Why Do We Use Variables in Algebra?
Date: 01/31/2004 at 19:38:16 From: Chrystyne Subject: Use of letters for numbers Why do we use letters to represent numbers in algebra? I think that the letter represents a number that we do not have the answer to.
Date: 01/31/2004 at 23:36:55 From: Doctor Peterson Subject: Re: Use of letters for numbers Hi, Chrystyne. There are several slightly different reasons for using variables, which are all variations on the idea you suggested: 1. We don't know the value, and we want to find what it is. For example, we might want to solve the equation 3x = 6 to find the height of a rectangle whose width is 3 units and whose area is 6 square units. We are finding what value makes that equation true. 2. We don't YET know the value, and want to be able to make a formula so we can use the value when we get it. For example, we might be preparing to find the areas of a set of different rectangles whose width is 3 units but whose heights will vary. We can write the expression A = 3x to remember how to find the area of each rectangle when we get its height. This is how computers usually use variables; when you enter a value for a variable, a program can calculate a result using an expression like this. 3. We know the value, but want to do all the algebraic manipulation (such as simplifying an expression) without using actual numbers, to save work. For example, rather than calculate (sqrt(1.234))^2 -- that is, the square of the square root of 1.234 -- I might first think of it as (sqrt(x))^2, replacing the known value with a variable temporarily so I can simplify the expression to "x", and not have to take a square root at all. That's too simple an example, but this sort of thinking is useful in many big problems, where solving the problem in terms of variables rather than numbers can save a huge amount of work and errors. 4. We want to prove that something is true regardless of the value of the numbers involved. For example, we can show that ab = ba for all numbers a and b; we don't need to ever know what those values are in order to talk about this fact. In summary, any time we want to talk about a number without having to know its value at the moment, we can give it a name and talk about it using that name. It's not much different from using your name rather than a picture of you when you write about yourself; or using the pronoun "she" or the noun "occupant" or "user" instead of your name in a document that might have been written with reference to anybody. It allows us to talk about things that are very hard to talk about in any other way. Here are some places where we have explained the usefulness of variables: Understanding Variables http://mathforum.org/library/drmath/view/57311.html Variables: Connecting Letters and Numbers http://mathforum.org/library/drmath/view/57312.html Why Use Letters in Math? http://mathforum.org/library/drmath/view/57020.html The Why of Algebra http://mathforum.org/library/drmath/view/57015.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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