I thought I'd share a post from my blog for anyone who enjoys fractal art, but hasn't studied much mathematics

The World of Fractal Art for People Who Hate Math

Excellent article, merrak!

Thanks for sharing that Merrick; very much understand complex math (as an EE, you had to just deal with it. lol). Also, really liked this video from the site. lol

Thank you for sharing it with us. Btw, I submit an answer to your June riddle. Please check it out

Nice article. Funny movie.
We are just not an intelligent enough species to get past fossil fuels.
We would rather war over oil fields than move on to cleaner fuels.

I predict in the year 2050 we still will be driving gas fed autos.Of course only the rich folk will own one because fuel prices will be 38 dollars per gallon by then.

Thanks




I'm thinking about doing a "part 2", going deeper into the math... but I try to write to a general audience as much as I can. At my college, the specialty of the math department is more math education (we train a lot of high school teachers), rather than pure math. I was always on the lookout for things I can share with my students that they, in turn, could share with theirs. Recently I realized - why not just publish my own articles?

I'm also always on the lookout for amusing videos on YouTube. I forgot to find a relevant one for this week - oops!




I just got it! I'll post the solution when I write the next one. I'm still working out what to do with the riddles. I might go in the direction of the "knots" from A Tangled Tale by Lewis Carroll - where the riddles are short stories instead of excerpts I find on the internet.

You overlooked something in your answer, though. Think about it some more




$38 wouldn't be too bad. Adjusting for inflation, $38 in 2050 would be more like $8 to $10 now. Even if we are still driving gas fed autos by then, it seems likely we'll figure out how to regularly get 60 - 100 MPG out of the engines, which essentially brings the cost back to $3 to $4 per gallon. The more things change, the more they stay the same

This is exactly what I need Merrak. Thank you very much. I've always found complex numbers odd and can't fathom a real world use for them. i squared =-1 is clearly impossible and yet you say it has practical uses. I can only take your word for that but I will study this further. I'm always intrigued by this kind of strangeness and have an active lay-interest in quantum mechanics but I've never understood the maths but then I've never understood why in decimal 3 thirds don't add up to one.

i² = -1 isn't impossible, for the very reason that I (and you) just said it

It isn't "natural", but nothing in math is - everything in mathematics was defined by somebody, none of it is naturally occurring.

Why would 3 thirds not equal one? You were told 1/3 + 1/3 + 1/3 =/= 1?

most of the oil will be gone by then...and certain way to expensive, either to extract or to buy

We humans are pretty resourceful; we'll probably figure out something.

Ah really? Ok I will think again

Hello Erisian,

the 3 thirds in decimal DO add up to one. I mean, three thirds in decimal form equals to 0.9999999.... and this is equal to one

Proof:

x = 0.9999....(1)

Multiply both sides of equation (1) by 10, you get:

10.x = 9.9999....(2)

Substract equation (1) from equation (2), you get 9.x = 9, which means x = 1. But from equation (1), x is also 0.999... which means 1 equals to 0.9999..... (wait a second, since when this forum becomes math forum?)

If you are wanting to share an example in your class where complex math is used in the real world, then you can give them a short lesson in Power Factor. Which is the ratio of real power over Apparent power (real power is measured in watts and Apparrent power in VA (volt-amps). The higher the power factor the better since it's real power that feeds a true resistive load. Imaginary power is what you want to avoid. UPS output is measured in Apparent power (which is quite misleading since PF for these devices tend to be terrible; lol). I always look at the watt capability and not the VA that they give you when I get a UPS.

I use .999... = 1 all the time as an illustration of series.






I'll have to hang onto that

I never paid attention to VA on UPS devices. My computer doesn't draw much power, anyway, and the most it's ever out for is a moment or two... unless we have a hurricane - and then I'd need a generator.

0.3 recurring X 3=0.9 recurring not 1. I don't know what you mean by possible with regards i squared =-1 because any number multiplied by itself = a positive number. I have to accept that this has practical uses but I still find it confusing. 0.999 might = 1 in maths but logically it's short of 1. I think I'll stick to philosophy, it makes more sense.

