Variables: Connecting Letters and Numbers
Date: 12/06/2001 at 12:48:12 From: Elizabeth Subject: Understanding the connection between letters and numbers in algebra No matter how many books I read, I can't understand how x = whatever. I'm still trying to know how you make a connection. Is there some sort of table you can use to show what letter is equal to what number?
Date: 12/06/2001 at 14:10:56 From: Doctor Ian Subject: Re: Understanding the connection between letters and numbers in algebra Hi Elizabeth, A variable is just a name you use to talk about a number whose value you don't know yet. Suppose I tell you that if you add John's age (in years) to Joan's age (in years), you get 27, and if you multiply their ages, you get 140. Now, you know that John _has_ an age. You just don't know what it is. And you know that Joan has an age, but again, you don't know what it is. However, you know some facts about their ages, and in order to express those facts, you need some way to refer to their ages. You could write something like John's age + Joan's age = 27 John's age * Joan's age = 140 Now, this is more writing than you probably want to do, so you might choose shorter names. For example, you might use 'J' to stand for John's age, since the J would remind you of 'John'. But this means you can't use 'J' for Joan's age, so you need to think up another name. A somewhat natural choice would be the next letter of the alphabet, 'K'. Now we can write J + K = 27 J * K = 140 At this point, it's important to realize that these are just arbitrary names. And a couple of important points follow from that: 1) We could use any other names without changing the meaning of the equations. For example, we could use 'X' instead of 'J': X + K = 27 X + K = 140 And we could use 'Y' instead of 'K': X + Y = 27 X * Y = 140 The particular variable names that we use don't make any difference, so long as within a given problem, the same variable name always has the same meaning. (It's sort of like in a novel. It doesn't matter what name is given to any character, so long as each name stays with the same character throughout the book.) For that matter, you don't even have to use letters for names. You could use pictures if you wanted to. But letters have a few nice things going for them. For one, most people can recognize them, and can agree on how they should be pronounced. Also, most people already know how to write them, or how to type them with a keyboard. Also, they aren't used for operations (like addition or multiplication), so if you use a letter in an equation, people reading the equation can have a lot of confidence that it is a variable name. 2) The value that turns out to be associated with a particular variable name can be different from problem to problem. For example, in this problem, it turns out that J and K have the values 20 and 7: 20 + 7 = 27 20 * 7 = 140 But here is another problem that uses the same variable names: John has twice as many cookies as Joan. If he gives her two cookies, they'll have the same number of cookies. Let J stand for the number of cookies that John has, and let K stand for the number of cookies that Joan has. J = 2K John has twice as many cookies J - 2 = K + 2 A transfer of two cookies makes them equal. In this case, it turns out that J has the value 8, and K has the value 4: 8 = 2 * 4 8 - 2 = 4 + 2 So there is no table, or other device, that you can use to simply 'look up' the value of a variable based on the name that has been given to it. Is J equal to 29, or 7, or 8? It depends on the particular problem being solved. (Again, it's sort of like with books. If you see the name 'Chris' being used as a character in one book, and you see the same name being used in another book, you don't assume that they refer to the same character. And you don't know anything about the character until you've started reading the book.) Because there is no connection between the name of a variable and the value that it takes, it's common to just use the same names over and over - x, y, and z, for example; or n, or k; or a, b, and c. As a way of getting used to the idea of variables, you might consider using entire names instead of letters. For example: A garden is 6 feet longer than it is wide. The perimeter of the garden is 30 feet. What are the dimensions of the garden? 1. Write an expression for the perimeter: 30 = perimeter = length + width + length + width = 2*length + 2*width = 2(length + width) 2. Write an expression for length in terms of width: length - width = 6 length = 6 + width 3. Substitute and solve 30 = 2((6 + width) + width) = 2(6 + 2*width) = 12 + 4*width 18 = 4*width 18/4 = width Now, when you see something like this, you don't get the feeling that you can just 'look up' the value of 'width', do you? It's pretty clear that the width could be any value, and your job is to find out what value makes the equations true. Once you get used to working with words as variable names, you'll get more comfortable with the idea of variables in general; and eventually you'll decide, in your own time, to start using letters instead of whole words. It's a little like learning to stand up before learning to walk. You might also find it useful to read these answers from the Dr. Math archives: What is Algebra? http://mathforum.org/dr.math/problems/jason.07.20.01.html Letters for Variables http://mathforum.org/dr.math/problems/spencer.11.19.01.html I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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