Well, as I said I’m not a math person. Making an effort to answer questions helps me regain some of what I’ve lost over the years.
So. Pi. Pi is a constant, just like 2 or 87 or whatever. It just happens that it’s an irrational number, so it’s easier to write π than 3.14159… all the way out to infinity. What makes it special is that for any circle, the circumference is equal to π times the diameter. It could be a circle the size of an ear ring, or a circle the size of a planet’s orbit (assuming a circular orbit, of course ). It doesn’t change. Divide the circumference by the diameter, and you’ll always get π. Similarly, the area of a circle is always π times the square of the radius.
What good is it? It’s good for all sorts of things. What is the volume of a cylinder? Will your marble collection fit in it? How much moulding do you need to buy to go around the rim of a table? If you buy too little your table won’t look good. If you buy to much you’re wasting money. How big a diameter can a propeller be for your airplane before the tips exceed the speed of sound at max throttle? What size gears do you need for your widget-making machine? How much filling do I need for the pie I’m not going to not eat?
I used to get frustrated with math. What good is it? How will I ever use this stuff? Turns out it’s actually very useful.
So I have loved math my whole life (well, once I knew what it was) and minored in it in college, and this concept eluded me my entire grade school career until I took chemistry and my wise and wonderful chemistry teacher taught us how to do these (I think she called the method “factor labeling”).
I’m not quite sure what the “volume” units you’re using are, but I’ll assume for the purposes of example that we’re converting between dose/in^3 and dose/cc (I’ll get to a more complicated example in a sec). The idea is that you keep multiplying by fractions that are actually “1” and watch the units cancel each other out, like this:
dose/in^3 * (1 in/2.54 cm) * (1 in/2.54cm) * (1 in/2.54cm) * (10 mm/1 cm) * (10 mm / 1 cm) * (10 mm / 1cm) * 1 mm^3/1 cc
= dose/in^3 * 60 in^3/cc = dose/cc.
Each of the things in parentheses is 1 (e.g. 1 in = 2.54cm) and you can cancel out the in^3 with the inches in the numerator, the cm’s with the mm’s, and so on. I went through that many steps to show the thought process: you only have to know one length conversion ever (inches to centimeters) and you can convert miles per hour to kilometers per second or whatever.
But, you say, what if I have to do something like convert from mass (kilograms) to volume (m^3)? For example, let’s say I’m trying to figure out the dose per volume of liquid (in in^3) per kilogram of body weight (or something like that…) and I’d like to convert to dose per volume of liquid (in cc) per volume (in m^3) of body weight. Well, in that case you have to use the density of whatever you’re using. The density of water (which is probably close enough to the body composition) is 1000 kg/m^3. That is, for water, 1000 kg = 1 m^3 of water.So
= dose/in^3 liquid/kg body mass * 1000/60 in^3*kg/cc/m^3 = dose/cc liquid/m^3 body volume
(see what i did there… I used the result from above… but you could do the whole above calculation again if you wanted to)
I realize that when I write it out like that it looks kind of imposing, plus confusing because of the powers of 3 involved in volumes, and I’m sorry for that. You might try writing out the fractions I give on two lines each, and it should be much easier to see how everything cancels out. If I could sit down with you with paper and pencil I think you would see it easily.
Also, there’s no way around having to learn things like “The density of water is 1000 kg/m^3,” but you can be happy in the thought that, wherever you measure water in the world, if it’s at standard temperature/pressure that density number will be the same – as Malacandra said, that’s what it means to be a constant.
Also, I apologize if you already know this, but the metric system is totally awesome because you don’t need to remember anything except the prefixes – centi always means 1/100, milli always means 1/1000, kilo always means 1000 (that is, centimeter is 1/100 of a meter, etc.). I apologize if this is obvious, but it was SO not obvious to me in grade school when we learned it-- this was another thing where I unsuccessfully tried to learn conversion factors for YEARS, literally, and was miserable and confused the entire time until a helpful teacher pointed out this fact.