Infinities sometimes like to play little tricks

The proof is a common exercise for algebra students, and again in linear algebra (hanciong's proof), 2nd-semester calculus students (geometric series) and 4th (epsilon-delta proofs).

Here's one way to think of it. If you claim it is short of 1, then how short is it? It can't be .001 short, because .9999 is closer. It can't be .0001 short, because .99999 is closer. No matter how short you think .999... is from 1, you can tack on more decimal digits and realize .999... is closer. Therefore, there are no numbers between .999... and 1, so the two must be equal.


"any number multiplied by itself = a positive number" is an observation, not a definition. Technically, it's also not correct. To be precise, you'd say, "any real number multiplied by itself = a positive real number."


If I could go back to school, I'd study more philosophy. I started to really get into philosophy courses my senior year, but it was too advantageous for me to graduate a semester early and transfer into the graduate program (free tuition + stipend, vs. taking out more loans). Once in the grad program, I could only take math, or sometimes computer science, courses.

So being a mathematician is a bit like being a quantum physicist - you have to be able to think inside out and upside down or do you just accept that it's a fact and work with it. I wasn't disputing your definitions, just trying to understand how it all works. I can almost see what you mean about 0.999 but how would you work out an equation who's answer is the square root of -1? What figures do you start with?

Hi Erisian,

Mathematics is a "game". You have set of rules, and if you follow them, you can "play" with them. Sometimes the rules "make sense" (intuitive) to us, sometimes they don't. But surprisingly, the rules that don't make sense to us (less intuitive), sometimes have applications in science and engineering. For example, we always think that two equal numbers, if multiplied, will give positive numbers. But that's because we don't deal with imaginary number in everyday life, that's why we have difficulty to accept it. People in the past also have difficulty to accept zero or negative number.

anyway, maybe the most useful application of i is this:

e^(i.x) = cosine x + i.sine x

where e is the euler number e = 2.718....... And so, anything which is periodic in nature, can be expressed as sine or cosine (if you remember, sine and cosine plots are periodic with period 2*pi), hence can be expressed in e^(ix). That's why imaginary number appears in science branches which deal often with waves/vibration. For example, lylejk said about power in ALTERNATING current (some kind of vibration). and then, quantum mechanics also uses imaginary number because it deals with some entity called WAVE FUNCTION (anyway it is some kind of wave). etc. Wave and vibration are everywhere, that's why i is used a lot.

You ask, an equation who's answer is the square root of -1? easy:

x^2 + 1 = 0

Then x = i. If I have this equation:

x^2 + 4 = 0

then x = 2.i. And so on

We invent new numbers system according to our needs. Prehistoric people use maths mainly to CALCULATE something. Hence they invent natural number: 1,2,3,..... But then what happens if I have 5 rabbits, and all of them are eaten by a lion? Hence they invent zero. But how do we define five minus ten? Hence negative numbers. but then if I have a rabbit, and I cut into two, I give one part to my friend, and I keep another part, how many rabbit that I have now? Hence fraction (rational numbers). Then irrational numbers. Then complex numbers, and so on, and so on; with each step is more sophisticated than before and less "intuitive".

Anyway, in mathematics, there are other numbers system which are more "weird" than imaginary numbers. For example, there is a system where a times b is NOT equal to b times a. There is a system where a times b is equal to MINUS b times a (the last one is called grassmann number, it is used in quantum field theory, which is one step more sophisticated than quantum mechanics) and so on

we already have several alternatives.....it's only the very rich oil companies who don't want us to go in that direction

This is an example I use a lot in introductory classes (college algebra, precalc) to convince students this.

When is 11 + 2 = 1?

Scroll down for the answer

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A: On a clock, we include one additional rule to the usual set of rules for adding and multiplying of integers. 12 = 0.

e.g. If it is 3:00, move the hour hand 12 spaces, and it is still 3:00. So 3 + 12 = 3.

We added 12 to a number and got the same number; 12 behaves just like 0.

So if 12 = 0, a lot of little things change. For instance, it is possible to multiply two nonzero numbers and get zero (3 x 4 = 0, 6 x 2 = 0, etc)