The white circles albeit traveling in a straight line across the circle exhibit a more collective circular behavior.
Here’s a much more real world scenario which follows similar guidelines:
If you notice, the motion of all the workers individually are also periodic in nature, but each of their motion is slightly out of phase leading to this beautiful symmetric behavior that constitutes this gif.
Truly mesmerizing!
The mathematical sciences exhibit order, symmetry and limitations; and these are the greatest forms of the beautiful
A lot of times I have fallen into this pitfall where I seem to
completely understand how to methodically do something without actually
comprehending what it means.
And only after several years after I first
encountered the notion of cross products did I actually understand what
they really meant. When I did, it was purely ecstatic!
Why on earth is the direction of cross product orthogonal ? Like seriously…
I mean this is one of the burning questions regarding the cross
product and yet for some reason, textbooks don’t get to the bottom of
this. One way to think about this is :
It is modeling a real life scenario!!
The scenario being :
When you try to twist a screw (clockwise screws being the convention)
inside a block in the clockwise direction like so, the nail moves down
and vice versa.
i.e When you move from the screw from u to v, then the direction of the cross product denotes the direction the screw will move..
That’s why the direction of the cross product is orthogonal. It’s really that simple!
Another perspective
Now that you get a physical feel for the direction of the cross product, there is another way of looking at the direction too:
Displacement is a vector. Velocity is a vector.
Acceleration is a vector. As you might expect, angular displacement,
angular velocity, and angular acceleration are all vectors, too.
But which way do they point ?
Let’s take a rolling tire. The velocity vector of every point in the
tire is pointed in every other direction.
BUT every point on a rolling
tire has to have the same angular velocity – Magnitude and Direction.
How can we possibly assign a direction to the angular velocity ?
Well, the only way to ensure that the direction of the angular velocity
is the same for every point is to make the direction of the angular
velocity perpendicular to the plane of the tire.
And it has been known since as early as 1350 that this series diverges. Oresme’s proof to it is just so beautiful.
Now replace ever term in the bracket with the lowest term that is present in it. This will give a lower bound on S1.
Clearly the lower bound of S1 diverges and therefore S1 also diverges.
But it interesting to note that of divergence is incredibly small: 10 billion terms in the series only adds up to around 23.6 !
On Monday, the Onion reported that the “Nation’s math teachers introduce 27 new trig functions”. It’s a funny read. The gamsin, negtan, and cosvnx from the Onion article are fictional, but the piece has a kernel of truth: there are 10 secret trig functions you’ve never heard of, and they have delightful names like ‘haversine’ and ‘exsecant’.
Trigonometry and its weird names have baffled me for ages. I decided to put an end to this madness by trying to understand them ;P
Here is my interpretation on what thy mean. You can find a detailed etymology here.
Sine
Sine ( meaning ~ bowstring in sanskrit )is ratio of the length of the bowstring with the diameter of the circle that the bow makes.
Sin(theta) because this ratio is solely dependent on the angle ‘theta’. Even the slightest change in theta messes up this ratio.
So, when theta = 90 (or 2*theta = 180) , we can see that x = d. This is why sin(90)= 1.
Cosine
‘Compliment sine’ ( ratio in the perpendicular direction of sine )
Cosine is the ratio of the distance between the bowstring and the center of the bow circle to the diameter of the circle.
So, when theta = 0 , we can see that y = d. This is why cos(0)= 1.
Sine2 + cosine 2 = 1 ?
Well.. thats because of our old friend Pythagoras
Have a great day!
Tangent (’touching’)
Imagine you are the center of the circle and shoot an arrow through
the point (x,y). Your friend is on the circumference of the circle and
starts running perpendicular to the circle.
Tangent is the distance that your friend needs to cover in order to catch that arrow.
Heres why the tangent is sin/cos
Why so many functions ?
It is important to realize that the reason why we have so many trigonometry functions: Trigonometric functions are merely ratios of distances.
Before the days of calculators, these ratios were manually calculated, and as a result, many functions were introduced and made into tables to ease the computation.
Dividing by 7 yields this fascinating play of the numbers 1,4,2,8,5 and7. I am not going to spoil the fun by letting you on the pattern that emerges and other captivating properties that you might discover along the way.
Although feel free to write to us if you found anything that marveled you and we would definitely share it with the world.
Who knew division could be this much fun. am i right?
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.
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 )
Ecstasy Shots 