! So, you know…stay tuned!

Have a great day everyone!

By the way: I’m Chris and I’m a guest poster here on FYP! I’m a doctoral student in mathematics (I study a branch called topology) and I have a main blog (source) and a side math-type blog (source); the latter needs some freshening up, though. ^_^

By virtue of everyday usage, the fact that (-1) x (-1) = 1 has been engraved onto our heads. But, only recently did I actually sit down to explore why, in general negative times negative yields a positive number !

Intuition.

Let’s play a game called “continue the pattern”. You would be surprised, how intuitive the results are:

2 x 3 = 6

2 x 2 = 4

2 x 1 = 2

2 x 0 = 0

2 x (-1) = ??  (Answer : -2 )

2 x (-2 ) = ?? (Answer : -4 )

2 x ( -3) = ?? (Answer : -6 )

The number on the right-hand side keeps decreasing by 2 !

Therefore positive x negative = negative. 


2 x -3 = -6

1 x -3 = -3

0 x -3 = 0

-1 x -3 = ?? (Answer : 3)

-2 x -3 = ?? (Answer : 6)

The number on the right-hand side keeps increasing by 3.

Therefore negative x negative = positive.

Pretty Awesome, right? But, let’s up the ante and compliment our intuition.

The Number Line Approach.

Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y.

image

Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps.

image

So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.

image

Ergo, -1 x -1 means to stand on 0, face in the negative direction, and then take 1 backward step. This lands us smack right on +1 !

The Complex Numbers Approach.

The “i” in a complex number is an Instruction! An instruction to turn 90 degrees in the counterclockwise direction. Then i * i would be an instruction to turn 180 degrees. ( i x i = -1 ). where i = √-1

image

Similarly ( -1 ) x i x i = (- 1 ) x ( -1 )= 1. A complete revolution renders you back to +1.

We can snug in conveniently with the knowledge of complex numbers. But, complex numbers were established only in the 16th century and the fact that negative time negative yields a positive number was well established before that.

Concluding remarks.

Hope you enjoyed the post and Pardon me if you found this to be rudimentary for your taste. This post was inspired by Joseph H. Silverman’s Book – A friendly Introduction to Number Theory. If you are passionate about numbers or math, in general it is a must read.

There are several other arithmetic methods that prove the same, if you are interested feel free to explore.

Have a Good Day!

PC: mathisfun

First Order LDE and that mysterious Integrating Factor.

The integrating factor can be deeply confusing if you don’t understand what it does. This was an attempt to explain the first order linear differential equation to a friend. 

(Sorry about the image quality though. I had to make these in a hurry )

Types of Damping

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations.

There are 4 types of damping:(in the order of the animations shown)

1. Under Damped System.

The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.

2. Critically Damped System.

The system returns to equilibrium as quickly as possible without oscillating.

3. Over Damped System.

The system returns to equilibrium without oscillating.

4. Un-Damped System.

The system oscillates at its natural resonant frequency

( Sources: xmdemo, timewarp,wikipedia)