#Admin - changes are coming...

Hi Everyone,

Over the course of the two or so months now that I've been doing this blog, two directions for where it would go developed:

1. A Q&A site that covers specific topics in a given post, and
2. An interlinked learning resource where posts support subsequent posts.

At first I tried doing both within the same blog, but that just made an organizational mess that was hard to navigate. 

And so I've decided to start another blog - one that will be that interlinked learning resource (and still maintain this one, where I do the Q&A). It'll be called Fact-orials and will take the initial posts here in Math Fact-orials (they'll move and get an edit in the process) and build upon them to explore topics as wide ranging as various fields of math, physics, other sciences, accounting, business, and so much more. 

In the end, if it seems like Math Fact-orials has gone rather quiet, that's because I'm hunkered down working on the other blog. 

I'll put in all the links and everything into both blogs and hopefully they'll work together to help everyone everywhere learn some cool stuff.

Many many thanks to both Stefan Velicu, a former Socratic.org moderator, Hero, contributor, and overall doer of all things, and my wife Aleesha, for helping to make Fact-orials something that can become reality, one entry at a time.

#Admin - Dealing with the moving goalposts...

Hi all,

After searching around for a bit to house this new brainchild, crazy idea of mine (to build a hyper-linked, non-linear website of a thing for all things... well... all things (but in particular math), I've finally decided on doing it right here on Blogger (mostly because I'm getting more familiar with it and it's free).

The idea is that there are going to be two kinds of posts - one that focuses on answering questions, and the other that focuses on building the online resource.

The ones that answer questions will follow a more traditional, blog-like scheme of being rather linear, dealing with a particular topic/question, utilizing the definitions pages and such when needed, but otherwise being stand-alone posts.

The ones that help to build the resource are going to be "living posts" - they are going to be edited and change, sometimes at great length, as content is added and the world of... well... everything (but in particular math) expands. As such, those posts won't remain static at all (or there are odds that they won't be. Perhaps I'll do a post about what those odds are???).

My hope is that this won't end up being a giant mess and instead will allow for integration of the resource, asking of questions, an ability to collaborate with readers, and overall be an enjoyable read for you and an enjoyable creation for me.

The place where the resource starts is the Index page - you can hyperlink from here or click on the link on the left side of the blog where it says "Start Here".

And with that, it's time to start writing!

(If there are areas that you feel have been missed or should be covered, please do drop me a line and I'll do what I can to make the needed content!)

#Admin - The goalposts just moved...

Hi all,

When I first started the blog back in late July, I had it in mind that I would answer questions, create content, and do the things I used to do at Socratic.org. I've been having fun doing it and plan on continuing to do so.

At the same time, however, it struck me today that through the use of definition pages and other supporting structures for the blog, something else can arise and it's that thing that I'm going to start building. It's a resource for the information age.

You see, I have often asserted that assembly line education is dumb, that teaching everyone exactly the same thing at the same time just because the child's age is x years old, just doesn't work at all  (echoing the thoughts of the speaker on a TedTalk I listened to raptly many years ago). I think that no matter what subject it is within mathematics (and probably within all of education itself), we can start with something that is of interest and move into domains that are related and help students (of all ages) be curious about math, science, and all that. For instance, if a student is really into triangles, that can lead to discussions about shape, line, area, perimeter, ratios, trig functions, graphing, and so much more. Any particular topic can lead to a whole heap of other topics.

And so that is the resource I want to build. I don't think I've seen its like in existence and perhaps I'm too naive to know that what I'm looking at building is crazy. Well... that's ok. A bit of crazy, applied long enough, can result in amazing things. Like the biggest ball of string in the world. Or a scientific breakthrough. Or perhaps, just perhaps, a useful online resource that will help index, categorize, and bring a flow to knowledge that doesn't exist yet.

I'll continue to write the blog as I start to build this new resource, so keep the questions coming!

#PreAlgebra - What are the different kinds of numbers (Natural, Whole, Integer,...)

Question

What are the different kinds of numbers (Natural, Whole, Integer,...) and where do they come from?

Answer

See below for a discussion:

Analysis

The groupings of numbers and the development of them is a fascinating walk through history. Let's take a brief walk down that path:

• We can start with the number 1

• One shows existence, it shows something I can point to - someone can hand something to someone else and say "here's this thing". You have to think really really basic representation stuff here (which operates below where essentially all of us do), but even if it never really happened, one serves as that bedrock number from which everything else is going to arise. As a side note, the development of the number 0 took much much longer to form (like thousands of years after the development of the number 1).