Anyway, the whole point is, the reason why you have to multiply and divide by weird numbers is because you’re going from one way of measuring things to another way of measuring things. Sometimes this is due to the stupidity of the English measuring system (I STILL cannot convert from teaspoons to ounces or pints without looking it up) and sometimes it’s due to wanting things in a different format (similar to the above, for example you have something that weighs a particular amount and you want to know what its volume is… well, volume and weight are two different entities!)
raspberry hunter’s method is how I learned it (from my chemistry teacher) as well.
I would add that multiplying by 1 doesn’t change a number’s value. 17 x 1 = 17. -8 x 1 = -8.
So to take a simpler example, how many inches are there in two feet?
12 inches
_______ = 1, correct?
1 foot
12 inches
_______ x 2 feet = 24 inches
1 foot
(Notice the feet/foot cancel out. I’d like to show the 2 feet at the same height as of “12 inches” but the board reformats, removing the space and putting it right next to it).
Likewise, if you wanted to calculate from miles per hour to feet per second, “per” is the line separating miles/hour, so how many feet/mile, how many hour/minute, how many minute/second…set up the fractions and let the units cancel. As raspberry hunter pointed out, you need to be mindful of “cubic” or “square” measurements.
I picked this one because it is the one I could truly follow, but I think I discovered something about myself in this thread. I don’t figure things out the same way.
Why would you take the time to set up an equation (not as an example, but to really solve something) in such a complicated way? I would do it this way:
How many inches are there in 2 feet? 12 inches/foot x 2= 24 inches/per 2 feet. End of story… You could say I’m so familiar with that particular conversion that I skip many steps, but honestly? That is how I would solve it.
The drug calc I used is one for IV drip dosages. Kgs refers to a pt’s weight; dose would be something like 400mg in 250cc of solution; cc’s per hour is how fast that drip needs to run in order for that pt to get the appropriate drug dose. Does that help? I still don’t know why 60 is in there. Something to do with 60 minutes in an hour, but how does one know to set it up that way? Forever a mystery to me. I hope this doesn’t frustrate anyone here–I’m not really asking to learn it in one math thread. I fear I would need a Prof Henry Higgins type person over many, many months… Please know I admire all who can whip up such problem solving and such in their heads. (but, and this is the little resentful kid in me, please know that I have skills that you probably lack. IOW, we all have different kinds of intelligence and one is not “better” than the other–can you tell I was teased about my lack of math ability?).
I LOVE the math quotes and they comfort me beyond measure. It’s all arbitrary as hell once you really think about it (all you have to do is accept some basic ground rules. If you don’t know the code, you can’t speak the language or solve the puzzles.)
One last thing: what is pi to a unit circle and WTF is a unit circle? My Trig teacher drank and stank of Newports–he had the shakes pretty bad in every class. Trust me, Trig remains a mystery. I did develop a deep and lasting hatred of it. I’m sure I use things that have been created from Trig and all that, but someone please tell me how or what Trig is in the real world. Thanks.
Thanks lobotomyboy63 for giving a more comprehensible example! (I got a little carried away there… I’m still a little confused as to where the minutes and hours go into your drug dosage thing, sorry)
Yeah, that’s the thing. It seems extremely complicated when you do a simple example, and you think, “Why the heck would you do it that way?” I remember thinking that way when I learned about balancing equations (if x + 5 = 7, of course x = 2, why would you bother to do the extra step of subtracting 5 from both sides?) – but when you do something complicated it really makes following each step a lot easier (for example, even as a trained scientist there’s no way I could do the conversion I showed above without the factor label method, I’d get lost in the middle).
I really feel, actually, given that I identify with so many of the things you brought up, that a lot of the fault probably lies with the way you were taught math (I think math teaching, in general, is pretty amazingly awful), and not necessarily with your math ability or lack thereof. I was quite lucky to have a couple of good teachers in with all the chaff to teach me some of this stuff.
The link I gave earlier was to a good, user-friendly, general explanation of dimensional analysis, which is the same thing that raspberry hunter and lobotomyboy63 were talking about. That page has a link to a page that is specifically aimed at nursing students and shows how to do the kinds of problems and conversions that they would face. Here’s the main page, and here’s a link to the part of the page that has some worked-out examples. In particular, #6-8 look similar to the kind of thing you’re dealing with here.
A unit circle is the circle whose radius is 1. That is, it’s the set of all points that are exactly 1 unit away from the origin in all different directions (the “origin” being the point (0,0) at the center of the x-y plane).
