Millions saw the apple fall, But Newton asked why.

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Rockets 101 – How to turn during flight ?

To be able to control is what distinguishes a toy rocket from a real one. And it is of quintessence to be able to channel the rocket’s direction. To be able to fly is cool, but you what is ever more cool, to be able to pinpoint the destination and its trajectory.

In most modern rockets, this is accomplished by a system known as Gimbaled Thrust.

In a gimbaled thrust system, the exhaust nozzle of the rocket can be swiveled from side to side. As the nozzle is moved, the direction of the thrust is changed relative to the center of gravity of the rocket and a torque is generated. As a result, the rocket changes direction. After necessary corrections are made, the exhaust nozzle is brought back to its initial state.

The angle by which the rocket’s nozzle swivels is known as the Gimbaled Angle.

This is what happens when Two Black Holes Collide.

This is the animation of the final stages of a merger between two black holes. What is particularly interesting about this animation is that it highlights a phenomenon known as Gravitational Lensing.

What is Gravitational Lensing?

Mass bends Light. What?

Yeah, mass can bend Light. The gravitational field of a really massive object is super strong. And this causes light rays passing close to that object to be bent and refocused somewhere else.

The more massive the object, the stronger its gravitational field and hence the greater the bending of light rays – just like using denser materials to make optical lenses results in a greater amount of refraction.

Here’s an animation showing a black hole going past a background galaxy.

This effect is one of the predictions of Einstein’s Theory of General Relativity

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.

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.

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

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

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.

I was in High School when the notion of complex numbers was fed into my vocabulary. None of it made sense. “ Why on earth did they have to invent a new Number System? Uh.. Mathematicians !! “, One of my friends remarked. And as distressing as it was, we weren’t able to comprehend why!

There are certain elegant aspects to complex numbers that are often overlooked but are pivotal to understanding them. Of the top of the chart – the events that led to the invention / discovery of complex numbers. To shed some light on these events is the crux of this post.

A date with history.

There were quotidian equations such as x² + 1 =0 which people wanted to solve, but it was well known that the equation had no solutions in the realms of real numbers. Why, you ask ?. Well, quite intuitively the addition of a square real number ( always positive ) and one was never going to yield 0.

And also, as is evident from the graph, the curve does not intersect the x- axis for a solution to persist.

For the ancient Greeks, Mathematics was synonymous with Geometry. And there were a legion geometrical problems which had no solutions, peculiar quadratics like x² + 1 =0 were branded the same way.

“ Why make up new numbers for the sole purposes for being solutions to Quadratic equations? “. This was the rationale that people stuck with.

The Real Challenge.

Quadratics, per se were easy to solve. A 16th-century mathematician’s redemption was confronting a cubic equation. Unlike Quadratics, cubic equations pass through the x axis at least once, so the existence of solution is guaranteed. To seek out for them was the challenge.

The general form of a cubic equation is as follows:

f (x) = au³ + bu² + cu + d

If we divide throughout by “a”, it simplifies the equation and substituting x = u – ( b / 3a ) gets rid of the squared term. Thus, we obtain:

x³ – 3px – 2q = 0

A mathematician named Cardano is attributed for coming up with the solution for the above equation as :

This equation is perfectly legit. But when p³ > q² it yields incomprehensible solutions.

Bombelli’s “Wild Thought”.

The strangeness of the formula enticed Bombelli. He considered the equation x³ = 15x + 4. By virtue of inspection, he found out that x = 4 satisfied the equation. But, plugging values into the cardano’s equation, he obtained:

x = ³√ ( 2 + 11 i ) + ³√ ( 2 – 11 i ) where i = √-1

Wait a minute! The equation is hinting that there exists no solution in the real numbers domain, but in contraire x = 4 is a solution!!

In a desperate attempt to resolve the paradox, he had what he called it as a ‘wild thought’. What if the equation could be broken down as

x = ( 2 + n i ) + ( 2 – n i )

This would yield x = 4 and resolve the conflict. It might sound magical, but when he tested out his abstraction, he was indeed right. From calculations, he obtained the value for n as 1. i.e

2 ± i = ³√( 2 ± 11 i )

This was the birth of Complex Numbers.

By treating a quantity such as 2 + 11√-1, without regard for its meaning in just the same way as a natural number, Bombelli unlike no one before him had come up with a modus operandi for dealing with such intricate equations, which were previously thought to have no solutions.

While complex numbers per se still remained mysterious, Bombelli’s work on Cubic equations thus established that perfectly real problems required complex arithmetic for their solutions.This empowered people to venture into frontiers which were formerly unexplored.

And for this triumph Bombelli is regarded as the Inventor of Complex Numbers.

Fun fact

A moon crater was named after Bombelli, honoring his accomplishments.

Thanks for reading! Hope you enjoyed reading it as much as I enjoyed writing it.