• Counting/Natural/Whole numbers

• Once we had the number 1, people would be able to count things with tick marks (essentially little number 1s) and add them up. If there's a tick mark and a tick mark... well that's tick tick, right? At some point, it'd be simpler to count, not by saying tick tick tick tick tick.... but by having a word that associated with the number of ticks. Thus came 2, 3, 4, etc... These are therefore called the Counting Numbers. And I believe they get the name Natural Numbers because they naturally arise from the number 1. Whole Numbers are the counting numbers with the number 0 included. The symbol for these kinds of number is .

•  Integers 

• Integers are a result of subtraction. The basic thought goes like this: I have 2 things. What happens if I subtract 3 things? In the world of "I have 2 chickens. And you want to take away 3?", 2 - 3 makes no sense. However, in many other applications, the use of negative numbers is very useful. And that is what the integers are - the counting numbers that are both above zero (positive) and below zero (negative). The symbol for these kinds of numbers is .

•  Rational numbers 

• Rational numbers are a result of division. We start with an integer (say 4) and divide it into a number of pieces (say 7). And in fact this is the definition of a rational number - any number that can be expressed as a fraction of integers. The symbol for these kinds of numbers is .

•  Real numbers 

• Real numbers are all the numbers that sit on a number line and are symbolized with . There are two types of numbers on the number line - Rational (discussed above) and Irrational numbers. 
• Irrational numbers are those numbers that can't be expressed in terms of a fraction using integers. These kinds of numbers have endless strings of digits, such as more famous ones like  and others like . Irrational numbers are symbolized by .

•  Complex numbers 

• Complex numbers consist of two parts - a real part and an imaginary number part that is a real number multiplied by the square root of -1. An example of a complex number is.

• Imaginary numbers came about from needing to take the square roots of negative numbers. Consider - the square root(s) of a number is/are the numbers we can multiply together to find the number under the square root. For instance, take . We can multiply 2 by itself, so 2 X 2 = 4, or we can multiply -2 by itself, so -2 X -2 = 4. We're good so far. So what happens if we have . There are no two numbers that we can multiply together that will get us to -4. So what to do? Well, if we were to rewrite it this way, , we know what the square root of 4 is - it's 2. So what to do with the square root of -1? Assign it a symbol and move on, and the symbol is the small letter i. 

 ~~~~~

Questions and comments always welcome!

#Algebra - the folding of A0 paper and the square root of 2...

Question

I was listening to a podcast (it's podcast 10, and here's the transcript on his blog) by the artist Danny Gregory (ok - more like I overheard my wife listening to it and my ears perked up when Danny started talking about the mathematical development of the A, B, and C paper sizes). He said that the paper sizes (A, B, and C) as they become smaller, a ratio of square root of 2 : 1. With the paper getting folded in half (the longer side) each time, how does that ratio hold? And are there any other ratios that would work just as well as square root of 2 : 1?

This question is related to this one which looks at finding the measures of the sides of the paper. 

Answer

See below on how it is that the ratio holds. There are no other ratios that will hold.

Analysis

Ok - so how does the ratio manage to stay the same?

Let's first consider a situation where the ratio does not stay the same. Let's set up a piece of paper where the ratio is 2:1 and we fold the long side over (the 2 side) and now we have a piece of paper that has a ratio of 1:1 - not the same at all. So how does the ratio keep maintained?

We start with a ratio of:



We fold the long side in half (that's the square root side) and now we have:



We started out with the "short side" being the "1" side, so let's scale up the numbers so that the short side is 1 again and we'll see where the long side is:





We're almost done. Let's rationalize the denominator of the left hand fraction:





And taa daa! It's the same ratio.

Is there any other ratio that will hold like this? To find out, we can set up a ratio:



We know that the "short pre-fold" will be 1 as will be the long post-fold (they're the same side). The long pre-fold will be some value, x, and the short post-fold will be one half of x:



We can cross-multiply:







And so the square root of 2 : 1 ratio is the only one that will work.

~~~~~

As always, questions and comments always welcome!

 

#Algebra - rectangle area and side ratios...

Question

I was listening to a podcast (it's podcast 10, and here's the transcript on his blog) by the artist Danny Gregory (ok - more like I overheard my wife listening to it and my ears perked up when Danny started talking about the mathematical development of the A, B, and C paper sizes). He said that an A0 piece of paper is defined as being 1 metre squared with the sides of the paper having a ratio of square root 2 : 1 (rounded to the closest millimetre). What are the sides of the paper?