If the radius is 1, the diameter is 2 and the circumference is 2*pi.
How this relates to trigonometry is that one way of defining the trigonometric functions sine, cosine, etc. (and hence, one “starting point” for trigonometry) is using the unit circle. If you start at the origin and go off in some particular direction (say, at an angle of 69 degrees), the sine of 69 degrees is the the y-coordinate of the point where you cross the unit circle, and the cosine is the x-coordinate of that point.
Trigonometry is about angles, and it has lots of uses when you’re dealing with problems involving angles, directions, triangles, and things like that. To give just one simple example, if you have a 15-foot ladder leaning against the side of a building at a certain angle, trig can tell you how far up the building the ladder reaches. (Or, if you know how far up, you can use trig to determine the angle.)
Trigonometric functions (sin, cos, etc.) are also used to mathematically describe things that are periodic (like sound waves or other kinds of waves, that go up and down and up and down in a repeating pattern). And they also show up in all sorts of advanced mathematical contexts where you wouldn’t think they had anything to do with angles at all.
A unit circle… depends what you’re trying to do at the time. If you want to talk about radius, then a unit circle has a radius of one $WHATEVER - inch, metre, mile, astronomical unit, parsec. If you want to talk about circumference, you might start with a circle of circumference = one $WHATEVER (in which case its diameter will be one divided by pi - so that pi times diameter equals circumference). And so on. If I was following what you were saying earlier, your prof was saying something like “Let’s start off with a circle of this size” - and, as often happens on the Dope, if the first thing you do is fight the hypothetical (“Wait, what? Why that size? What if it isn’t? Who said it was?”), you never develop the argument.
Trigonometry is founded on one highly important fact - a given set of sides always gets you the same triangle. Triangles can’t wobble at the corners if the sides stay the same length. A given set of four sides, though, could define a rectangle, a parallelogram, or even a kite. This means that it’s worth spending extra time and trouble looking at triangles, because there are ways to relate the sides and the angles (whereas a rectangle can easily be squished down into a parallelogram with more or less any angles you like).
One practical application of trig is in mapmaking. If it’s five miles from Herecombe to Therecombe, and from both those villages I can take a compass bearing on Bald Hill summit, then I need only do a sum or two to work out how far it is from each of them to Bald Hill, without having to actually go there and measure the distance. Repeat this procedure enough times, and I get an accurate map of Loamshire with all the inaccessible spots correctly plotted. And this works only because I need only two angles and a side to uniquely define a triangle.
Then there’s a whole load of extra-special stuff we get from considering right-angled triangles… including how to keep your washing machine from shaking itself to pieces.
Your method is the exact same as mine, really. I just set it up visually a little differently. With “nice numbers” like two feet they can be done in your head many times. However it’s also useful to set up the equation if you need to calculate using weird numbers, like 2.34 feet = how many inches exactly?
Another reason for doing it that way is that you’re often converting more complicated things, say “miles per hour” to “feet per second.” You convert the miles to 5280 feet/mile, then 12 inches/foot and you can see the units canceling. You convert from hours to seconds by multiplying by 60 minutes/hour, then 60 seconds/minute.
So
60 miles x 5280 feet x hour x minute
1 hour x 1 mile x 60 minutes x 60 seconds
The remaining units, which haven’t cancelled, are feet per second.
If you came up with a wrong answer, you’d probably be able to look back at your units and say, “Wait, the answer I have is feet per hour” or “No, that’s miles per minute,” that sort of thing.
The equation is set up for units to cancel out…remember these?
6x7x3x4
4x7x3
You just cross off 7s, 3s, and 4s to simplify and the answer is 6. Only here, it’s getting to the units (feet, miles, whatever) that you want.
Just keep in mind where you’re starting and where you want to end up. If you’re starting at 10 cc per minute and want to convert to hours, multiply by hour/60 minutes and your minutes will cancel out. OTOH if you’re given 10 cc per hour, multiply by 60 minutes/hour and your hours will cancel out.
Mal–that is the first time anyone has ever been able to tie Trig to something concrete and useful for me. I now have a new respect for it. I still don’t know how to do it, but I don’t need to. Thank you!
