This Beef Just Got REAL!

Back when I was in school, this concept was really hard for me to grasp my head around, and I must say, this was a compelling way to understand the function of multiplying with negative numbers!

Thanks!

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.

Have a Good Day!

PC: mathisfun

Wow, “ For a photon, its entire existence is instantaneous. “ , that put things in perspective! 

Ask Ethan #109: How do photons experience time?

“[L]ight takes about 8 minutes to travel from the sun to earth. Light travels at the speed of light. If you do that relativity kicks in. So my question is, how much time passes for the photons traveling? In other words, how much have the photons aged when the reach the earth? Thanks for considering this.”

Travel at any constant speed, at rest, slowly, or near the speed of light, and you’ll experience time passing at the same rate it always does: one second per second. If someone else is moving relative to you, they’ll see your clock run slow (and you’ll see theirs run slow) depending on how quickly you move relative to one another. But what about a photon, which moves at the speed of light? From your point of view, no matter what your motion is, you won’t see time pass for it at all, and it won’t even experience time! For a photon, its entire existence is instantaneous.

This slide is really something! :)

Video of my talk last week at ICERM, where I discussed a bit of my process for making GIFs. You can also access my slides here.

"Oh captain, my captain."

Wow, this story is my hope for humanity! Keep it up and keep doing what you do, Mathed-potatoes! :)

People never cease to amaze me in every way. 

Today, we had an emergency response drill. This involves flashing lights and loud sounds. I have a neurological disorder that makes this type of even very uncomfortable. 

The hour before this drill, I had an incredibly disappointing interaction with one of my work supervisors. My reaction was much stronger than I might have anticipated on a theoretical level, and I was still reeling when the drill started… so I didn’t take steps to avoid the sensory barrage. The loud noises happen first, and I was already starting to spiral the drain. 

And then…

And then one of my students caught my attention. She told me the lights were about to start flashing, and asked if I needed to go outside. I was so far gone I couldn’t really process how to do the ‘get outside’ thing. Like, I couldn’t find ‘outside.’ So she sherpa’d me to safety, just in time to avoid the flashy lights that would have triggered a full meltdown. 

So, here’s the thing: I have mentioned my disorder to her once, in passing, by name only. She literally remembered that I had a condition, and took steps to know what emergency protocol I might need in certain situations. I thanked her profusely, and asked how she had learned to be so sensitive to stuff like that. She told me she has a younger sister with a disability, and so she understands how significant the timely attention of a knowledgeable ally can be. 

It is significant. It makes a world of difference. 

She saved the day for me. 

She kept me from having to leave in the middle of the day, from having to cancel my afternoon discussions. 

And, at that particular moment, when I was still shocked by an instance of the human capacity to harm, she reminded that their capacity to help, to be compassionate, to care about one another, is far greater still.

Beautiful, where can I get my hands on one of these? :) 

Dragon Fractal - Level 5

If you take each of the 4 sides of a square and modify the angle of the dragon fractal as you go up, you get this shape. I’m still fighting the 3d models to get a nice smooth version.

Important topic!

Spread the word! 

Science is being defunded around the world. Help us spread the importance of science, by sharing this video!

Despite our achievements, science and society are often at odds. Scientific discoveries like the Earth is round, that our planet revolves around the sun, or that diseases are spread through germs were all once ideas that were rejected by society. So much so that Giordano Bruno was burned at the stake for suggesting the Earth wasn’t the centre of the universe, and even Galileo was sentenced to house arrest for supporting the theory. Today we see the rejection of scientific evidence for vaccinations - leading to preventable disease like measles coming back, after being wiped out in the year 2000; or the rejection of scientific evidence for climate change, despite the vast consensus among scientists.

Let’s spread our love of science further, so others can learn to understand and appreciate it! Your sharing is appreciated

Welcome to my world of dyslexia... :)

When I'm halfway through a sentence and suddenly realise there is no way out of the grammatical crater I have dug myself

Damn, yes, why wouldn’t you! Looks great!

idk, why wouldn’t I make these and post them