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.

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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.

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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

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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

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Perceiving Light at Trillion Frames per second !!

I have shown these gifs time and again to people, but they don’t believe me at first when i tell them that this is indeed Light, at a staggering Trillion Frames per second! This was achieved by a technique known as femto-photography at MIT Media Lab.

Is that really how a single photon of light looks?

Unfortunately no! If the experiment was conducted with just one single photon, or with a very small amount of photons, a complete picture of the interaction of the light could not be achieved.

So, what you are actually seeing are Pulses! Pulses contain many photons and the photons interact with each other as well. 

Ergo, this is indeed Light- Light that travels at 299 792 458 m / s captured by Human Intelligence! Wows me every single time!

Thirsty for more?

Check out the TED Talk by Ramesh Raskar.

Exploring the Realms of water – The Leidenfrost effect ( #1 )

The skittering of water droplets of a hot surface that you just witnessed is known as the Leidenfrost Effect.

The effect can be seen as drops of water are sprinkled onto a pan at various times as it heats up. There is no specific temperature, beyond which the Leidenfrost effect kicks in.

Unveiling the mystery.

image

The bottom part of the water droplet vaporizes immediately on contact with the hot plate. 

The resulting gas suspends the rest of the water droplet just above it, preventing any further direct contact between the liquid water and the hot plate. 

As steam has much poorer thermal conductivity, further heat transfer between the pan and the droplet is slowed down dramatically. This also results in the drop being able to skid around the pan on the layer of gas just under it.

Mesmerizing, isn’t it ?

(Sources: The Leindenfrost Maze , Strange Water

Meet Oobleck ( Corn Starch )- A Non- Newtonian Fluid!

It is a fluid, but not the conventional type. You can walk or even dance on it and yet not sink. Under the percussive action of the speaker, it abstractly behaves as a solid! 

You can dip your hands in it and play around with it like any other fluid. But if you try to punch it ( see animation ) it becomes hard ( like a solid).

Bewitching isn’t it?

PC: Slowmoguys, Discovery, Hong Leong Bank.

Pun on ‘mew’- explained. 

Okay, this joke deserves an explanation for so many different reasons.

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. The coefficient of friction (symbolized by the Greek letter mu (μ)) describes the degree of interaction between two surfaces.

Higher and Lower ‘mews’.

A higher mu (mew) indicates that two surfaces in contact have a greater resistance to relative motion. i.e to pull one surface off the other would be hard.

A lower mu (mew) indicates that the two surfaces in contact have less resistance to relative motion. i.e to pull one surface off the other would be easier.

Ergo the cat with the lesser “Mew” will fall off first due to the lesser resistance offered between the cat and the roof.

Meow!

Nitrous =  Boost, but Why?

It is safe to presume from the increasing number of Car racing games and the profound popularity of the Fast and Furious Franchise that you know what nitrous does. (It gives a significant power boost to the engine)

Nitrous Oxide or popularly known as Nitrous in the racing world is N2O. When you heat nitrous oxide to about 570 degrees F (~300 C), it splits into oxygen and nitrogen.

image

When Nitrous is supplied into the engine, you are increasing the amount of oxygen inside the engine.

And due to the presence of that excess oxygen you can inject and burn more fuel. As a result, you get a significant power boost!

Have a great weekend!

EDIT: 

Why can’t you use pressurized Oxygen ?

Well.. You need controlled combustion! Nitrous Oxide splits into oxygen and nitrogen only when you heat it to 300 Celsius… Whereas the entire oxygen tank would explode almost immediately when administered into the engine! Boom!