This question is related to the one regarding the folding of A, B, and C series paper and the holding of the square root of 2 : 1 ratio. 

Answer

1189 mm X 841 mm

Analysis

Let's first remember that the area of a rectangle can be found by multiplying the base times the height:

A = bh

We're told the area of the paper in metres. Let's convert that to millimetres:

1 m^2 = 1 m X 1 m = 1000 mm X 1000 mm = 1,000,000 mm^2

With the sides of the paper, we know the ratio is square root 2 : 1. We'll need a variable, x, to make sure we count the same amount of 1's as we do square root 2's:







The short side is 841 mm. The long side is:



These are both in millimetres.

And now let's see just how big the paper really is (in millimetre squared):



and so just a wee bit smaller than the defined size!

~~~~~

Questions and comments always welcome!

#English - Socratic Sunday - Question on the perfect tense...

This is another Sunday and so another Socratic Sunday throwback - a question/answer I worked on from the now read-only site Socratic.org. This question deals with verb tense...

Question

"I have to clean up after him." Is this a perfect have?

Answer

No. See below for more:

Analysis:

We're dealing with simple tenses vs perfect tenses.

In a simple tense , we're told that something can occur. For instance:

I study (present simple)
I studied (past simple)
I will study (future simple)

whereas in a perfect tense, something that was ongoing is now complete:

I have been studying (present perfect)
I had been studying (past perfect)
I will have been studying (future perfect)

And I think this what the question is referring to - if the use of "have" in the sentence "I have to clean up after him" is a use of a perfect tense.

The answer is no - the verb "have" can also be used in a simple tense , such as:

I have a cold (present simple)
I had a cold (past simple)
I will have a cold (future simple)

One of the ways "have" can be used is to indicate a necessity , a need, or something that you are required to get done. This can be accomplished by using "have" in a simple way:

I have to do my homework (present simple)
I had to do my homework (past simple)
I will have to do my homework (future simple)

Or to use the wording in the question:

I have to clean up after him
I had to clean up after him
I will have to clean up after him

If the use of "have" were in a perfect tense, we'd have something like this:

I have been cleaning up after him
I had been cleaning up after him
I will have been cleaning up after him

~~~~~

Questions and comments always welcome!

#English - What is "snuff"?

Question

I like reading. I've started reading books by Charles Dickens and there are references in it to characters "taking a pinch of snuff" and then sneezing. What is "snuff"?

Answer

Analysis

It took a few minutes to wade through google searches that only brought up more modern readings of the word "snuff" (as is related to death and killing things) to get to the meaning used by Dickens and other authors of that era.

According to www.geist.com and the article they did on snuff, it was pulverized (i.e. ground) tobacco leaves that people would inhale. The basic idea was that it was a way to have tobacco and get the mild buzz from it (the nicotine being absorbed by the mucous membranes of the nose) and also the (apparently) pleasant odour of it. It was seen as being better than smoking and chewing because snuff required nothing further to enjoy it (as opposed to say smoking, which requires either papers to make cigarettes or a pipe) and wasn't as messy.

The Geist article has much more about its history, use, and references in literature.

~~~~~

As always, questions and comments are welcome!

#Household math, #Investing - Paying in instalments or all-at-once...

Question

I pay $1,300 rent per month. If I pay a year's worth of rent at the signing of the lease, I'll pay $15,000. Is it worth it?

Answer

It depends on your particular circumstances. See below for a discussion:

Analysis

Let's first look at the numbers.

You are paying $1,300 per month, which over 12 months is $15,600 per year. If you pay $15,000 all at once, you end up saving $600 over the course of the year, or $50/month.

The question is - is it worth it?

I'm going to assume that there's little to no risk of wanting to leave or break the lease or anything like that that would cause you to want to no longer live in your apartment. And so I'll further assume that you'll live in the apartment for the full year.

One thing people will often look at is the "dollar value of savings", which in this case is $50 per month. A lot of people would stop right there and say that paying $15,000 all at once to save $50/month isn't worth it - in fact it makes no sense at all. However, I disagree with that assessment - what's being saved is $600/year (it's important to make sure the time periods with both the cost and the savings are the same time periods!)



Let's look at the percentage savings, which is the $50 savings per month divided by the $1300 monthly rent (we could also do this on a yearly basis with $600 and $15,600 - again, we make sure the time periods on cost and savings is the same):



So roughly a 4% savings.

Is a 4% savings worth it?

Keep in mind that a savings account only earns about 1.5% or so and a checking account even less, so if you have the money and it'll sit in the bank otherwise (or worse yet, it'll just get spent), go ahead and pay the rent up front.

However, if you have the ability to invest the money at a rate that is above 4%, and you have the ability to otherwise make your rent payments, then I'd say invest the money and pay the rent in instalments.

So let's walk this through:

• if you have the $15,000 available, can invest it at above 4% per year, and have a source of income that will allow you to pay your rent (and all your other expenses), then invest the $15,000 and pay the rent monthly, and
• if you have the $15,000 available, but either can't invest it at above 4% per year or don't have a source of income that will allow you to pay your rent (and all your other expenses), then pay the lump-sum rent (you'll effectively be getting a 4% return on that money)