Only Time will tell – A Complex Number Tribute.

I was in High School when the notion of complex numbers was fed into my vocabulary. None of it made sense. “ Why on earth did they have to invent a new Number System? Uh.. Mathematicians !! “, One of my friends remarked. And as distressing as it was, we weren’t able to comprehend why!

There are certain elegant aspects to complex numbers that are often overlooked but are pivotal to understanding them. Of the top of the chart – the events that led to the invention / discovery of complex numbers. To shed some light on these events is the crux of this post.

A date with history.

There were quotidian equations such as x² + 1 =0 which people wanted to solve, but it was well known that the equation had no solutions in the realms of real numbers. Why, you ask ?. Well, quite intuitively the addition of a square real number ( always positive ) and one was never going to yield 0.

And also, as is evident from the graph, the curve does not intersect the x- axis for a solution to persist.

For the ancient Greeks, Mathematics was synonymous with Geometry. And there were a legion geometrical problems which had no solutions, peculiar quadratics like x² + 1 =0 were branded the same way.

“ Why make up new numbers for the sole purposes for being solutions to Quadratic equations? “. This was the rationale that people stuck with.

The Real Challenge.

Quadratics, per se were easy to solve. A 16th-century mathematician’s redemption was confronting a cubic equation. Unlike Quadratics, cubic equations pass through the x axis at least once, so the existence of solution is guaranteed. To seek out for them was the challenge.

The general form of a cubic equation is as follows:

f (x) = au³ + bu² + cu + d

If we divide throughout by “a”, it simplifies the equation and substituting x = u – ( b / 3a ) gets rid of the squared term. Thus, we obtain:

x³ – 3px – 2q = 0

A mathematician named Cardano is attributed for coming up with the solution for the above equation as :

This equation is perfectly legit. But when p³ > q² it yields incomprehensible solutions.

Bombelli’s “Wild Thought”.

The strangeness of the formula enticed Bombelli. He considered the equation x³ = 15x + 4. By virtue of inspection, he found out that x = 4 satisfied the equation. But, plugging values into the cardano’s equation, he obtained:

x = ³√ ( 2 + 11 i ) + ³√ ( 2 – 11 i ) where i = √-1

Wait a minute! The equation is hinting that there exists no solution in the real numbers domain, but in contraire x = 4 is a solution!!

In a desperate attempt to resolve the paradox, he had what he called it as a ‘wild thought’. What if the equation could be broken down as

x = ( 2 + n i ) + ( 2 – n i )

This would yield x = 4 and resolve the conflict. It might sound magical, but when he tested out his abstraction, he was indeed right. From calculations, he obtained the value for n as 1. i.e

2 ± i = ³√( 2 ± 11 i )

This was the birth of Complex Numbers.

By treating a quantity such as 2 + 11√-1, without regard for its meaning in just the same way as a natural number, Bombelli unlike no one before him had come up with a modus operandi for dealing with such intricate equations, which were previously thought to have no solutions.

While complex numbers per se still remained mysterious, Bombelli’s work on Cubic equations thus established that perfectly real problems required complex arithmetic for their solutions.This empowered people to venture into frontiers which were formerly unexplored.

And for this triumph Bombelli is regarded as the Inventor of Complex Numbers.

Fun fact

A moon crater was named after Bombelli, honoring his accomplishments.

The 2015 Ig Nobel awards have been as comical as usual.

Image re-composed in Gimp from the official poster (PDF).

All have been very well-deserved

recipients, but my favorites are:

MATHEMATICS PRIZE — Elisabeth Oberzaucher [AUSTRIA, GERMANY, UK] and Karl Grammer [AUSTRIA, GERMANY], for trying to use mathematical techniques to determine whether and how Moulay Ismael the Bloodthirsty, the Sharifian Emperor of Morocco, managed, during the years from 1697 through 1727, to father 888 children.

BIOLOGY PRIZE — Bruno Grossi, Omar Larach, Mauricio Canals, Rodrigo A. Vásquez [CHILE], José Iriarte-Díaz [CHILE, USA], for observing that when you attach a weighted stick to the rear end of a chicken, the chicken then walks in a manner similar to that in which dinosaurs are thought to have walked.

PHYSICS PRIZE — Patricia Yang [USA and TAIWAN], David Hu [USA and TAIWAN], and Jonathan Pham, Jerome Choo [USA], for testing the biological principle that nearly all mammals empty their bladders in about 21 seconds (plus or minus 13 seconds).

CHEMISTRY PRIZE — Callum Ormonde and Colin Raston [AUSTRALIA], and Tom Yuan, Stephan Kudlacek, Sameeran Kunche, Joshua N. Smith, William A. Brown, Kaitlin Pugliese, Tivoli Olsen, Mariam Iftikhar, Gregory Weiss [USA], for inventing a chemical recipe to partially un-boil an egg.