Lobotomy–but here is my question: HOW does one know which to put in where? Minutes or hours, ounces or pounds, whatever, whichever. I know you’ll say it all depends on what one is solving for and that is also the rub: I often didn’t know… Thudlow–thanks again for your help. I don’t need to know the formulas anymore for nursing for 2 reasons: I no longer work in ICU and so I don’t titrate drips and also RNs now have IV pumps with computers in them. We have to check to make sure the pump is programmed correctly. But also, re the unit circle: unit of WHAT? inches, centimeters, miles, light years, pubic hairs, WHAT? And with that, one what? It is a circle. Circles describe something–so what does the unit and/or the 1 refer to? Puzzles me no end.
Raspberry: I agree that I probably had really bad math teachers. Combine that with major math anxiety AND “the new math” (which never, ever made sense to a very young me), and I was not destined for Mathletes. I do like orderly, precise things, though. I have often wondered, what with all the testing that goes on now and kids being diagnosed with this or that disorder, if I don’t have some kind of learning disability re math, but that’s another thread. I didn’t want this to become all about me.
Thanks again for your patience and kindness. Who wants to balance my checkbook?
Dopamine comes in 400mg per 250cc D5W (the D5W is unimportant). You receive a pt with a Dopamine drip infusing at 15cc/hr*. Your pt weighs 112 kgs. You need to figure out how much Dopamine this pt is getting. The doctor wants the pt to get 5mcg/kg/min.
(dose/volume) x 1000 (somethings**)
400/250=1.6mg/cc x 1000= 1600mcg(somethings)
1600/60=26.6 somethings
26.6/112kgs=0.23mcgs/cc
0.23 x 15 = 3.45mcgs/kg/min, which is not enough. This pt needs to get 22cc/hr of the Dopamine drip to get 5.06mcgs/kg/min. 21 is not enough (there usually is no getting of exactly 5 or whatever number is designated).
The way I figured out how much to increase the dose was by just plugging in numbers until I got the one I needed. I USED to know the formula to calc that type of thing straight away, but it is lost in the mists of time (I lost my little notebook).
*No nurse should accept a pt with this type of “report”. Knowing how fast the drip is going is fairly useless in the relaying of info-- a nurse needs to know the current dosage infusing, not the current rate of infusion. For other drips, other things matter. Anyhoo, the nurse should say, “this pt is on Dopamine at 5 mics”, but I digress…
**I used “something” for where I have no idea what unit should be there to describe the calcs I’m making.
I have looked at formulae online and most of them use a constant. That is not how I learned, but I see the advantage of that shortcut. Anyway, there it is: a small window into the world of an ICU nurse, circa 1986.
First question: it does depend on what you are converting to what. Usually you should be told this information, like you were with the kg and cc/hr-- I think that there must be a piece of missing information in your ICU formula, which is “how many mg of drug does the patient need per minute,” which is part of why it’s hard to figure out what to put where. That’s also a pretty intricate problem as conversions go.
The unit circle… let me take a whack at it. If you think about a triangle with angles 90, 45, and 45, the sides always have the same ratio, 1 to 0.7 to 0.7 (roughly), where the 1 is the hypotenuse of the right triangle. If you had a 45-45-90 triangle with a hypotenuse of 3 in., you could work out that the non-hypotenuse sides would have side lengths of 0.7*3in. = 2.1in because you know the sides have to have this ratio. (So far I have said exactly what Malacandra was saying, only with a specific example.)
Using a “unit circle” is a simple tool that lets you keep track of what these ratios are. If you draw a circle with radius one, and draw a line from the center of the circle that makes a 45 degree angle with the x-axis, the (x,y) coordinates of where that line intersects the circle will be exactly 0.7 and 0.7 (the ratios I alluded to above). (These ratios are called the “cosine” and “sine,” if you care.) So this can be thought of as a quick way of “automatically” drawing a triangle with the correct angles to get those useful ratios. (There are more complex reasons as well why this is a useful picture for mathematicians, but we’ll skip those.)
Also, I never had to deal with the new math as a child, but I could imagine that someone whose introduction to math was that would be scarred for life.
Oops, as I was typing that I see that you supplied that information! So actually this is much simpler than I thought because you have all the information you need.