~~~~~

Questions and comments always welcome!

#Current Events - Why was the decision Roe v. Wade important for feminists?

While I normally do a "Socratic Sunday", where I bring over a Q&A from Socratic.org that I worked on and found interesting or important, today I'm doing one that is vitally important given the hearings for the proposed new Supreme Court justice. While the question starts out by asking a rather limited question, the answer is quite expansive.

Question

Why was the decision Roe v. Wade important for feminists?

Answer

Specifically, the decision granted the ability to women to legally obtain an abortion. However, in the larger view, I believe this paragraph sums up my view: 

"When government dictates how someone should treat their body, it strips away a basic human right. It strips away a piece of our humanity. And when government dictates how some people should treats their bodies, it strips away a bit of humanity from all of us."

Analysis

There are basic human principles that collide on occasion and it's up to the Supreme Court of the US to determine how those principles should be applied. And in my view, there is no bigger principles that collide than:

• the right of a person to determine what they do with their body vs
• the right of a person to live

These two principles collide in the case of a woman who is pregnant but doesn't want to be. Can that woman legally terminate her pregnancy?

• If she does terminate the pregnancy, she has exercised her right to do what she wants with her body, but has terminated a fetus - which is arguably a life
• If she isn't allowed to terminate the pregnancy, her right to do what she wants with her body has been violated but the fetus will probably (it could be stillborn, etc in later stages of the pregnancy) survive

In Roe vs Wade, these two principles were interwoven. It states that as a fetus grows older, it becomes more and more likely that it will be a child. And so for the first trimester of gestation, abortion is legal - that is, a woman has the absolute right to terminate her pregnancy. As the fetus develops through the second and third trimesters, individual states have the right to restrict abortions.

So why is this important - and not just to feminists, but to women in particular and to humanity in general? Because it grants a person power over the use of their own body. It takes away the power of others to make determinations as to how someone uses their own body.

The right of women to make this choice for themselves has been under constant attack from those who view this debate in terms of "terminating a pregnancy = murder of the unborn fetus". Those people believe that a fetus is imbued with life at conception and that a woman is morally, ethically, and should be legally responsible for giving birth to the fetus and that after the child is born, it can be given up for adoption and the woman can have her life back.

https://en.wikipedia.org/wiki/Roe_v._Wade

And now I'd like to share my view on this issue - it's pragmatic and some may not agree with it and that's ok.

The decision to end a pregnancy is a big decision and is one that each and every woman needs to face in her own way and on her own terms. For government to get involved and force a woman to carry an unwanted pregnancy to term is to violate that woman.

Different sides on this debate will rephrase the issues in different ways to try to emphasize their own views. And I'm about to do the same here! For me, it's about the:

• humanity of the woman, vs
• the potential life of the fetus

When government dictates how someone should treat their body, it strips away a basic human right. It strips away a piece of our humanity. And when government dictates how some people should treats their bodies, it strips away a bit of humanity from all of us.

