Aren't all but maybe the natural numbers "impossible"? You cannot have -5 sheep. You can even less have 5/3 trees, even less so -pi hamsters. Why is imaginary numbers suddenly the issue?

Well, it simply isn't. All those extension of the numbers made sense, do not cause contradictions, and most importantly, turn out to be useful in everyday life, engineering (electricians, for example use them) and physics (all over the place). Notwithstanding the immense effect on mathematics itself.

There are billions of bacteria living in your mouth all the time. They're all constantly reproducing, peeing, and pooping. That waste smells bad.

During the day you are generating spit and swallowing, which helps to keep the bacteria populations down, and continually washes the waste away as it's produced, or at least before it can build up too much.

At night, you stop generating as much spit and you don't swallow. So all the bacteria can multiply and their pee and poop can also accumulate. That's what morning breath is. Bacteria colonies and their waste.

First, pure alcohol (ethanol) is tasteless. However, the process of making and distilling alcohol includes congeners - other chemicals that are the result of the fermentation process and then selected (and in some cases modified) during distillation.

The congeners depend on the raw material (cane sugar, beet syrup, barley, malted barley, grain, corn, or potato) and the yeast strain that feeds on the sugars to produce alcohol.

Distillation then extracts the alcohol from the water in the ferment. Along with the alcohol comes the methanol (the heads and tails, which are discarded), some water, and some of the congeners. The heat of distillation also chemically modifies the congeners.

The temperature, type and material of the distillation still further modifies the congeners - copper stills react with and remove sulphur-containing congeners, for example. Modern column and vacuum stills can have very selective distillation. Cycling the distillate through multiple distillation and filtering stages allows for highly selective congener selection and end-product.

However, better source sugars, distillation stills and multiple distillation stages result in more loss and wastage. This is expensive. So cheap spirits use cheaper raw materials and cheaper distillation methods, and their products have more congeners that generally result in a harsher taste. Better starting materials and distillation/filtering smooths out those congeners for a better raw spirit.

Then comes the flavouring process (for gins) or cask aging (for bourbons and whisky/whiskey), which have been discussed by others. Again - more expensive starting materials at this stage produce better results. Proper barrels vs wood chips, burnt casks vs charcoal filters, more cheap vs fewer expensive botanicals - it all adds up to better flavour in the end result.

However, as also pointed out, selection should be based on blind tasting and not on price alone - the human brain is hard-wired to appreciate more expensive tastes.

Why do negative numbers even need to exist?

Because the math is usefull to describe real world situations.

For imaginary numbers, for instance to describe both amplitude and phase of waves.

Because the maths is consistent, so it's interesting to mathematicians to explore it. And like so many "purely academic" things in maths, it turns out that it's no such thing; it has practical applications in things such electronics.

Would multiple distills make a better tasting vodka? Example would be Tito's (6x) compared to Kettle One (3x).

IMO, Tito's is way better.

I understand one is by corn the other is by wheat. Dont know if that makes a difference.

Not vodka, but there was a [wine tasting in 1973] (https://www.vivino.com/wine-news/the-day-california-wine-beat-the-french-and-shocked-the-world) that proved that telling the difference between “lesser quality” California wines and the “obviously superior” French ones was a lot less cut and dried than they thought.

And Grey Goose is an overly hyped bottle of crap. I can provide you a number of ~$40.00 bottles that are FAR superior. I live in Ontario, Canada.

If you have access to Beattie's Farm crafted potato vodka, give it a shot.

Well I can’t, but my wife once ordered a martini with a specific vodka, and if they didn’t have that specific vodka to come back and she’ll change her order.

They brought the martini and after 1 sip, she sent it back because she could taste the wrong vodka.

The bartender came over to ask how she could tell (didn’t deny, just didn’t believe anyone could tell)

I spent several years exploring an reviewing spirits. Vodka is meant to be "odorless, colorless and tasteless" by its very definition. So yeah, if you think the "average Joe" can tell them apart, you're kidding yourself.

​

Where taste REALLY comes into play is for things like Whisky and Rum, where aging IS a factor.

Aluminum is stupidly common. You know rocks? Aluminum is a significant fraction of the mass in rocks.

One of the big reasons that cheap alcohol can taste "off" has to do with the nature of the distilling process.

When you distill fermented "beer" into spirits, lots of compounds come over in distillation. Some of those compounds taste "good", and some taste "bad". At the beginning of a batch distillation run, the first compounds to come over are lighter-than-ethanol compounds like acetone. Those are called "foreshots" and are discarded or recycled. Next come the "heads", which have some ethanol, but also methanol, acetaldehyde and light esters (flavors). Next come "hearts", which are mainly ethanol and water, with a few esters in the mix. Last come "tails" which tend to include fusel oils and other foul-tasting compounds. Tails have a distinct wet dog smell and taste very bitter.

The art in distilling is knowing how much of the heads and tails to include with the hearts in the final product. More expensive spirits add just a little heads and tails to give the spirit character while still tasting good. Cheaper spirits will add back more of the heads and tails, because it increases the total amount of spirit produced, at the cost of including undesirable flavors. Barrel aging can temper some of those off-flavors a bit, but you'll still get some odd flavors/smells.

