But what I’m claiming is that the different meanings of rational and irrational both have the same origins.
rational 1. Ratio of two integers. 2. Reasoned.
irrational 1. Not the ratio of two integers. 2. Not reasoned.
I’d be very surprised if one of the definitions in each case was not derived from the other. I’m not sure what direction it goes in, however.
For example, with real/imaginary, it seems fairly certain to me that the common definitions of “real”/“imaginary” are what led to the mathematical definitions of “real”/“imaginary”–real numbers were thought of as “real”, imaginary numbers were thought of as “imaginary”.
It may have been the reverse with “rational”/“irrational”. RATIOnal numbers and irRATIOnal numbers were named that because they “are ratios” and “are not ratios”, respectively. The common definitions probably sprung from these definitions–historically, it was rational to accept rational numbers, irrational to accept irrational numbers.
Ditto to iampunha’s “define real life”. I’m a physicist, and use it frequently. In addition to things like impedance and wave amplitude, it comes in real handy for describing timelike dimensions.
Cabbage: The classical Latin word for logic or reason is “ratio” (ROT-ee-oh). I don’t think that folks commonly referred to the different types of numbers back then, although it was a Pythagorean who proved that sqrt(2) was irrational.
As to the “reality” of the imaginaries: They don’t make sense in certain contexts (you can’t have i dollars in your checking account), but then, neither do negative numbers (ever see -3 apples?).
And panamajack, you electrical engineers are silly. i obviously stands for “imaginary”, but I doesn’t stand for anything-- Why does the imaginary unit have to be the one to change?
We added a new building to our estate and of course had to put in electrical services.
We also generate our own power in parallel with the outside supply so we have a neutral earth resistor, which is somewhat like a tank with a solution of water/soda in it.It is also quite large say fifteeen feet tall and thirty feet round. The neutral earth resistor deals with unbalance currents from a 3 phase supply and it was getting hot, like very hot boiling, but no-one was sure why.
The new building was supplied using a new transformer on the 11000volt network .
I pointed out that there was an addition of 3rd harmonic current that had not been taken into account and that there was a large amount of out of phase power being caused by altering the network.No-one believed me till I did the calculations and using root -1 was by far the easiest way to do it.
Eventually a ballast bank(reactor ) was added.
You may be right about that, I don’t really know a lot about the specific origins of the meanings of the words “rational” and “irrational”. I would still be surprised, though, if the “common” definitions and “mathematical” definitions of the two words weren’t somehow intertwined. “Ratio” may very well have meant “logic” or “reason” originally, but somewhere along the way it also came to mean “ratio” of two numbers. I imagine one definition was derived from (inspired by) the other, I’m just not sure which direction it went.
Why is “i” any more imaginary than “1” or “2” or “pi” or “e” or anything else?
I mean, I can’t have “i” any more than I can have “1” or “2”. I can’t have “1”. “1” doesn’t exist. It’s a concept – it’s imaginary.
PeterB, the reason that imaginary numbers are deemed imaginary is that Mathematicians and Physicists have to come up with words to describe what there are no words for. Sooner or later, terms are going to start becoming arbitrary, and confused. Some of my favorite examples:
“Calculus” comes from the same root as “calculate”, but that branch of Math has no more to do with calculations than, say, Arithmetic.
“Logarithm” comes from the Greek words for “name of the number”. How much more arbitrary can you get than that?
The power company doesn’t sell Power - they sell Energy.
I once heard a yoga instructor refer to “five pounds of pressure.”
You use a cartesian coordinate grid in real life, and that’s essentially the same as setting up an imaginary axis, even if you don’t refer to square root of -1 directly.
Plus, any calculations involving polar coordinates (I would imagine most NASA stuff is like this) relies on imaginaries.
Electrical systems engineers use r -1 quite a lot in closed loop systems.
These are systems were the output is sampled and used to act as a control referance, for instance, in an amplifier, negative feedback.
Feedback reduces the gain but increases bandwidth and a whole host of other required characteristics.
You will have heard a system such as public address seakers giving a horrible howl whenever the user puts his microphone too close to the speakers, positive feedback.
If you design a closed loop system then the negative feedback you have derived will change phase as the frequency changes, in amplifiers this usually means as the frequency rises, to such an extent that you wind up with positive feedback and likely as not the amplifier will oscillate.
We can and do use root -1 to predict this and then ensure that as the threshold frequency for oscillation is reached the gain of the amplifier is reduced by installing filter networks.
In fact there are few machines or systems about that are not closed loop, these can range from electronics to steam engines right the way to taking human reaction times into account in piloting aircraft.
Root -1 is used extensively in vibration analysis in bridges, engines, and an enormous range of engineering applications.
At work I am writing a computer program that uses imaginary numbers to a practical end. The program predicts radiowave propigation around cityscapes. The power from any individual ray from the transmitter to a receiver is recorded as a complex number, and the power values are summed and whatnot to give a real decibel loss. I can’t say I really understand the math involved, I just do the user interface. But the program does use those imaginaries.
To add to what CKDex said, you don’t have to use complex numbers with polar coordinates; it’s just that they’re usually used for the useful calculations.
Chronos, us enginerds can’t use i since we have to say j-omega all the time, and if we don’t everyone thinks we’re talking about zip drives. And it’s hard enough for me to dissuade people from thinking that because I’m an electrical engineer, I work with nothing but computers. (Yes, there are many elect. engineers who do, just not all of them.)
Going along with what Achenar said, it’s important to label imaginary numbers as such so as to make a distinction, since when you apply the math there is a distinction.
Chief Scott, you got some sticks? Okay, take one, and move it back and forth slowly in front of you. Then stick one on your record player …
panama jack
Imagination is more important to ensure the survival of mankind than a vegatarian diet. – A. Einstein
I’ve always liked irrational numbers. Just because they were something a little different from the numbers used in normal life. That, and it disproved one of the first things I learned about roots.
I guess imaginary stuff can be useful sometimes even if some people don’t understand it.
Try explaining to a 7 year old that meridians and parallels are imaginary lines on the surface of the globe and he’ll logically say if they are imaginary they are useless. Once someone goes out there and actually marks them, then they will be useful.
complex numbers are definitely very useful in engineering. I have no problem handling them.
I am always content with the idea that people who have to ask the usefulness of any intelectual pursuit or concept will never have to actually use that pursuit or concept. So be it. The ignorant have their place in the world as well.