And this stripping away of humanity is, for me, a bigger issue than ending a pregnancy. We are told by various sources that life is sacred and that therefore the life of the fetus trumps the humanity of the woman. But I don't think that way at all - the fetus has no knowledge of its life and its termination - it loses its potential to what it could possibly be. But a woman losing her humanity is done with her full knowing. That to me is the issue that should prevail.

 ~~~~~

Questions and comments always welcome!

#Physics, #Household math - Light reduction through screen doors and windows...

Question

When I have my window open, it's always so much brighter when the screen is open as well. Why?

Answer

Diffraction of light 

Analysis

First off - great question! I'd never really noticed just how much the light is dimmed in my house until I did a quick comparison of having the screen open and shut.

So ok - what's going on?

As far as I can tell, there are 3 things going on:

• The wires that make up the screen catch debris.

Now, I'm not claiming that your screens are dirty, but I'm realizing that mine are, so I'll be back in a minute after I wipe them down... and I'm back. And the light still dims noticeably. Another explanation is needed.

• The wires block light.

Depending on the type of screen you have with the size of the holes and the gauge of the wire, any calculation will vary in results. I did some searching online to see if there was a standard size for wire screens and I found that there are several different kinds of metal mesh screens. So what I'll do is use my screens as at least an approximation of what you may be experiencing.

I found that my screens have, per cm^2, 7 wires in one dimension and 9 in the other.

The width of each wire can be quite hard to find (I couldn't find a figure online). One way to find the measure of a small amount is to measure many of those things together and then divide by the number of elements. In this case, I am fortunate enough to have a cat who likes to climb the screens, bending them in places where I was able, to a rough degree to measure a number of wires compressed together. I found that roughly 5 wire widths are equivalent to 1 mm, or in other words each wire is roughly 0.2 mm per wire.

We can then do some multiplication to find the area of wires per cm^2:

7 X 0.2 X 9 X 0.2 = 1.4 X 1.8 = 2.52 mm^2

Keep in mind that 1 cm^2 is the same as 100 mm^2, so the area of wire to space is 2.52 / 100 = 2.52%

But let's face it - the amount of light that comes through the screen is less than a very small 2.52% reduction. We need another explanation.

• Diffraction

And here's where physics comes into play...

Light is a funny thing. You'd think that, knowing that only 2.52% of the area of a window is blocked by a screen mesh, that 97.48% of the light would just zoom on through. If light only acted like a particle, this would absolutely be the case and we'd have much brighter rooms when the wire screens are closed.

Here's the thing - light also acts like a wave and it's this form of light that gives rise to the light reduction. Here's how it works:

Ambient light consists of a whole bunch of different colours (or wavelengths) and they vibrate in all the ways possible (think of it this way - when you shine a flashlight at something, some light vibrates up/down, some left/right, etc. In fact, if you think of the circular shape of the flashlights' light, you can pick any given spot on the "clock" of that circle and the light will vibrate through that spot (say 2 o'clock) and also through the opposite side (8 o'clock).

All these vibrations come streaming towards your window and some of these vibrations "bump" into the wire mesh and scatter (and the tighter the mesh, the more apparent this effect will be). Gone are the lovely beams of light streaming into the room and instead the light heads off in a gazillion different directions. Let's face it - the screen mesh acts much like a cloud - it scatters light in a way so that we don't get lovely beams but rather a medium-glow. And just like a cloud, there doesn't need to be a high density of things blocking the light to scatter it.

If you are of a mind to learn more about diffraction (and this article in particular deals with laser light, which is different from ambient light in that the laser light is all the same colour and vibrates in same direction), here's an excellent paper on it:

http://www2.optics.rochester.edu/workgroups/berger/EDay/EDay2008_Diffraction.pdf


~~~~~

Questions and comments always welcome!

#Household math, #Investing - The Rule of 72

Question

I've heard of something called the Rule of 72 but I have no idea what it is.

Answer

It's a quick calculation used to figure out how what factors are needed (in terms of time and average return) needed to double your money.

Analysis

There are two basic factors that play into how an investment (or group of investments, such as a mutual fund portfolio) will do over a substantial number of years:

• the average rate of return, and 
• the number of years the investment is held.

So let's say for a minute that I have an investment and I anticipate making 6% per year on it and I plan on holding it for 12 years. Well... so what - is there some way to get an easy to understand number from these two details?

This is where the Rule of 72 comes in. It gives a quick answer to the question of "How long will it take to double my money". We take the average rate of return without the percentage mark (so in this case 6) and multiply it by the number of years to be held (in this case 12) and then multiply them together:

6 X 12 = 72

And so at the end of 12 years we can anticipate that our money would have roughly doubled.

We can work this another way - if we wanted to know the number of years we had to hold an investment for it to double at a 6% average return, we can divide 72 by 6 to get 12 - which is to say that every 12 years we can anticipate our investment doubling.

