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What is a variable, and how is it different from a constant? In algebra, letters like a, b, c... or x, y, z..., or n, or some special letters or symbols for special cases are used to represent numbers who's values are not determined yet. We do this because values in equations can change; sometimes we just want a "placeholder" for a number that we will decide upon later, and sometimes we don't even know what the number could be yet - we first have to solve for it. Letters and symbols that are used like this are called variables. A constant, on the other hand, a is a value that never changes. Numbers like the whole numbers, rational numbers, etc. are constants, as are symbols for special numbers, like π, which represents the number 3.1415... Using letters like a, b, c... is generally accepted in algebra to mean, "any number," or "some number." If you want to have your variable represent a specific kind of number, then you could say: a (where a is ...) and then say what kind of number you want a to be. then a would be accepted to be that kind. For instance, if we wanted a to stand for "any even number", we could represent it as: a (where a is any even number). Then we could use a in our equations without having to say "any even number" over and over again. For example, we could ask this question about the following mathematical expressions: Can you tell if each of the following expressions could possibly give an odd number as a result?
That would save us a lot of writing. If we couldn't use variables, we'd have to write:
By the way, you may have noticed that variables are generally written in italics. This is done so they are easier to recognize. The letters in the beginning of the alphabet, like a, b, c... are usually used to express numbers in formulae. (Formulae is the plural form of formula. You can also say "formulas", but it doesn't sound as cool.) We use a, b, c... to express generalizations of numbers that may be known before you do the problem. You'll find some examples below. Examples of variables in formulae that you might be familiar with You may know of the Pythagorean theorem, which is a2 + b2 = c2 . It is the formula to find the hypotenuse of a right triangle. I won't go into the details here, but in a nutshell: If you have a right triangle (one with one angle that is exactly 90 degrees - like the corner of a square) and you know how long two of the sides are, you can always find the length of the third side using Pythagorean Theorem. All you have to do is plug the lengths of the sides into the formula, where
Try it: Let's say we have a triangle that has side a at 4' and side b at 3'. What is the length of the hypotenuse? Just plug the numbers into the formula: 4 2 + 32 = c2, which
means: I think you get how to use a, b, c... in formulae by now. But when do you use ... x, y, z? Remember when I said "We use a, b, c... to express generalizations of numbers which may be known before you do the problem"? We use the numbers at the end of the alphabet, like ...x, y, z to represent numbers in equations that are unknown. For example, if I asked you, "what number, when added to 63, then multiplied by 7, will give you 483?" you wouldn't know what any variables at the beginning of the equation were. You would know some numbers, though, like 63, 7 and 483. You would use an x to represent the unknown variable. What's the difference between a
formula and an equation? An equation is a way of relating two mathematical expressions as having the same value. Equations are usually what you are using when you are "solving for x". They often are used for solving word problems. For example, "If Bob is 27 and his son is 6, how in how many years will Bob be twice the age of his son?" You could represent that as 2(6+x) = 27+x, which,
when you multiply the parenthesis is: We now know that it will take 15 years until Bob is twice as old as his son. In 15 years his son will be 21 and he will be 42. We have shown how we can use variables create an equation to solve a problem. Hotcha! A formula is a generalization into which you put known values to get a result. You know some of them already, even if you (nor most other people) know what they really mean. For example: E=mc2 (where E = energy, m = mass, and c = the velocity of light traveling in a vacuum, squared. You can't solve for anything until you get more information, so it's not an equation. You need to know the mass (m) before you can make an equation of it. Here's a question that will help you test if you really know your stuff about variables and constants: Is c (in E=mc2) a variable or a constant? (Scroll to the bottom for the answer if you aren't sure. But think about it first.) n is generally used for some unknown amount. Like if I said, "I have a bunch of bananas, and they each cost 45¢, how do I determine how much any group of them will cost?" I'd just multiply 45n. OK, but what about special variables? Like h for "height," t for "time" etc.? There are formulae for area of a triangle (A=1/2bh), for speed (S=d/t), etc. The variables in such equations are generally identifiable by their context (the way they are used). We use those variables because they seem to make sense; A for area, B for base, H for height, S for speed, D for distance, T for time, etc). It just helps to keep things less confusing. Of course you could use any variable symbol for them, but why make things tougher than they have to be? (Because your a stubborn teenager who has to rebel, right? OK, I can relate.) Capitalizing variables, like the "E" in E=mc2 Why is the E in E=mc2 often capitalized, and not italicized? The same goes for other formulae, like the F in Newton's Second Rule of Motion, which is F=ma. You can also see this in the formulae for the area of a triangle and distance traveled, mentioned above. As a general rule of thumb, it seems fair to assume that you capitalize the variable that is the name of the formula. For instance, The formula for the circumference of a circle would be C = πd, but the formula for the diameter of a circle would be D = c/π. In other words, you capitalize the variable that you are trying to find (it's usually the one on the left of the equal sign, and is alone.) The more precise reason is because we capitalize dependent variables, and we use lower case letters for independent variables. Rats! Now we have more stuff to define! Don't worry, it's cake. Dependent variables can't be determined until the independent variable is given. For example, in D = c/π, we can't figure out what D is until we know what c is. That means that the value of D is dependent on the value of c. c is an independent variable (also known as a free term) because we have to deal with whatever value for c that is given. In other words we have to use whichever c comes along. Independent variables are like the Ronin of variables. The Knights Errant, the Prodigal Sons, the hippies of the variable world... In calculus, the lower case letter represents the derivative of the capital letter. This seems pretty consistent with the rule for algebra. Also, when constants that are unspecified but fixed are represented as letters, they may be capitalized when combined with variables in linear equations, as in the case of Ax = By + C = 0. We are starting to get out of our range here, and I don't intend to confuse you with this any further. I just wanted to mention it here for those of you who appreciate it. As far as why the capital letters are italicized or not, your on your own with that one, too! Who made up these rules, anyway? Variables have been used since antiquity, but it was Renè Descartes in his La gèometie (1637) who set the rules about a, b, c ... and ... x, y, z. (So now you know who to flame on the message boards if you hate this stuff.) Remember, all these rules are generalizations. We use them for convenience. You won't go to hell if you switch them around, but you may confuse yourself and others if you do. It's a good rule of thumb to generally stick with these rules unless you have a good reason for not doing so. And when you don't stick with them, somewhere in your work you will probably want to make it clear why you didn't. Answer to the question about the c in E=mc2: Just one more thing: Teachers and testers who hold you responsible for following specific rules about variables, but didn't try to explain them too you are a... |
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