Source: I work with a distiller

Fair enough. Now i wanna see what this upstate Syracuse thing tastes like. Cheap vodka I've had tasted like acidic rubbing alcohol.

This is incomplete - electroporation relies on large voltages but little to no current to be effective. If there are enough ions (i.e. salt) in your bacterial culture to carry a current, it will kill the bacteria (based on my experience with E. coli).

Here's a survey that says people are split nearly 50/50 between top shelf and bottom shelf:

https://nypost.com/2013/05/26/post-taste-test-reveals-drinkers-cant-tell-good-from-cheap-vodka/

While I realize surveys are utter trash - I still think it's pretty common that the "average-person" will not be able to tell the difference.

Same with high quality bottled water vs low quality bottled water...

You can say "there is no solution." But that's boring.

Mathematicians don't like being told they cannot do something. Instead they try to come up with new rules or new definitions to do whatever it is there isn't already a rule for. And those rules can be anything, but generally we look for consistency (those new rules should complement or add on to existing rules), usefulness (the new rules should help us do something we couldn't do before), and interesting consequences (things that make us go "ooh, that's neat").

And the more useful, interesting and neat those rules are, the more likely they are to be used by other mathematicians, and adopted as standard.

With complex numbers, we take all our usual rules for numbers and throw in one more; there exists some number(s) i such that i^2 = -1. It is consistent with all our existing rules, and turns out to be really useful in a bunch of areas of maths and science (and leads to some really interesting results).

i isn't impossible. It doesn't appear on a standard number line, but that's not a huge problem. The number line is an interesting and useful tool, but not the end point of numbers. Interestingly the first mathematical paper to use something like a modern number line was published about the same time Newton was publishing his Principia Mathematica; there weren't number lines when Newton was learning maths. Number lines are fairly modern.

Some classical Greek mathematicians had a very different way of looking at numbers, seeing them more as a lose collection of concepts, with fractions being connections between the different concepts (so 1/2 was a connection between 1 and 2). This did cause them problems, though, when it came to irrational numbers...

I'm not talking about mid-shelf. I'm talking the absolute cheapest shit on the market.

Is this true?

How often can people truly distinguish between a mid-shelf and top-shelf vodka? Or even a well?

Most surveys I remember - no one can tell the difference.

I meant square rooted not squared

Others are saying that they're circular, which they kind of are, but only in 2D. Rainbows are cones, with you at the point (and the centre of the cone lining up with your eyes and the Sun).

When you look at a rainbow, you're seeing light being reflected through water droplets that are somewhere in that cone. So it could be that two parts of the rainbow you see that are right next to each other are from water droplets miles apart, if some are much closer and some much further away.

The reason why they are cones involves a bit of physics and a bunch of geometry. Essentially the light from the Sun hits the raindrops and gets scattered out in all directions at an angle of about 40 degrees (that is just a 2D slice, imagine it being rotated around). So the scattered light comes out in a cone from each raindrop, and by geometry and symmetry, the light from a bunch of different raindrops that hits any given point (i.e. your eyes) forms a cone of itself.

The End of the Rainbow

Taste may be subjective, but not so subjective as for there to be a sizeable portion of people thinking that the cheapest vodka tastes better than a good vodka. xD

What kind of alcohol are you talking about?

In alcohol such as wine blind taste tests have shown that there is little difference between the mid-level and expensive brands. A lot comes down to people's expectations instead of whats actually in the bottle. Although some of the truly cheap wines are god awful. Thats down to the quality of the grapes and processing.

For distilled spirits it depends on the type of spirit. Some require filtering to remove certain chemicals that can effect the taste. The extra filtration steps add cost to the production process and increase price. Aged whiskeys are more expensive because of the aging process. Its a heavy investment with a lot of risk to produce a product that you have to store for a decade or more before the first bottle hits the shelf. Higher quality spirits will usually have some extra level of processing/production that adds to the cost.

Beer tends to come down to the quality and variety of ingredients, as well as volume of production. A beer that is produced in gigantic vats a hundred thousand gallons at a time with basic cheap ingredients is gonna cost less than one brewed in a 500 gallon vat with a selection of premium ingredients. And the taste is purely a matter of personal preference. I know people that prefer miller lite to high quality micro-brews. They just have a taste preference.

Imaginary number exit because we define them to exit. Maths is an internally consistent system that is based on a few unprovable axioms..

So we define the imaginary unit i as i^2 = -1 and the rest of the usage follows from that definition

>I get that squaring a negative leads to a solution that’s impossible, but why do you have to make an impossible number into a number? Can’t you just say, “no solution is possible”?

I'm not sure what you mean by "impossible". Could you elaborate on your premise?

Subtracting a number from a smaller number is impossible within the realm of N (Natural Numbers). So why do we need Integers if they are "impossible"?
Well, sometimes we need to subtract a number from a smaller number.