This becomes a very powerful tool when young people are first looking at putting some money away for retirement. For instance, if at age 25 someone puts $10,000 into a retirement account and anticipates 6% return per year, we know that every 12 years the money will double. If the person wants to access that money at age 60, it'll double 60 / 12 = 5 times. And so the value of that 10,000 will have become $320,000.

~~~~~

Questions and comments always welcome!

#English - Socratic Sunday - looking at an e e cummings poem...

Every Sunday I like to do a post called Socratic Sunday - I take a post I worked on while a contributor with socratic.org (it's a read-only site now) and bring it over here (sometimes verbatim and sometimes reworked). I really enjoyed this next question and I think it'll add a new, and rather nice, element to the blog:

Question

What does this quote mean: "Life's not a paragraph and death, I think, is no parenthesis"? 

Answer

I don't think there is just one meaning. I explore some ideas below:

Analysis

 e e cummings, an American poet, author, etc (see the wiki article for a full list), wrote roughly 2900 poems - and this quote is from one of them. The poem is since feeling is first. It's a love poem (he wrote quite a number of erotic poems and works as well) and explores the fact that he sees love and the emotions surrounding it as far exceeding the rational and logical of the brain/mind.

https://en.wikipedia.org/wiki/E._E._Cummings

since feeling is first
who pays any attention
to the syntax of things
will never wholly kiss you;

wholly to be a fool
while Spring is in the world

my blood approves,
and kisses are better fate
than wisdom
lady i swear by all flowers. Don't cry
—the best gesture of my brain is less than
your eyelids' flutter which says

we are for each other: then
laugh, leaning back in my arms
for life's not a paragraph

And death i think is no parenthesis

http://coming2terms-fiction.blogspot.com/2013/05/for-lifes-not-paragraph-and-death-i.html

So what does it mean? Especially the ending, which is where the quote is pulled from? This, of course, is one of the purposes of poetry - to make you think, to consider what is being expressed and to engage you with it. So these are my thoughts and ideas, not "right ones" - because there aren't any.

First off, we see that he is talking to a woman (which makes sense, it being a love poem). So he is talking about how much more he values love and love's expression than anything of the mind.

Life's not a paragraph - I take this to mean a couple of things. First, he's already established that he values "the flutter of her eyes which shows that she loves him" over anything his brain can produce - and since he's a writer and poet, he works in paragraphs and pages, he states clearly it's far better to have "her in his arms".

Another way to understand this statement is to see that in terms of life, he could express it within a paragraph but instead chooses to express it physically with her.

And I think a third way to see it is that he views the rational part of life as being like a paragraph within the work, the book, the novel of life. There is so much more to life than what the mind can create.

And death i think is no parenthesis - Parentheses are usually used to help explain something, to give an aside or a comment that isn't necessarily important to the story or whatever all is being written. So he is saying here that death is not simply an explanation for life, not something that explains what or why we're here.

Life is meant to be lived and not commented on afterwards. He clearly has far more interest in loving and being loved than in anything he can express with paper and pen - and doesn't care a whit about what future generations will get from his work.

~~~~~

Questions and comments always welcome! 

#Physics - How does an induction stove top work?

Question

I just bought an induction stove top. I don't understand how it works! The pot gets hot but the stove top doesn't?

Answer

The cook top emits a magnetic field which excites the atoms of the pot and heats them up.

Analysis

There are a few ways we can make something (like a cooking pot) get hot:

• Conduction - When we use an electric stove (the ones that have the coil of metal that gets hot and the pot gets set on top of), there is a direct transfer of heat from the metal coil to the pot. This is conduction.
• Convection - When a stove, whether electric or gas, heats the air around it and then that hot air heats up the pot, that's convection.

And then there are less practical ways:

• Friction - When you rub your hands together long enough and hard enough, they'll heat up. And so if we were to rub the pot long enough and hard enough, it'll heat up. One of the downsides is that to achieve cooking temperatures, whatever it is that we're rubbing the pan with (our hands perhaps), it'll be at the cooking temperature too for however long we're cooking. Since that isn't practical at all, we have some other methods.
• Radiation - A nuclear reactor gets hot because the fuel (often uranium) experiences nuclear fission - in other words it releases energy as the uranium atoms break into smaller and smaller atoms. A pot could be bombarded with charged particles, say for instance at your local nuclear reactor, and heat it up.

To Induction. Induction uses a magnetic field in the cook top to excite the atoms in the pot. The pot has to be able to conduct electricity, which is why only certain types of cookware will work with the stovetop. For instance, stainless steel and cast iron will work but copper and aluminum won't).

~~~~~

Questions and comments always welcome!