The doctor has given you the starting point:
He wants 5 mcg/kg/min.
The ending point you yourself have given me: you want to know how much cc/hr/patient.
We can factor label/dimensional analysis as follows:
Again, each set of parentheses has something that multiplies out to 1 (1 patient = 112 kg) and you can cancel out units (the kg on top cancels the kg on the bottom of 5 mcg/kg/min). The way I figured out what things to put in the parentheses was something like “Okay, I have mcg and I want cc. I have kg and I want 1 patients’ worth of weight. I have minutes and I want an hour.”
No offense, but I can’t really follow your something-somethings above – no wonder you are confused!
(Or you can go the other way around: given 15 cc/hr/patient,
15 cc/hr/patient * 1 hr/60 min * 1 patient/112 kg * 400mg/250cc * 100mcg/1mg = 3.45 mcg/kg/min.)
I think the confusion is because the word “unit” can refer to a unit of measurement (like an inch or a second), but it can also be essentially a synonym for “one.” (For example, when you’re talking about place value in numbers, the column to the right of the “tens column” is sometimes called the “units column” and sometimes the “ones column.” The word “unity” essentially means “oneness.”) It is in this latter sense that the word “unit” in “unit circle” (or “unit square”) is to be taken. The unit circle is the “one circle.”
This is related to the former sense, because when you are measuring things, a line segment of “unit length” is one whose length is 1 in whatever units of measure you’re using (inches, centimeters, or whatever), and all the other distances are described in terms of how many times as long as that “unit length” they are.
In the case of “the unit circle,” the answer to “one what?” is “one of whatever you’re measuring in.” Or “one of those little squares on the graph paper.” Or just “pure one,” without any units attached. In pure math, as opposed to applied math, numbers are entities in and of themselves; you don’t have to worry about units. It’s like when you graph an equation (like y = 2x + 5) or plot points (like (3, 7) ) in algebra. What units are the 3 and the 7 in? None, necessarily; they just tell you how many “little squares” to count out on the graph paper.
<Keels over from mental overload.>
(actually, that does help a lot and probably would have made a difference in Trig back in 1979, but it’s too late now).
Ok, I’m not retarded: I know that “per” stands in for the “/” or dividing sign (or a ratio of sorts, no?).
Again, I CAN and HAVE solved the drug calcs equations many times, I just don’t don’t understand how all the steps were decided upon. Math (to me) is like driving a car, you can do so perfectly competently w/o knowing how the engine works.
I can’t convey just what my confusion is, so I’ll stop trying. This was fun in a weird way. I think I get (finally) that there are many ways to solve for x–something my teachers never shared with me… I think I am better at breaking it down into steps, rather than presenting the whole equation, if that makes sense.
Sorry, it wasn’t clear from your first question whether you wondered how to set up the initial fraction, the subsequent steps, or both so I took it from square one.
The big principle behind converting things is that you can multiply anything by one and the value of it doesn’t change. 24 inches and 2 feet represent the same length but expressed differently. Five pennies and a nickel are the same amount of money, though they look different. 1/2, 2/4, 3/6…the same amount.
Just as you decided to multiply by 12 to convert from feet to inches, someone decided to convert mg to mcg (multiply by 1000) and so on for every part that needed converting, then wrote up the equation for others to use. If that equation were lost somehow, mathematicians could figure it out again.
It’s like translating from English to Spanish. Translation doesn’t change meaning, so why do it? Well, you do it specifically because you want, for whatever reason, to change the presentation of that meaning. You start with “dog” but you need a Spanish word, so you switch over to “perro”, which means the same thing but is presented in a different format. Same thing with converting between units; you start with some number of feet, say, and you want to end up with some number of inches that means the same thing. If you keep multiplying by things which are equal to 1, you can be sure that you’ll end up with (something which means) the same thing; the point isn’t actually the multiplication, but the way you switch the presentation of the results while you do so. 4 feet * 12 inches/1 foot = 48 inches. Well, 4 feet and 48 inches mean the same thing; you only multiplied by 12 inches/1 foot because that “inches/foot” bit in there caused you to convert your presentation of that quantity from feet to inches. Does that make sense?
