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I Am the Walrus
Equations

 
 In science, we try to describe the actions and reactions 
 of stuff in the universe. 
 For the most part, that means we say things like: 
 "When A does this, B does that."
 
 We could write paragraphs about one of these action/reaction things 

 and someone who read it, might not get it exactly right. 

 When they told someone else, it might get changed even more. 
 Sooner or later, as the info was passed around through telling and retelling, 
 the description would be totally wrong.
 
 To make misunderstandings less likely to happen, 
 we need a way to describe these things that is very short 
 and uses only a few words or symbols 
 that everybody agrees on and understands.
 
 We needed equations.
 
 Equations describe the relationship between different things.
 The most basic equation says if one thing gets bigger by a certain percent, 
 the other thing gets bigger by that same percent. 
 If A and B are the two things, the equation looks like this ...
 

 A = B

 
 The first complication that we get is that unlike math, 
 science rarely uses numbers as just numbers. 
 In science, numbers almost always have units attached to them.
 
 The size of the units of stuff often turns out to be 
 a result of historical accidents.
 So units of some stuff might be tiny, 
 while units of other stuff might be gigantic.
 
 Example:
 
 There are two common systems used to measure distance.
 The metric system uses meters, centimeters, millimeters and the like.
 The older English system uses feet, inches, miles and stuff like that.
 
 Numbers from each of these systems can be put into an equation.
 We know that if something gets twice as far away in feet, 
 it also gets twice as far away in meters. 
 We should be able to put these into an equation 
 something like our A = B equation from above.
 
 If we try that, we have a problem. 
 One foot is not the same size as 1 meter.
 A meter is much bigger. 
 To make Feet and Meters "balance" we need to multiply feet by something. 
 That is, multiply 1 foot by some number so that it is just as big as a meter.
 
 That something is a number called a constant.
 Generally, we use the letter c to stand for the constant.
 Using that idea, we can say ...
 

 X meters = X feet c

 
 So what do we have to multiply a foot by to get a meter?
 You could go somewhere and look it up, but to save time I'll just tell you.
 
 The number is about 3.28. 
 To be a bit more accurate, it is about 3.28083989501 (and so on)
 
 This is our "c" for the equation.
 So we have ...
 

 X meters = X feet 3.28

 
 Since 3.28 is only approximately right, 
 using an equal sign here is not completely, exactly correct.
 Sometimes when we have an equation where the two sides 
 are only "about" equal, we use a wavy equal sign. 
 Like this ...
 

X meters X feet 3.28

 
 In some books, the wavy equal sign is printed as a regular equal sign 
 with one wavy line on top, 
 like this ...
 

X meters @ X feet 3.28

 
 The big point here is that the constant is just there
 to make the units work out right.
 
 The relationship between feet and meters says 
 that as a measurement in feet changes by some percent,
 the same measurement in meters gets bigger by that same percent. 
 We just need to multiply by the constant so the units work out right.
 
 Also, there's no rule that says we need to put the constant 
 on the side of the smaller unit. 
 
 We could put a constant on the side with the meters like this ... 
 

  X feet = X meters c

 
 This time, the constant would be EXACTLY 0.3048.
 That gives us ...
 

X feet = X meters 0.3048

 
 In Einstein's famous equation ...
 

 e = mc2

 
 e is energy, m is mass, 
 and c 2 is a constant to make the units work out right.
 The number we use for the constant c 2
 depends on the units we use for the energy and mass terms.
 
 But whatever the constant is, the underlying idea
 is that there is a direct proportional relationship between mass and energy. 
 The c 2 is ONLY there to make the units work out right in the equation.
 
 Sometimes when one thing gets smaller, something else gets bigger. 
 This is called an inverse relationship. 
 If we have air in a cylinder and made the volume of the cylinder smaller, 
 the pressure inside the cylinder would get greater. 
 If the cylinder volume was made half as big, 
 the pressure would be twice as great. 
 
 If we have V stand for the volume of a cylinder 
 and P stand for the pressure, 
 we can start to write the equation as ...
 

 
 Through some algebra shenanigans, 
 you should be able to see that we can also write this as ...
 

 
 Once we know what units V and P are measured in, 
 we can figure out what constant we need  
 to make the units we have work out right.
 
 Realize that the constant we need for this equation ...
 

 
 Is VERY unlikely to be the same number we need for this equation ...
 

 

 
 It is also very unlikely that the units all work out right 
 so that we don't need a constant at all. 
 This would be a case where the units all happen to balance.
 It's kind of like if you flipped a coin and it landed on edge. 
 
 It's like one of my favorite sayings. 
 In science, all things are possible, 
 it's just that some of those things are incredibly unlikely.
 

   copyright 2005 Bruce Kirkpatrick

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