Importance of Linear FunctionsDate: 02/06/2002 at 16:38:51 From: Abreeka Moore Subject: Importance of Linear Equations Hello, I was wondering what the importance of linear functions in the "real world" is, and how they are used. Thank you. Date: 02/07/2002 at 09:46:12 From: Doctor Ian Subject: Re: Importance of Linear Equations Hi Abreeka, Most 'real world' functions are approximations, as the world usually contains too many complications to be modeled exactly by mathematical functions. The really nice thing about linear functions is that they are easy to work with. They are easy to solve, easy to plot, and easy to understand. So when you're looking for a function to approximate the behavior of something in the real world, you usually try to use a linear function first; and only if that proves to be too simple a model do you look for other kinds of functions to use. Often, in order to use a linear function to model something, you need to find the 'best' line that models your data. For example, if your data look like | * | * * | * * * | * * * | * | +----------------- each pair of points that you could choose would give you a different line; but there is _no_ line that contains all the points. What you can do in a case like this is find the line that is 'closest' (in some sense) to all the points: | L | * | * * | * * L* The 'best' line is the one that passes | * * * through the L's. | * |L +----------------- This line is normally found using a process called 'least squares' analysis, or 'linear regression'. (The basic idea is this: Choose a line; find the distance of each data point from the line; add those distances up to get a score for the line. Try other lines, until you find the one that gives you the lowest score. That's the 'best line'.) Once you've found a linear function that you think models your data, you can use the function to make predictions. For example, if the data above showed the amount of wheat harvested as a function of inches of rain in past years, then knowing how much rain has fallen _this_ year can let you predict the amount of wheat that will be harvested, even before it's finished growing. Note that when you've reached this point, one little function wheat = m*rain + b replaces a whole collection of data, so it's more efficient in terms of space. Of course, there is a danger in doing this: It might turn out that the function doesn't consider all the relevant factors. (For example, interest rates and subsidies have as much to do with the production of wheat as rain does.) In which case, any predictions that it makes might be wrong. But this is always the trade-off - ease of use versus precision - that you make when you try to use math to describe the real world. As someone once said, you want to make your mathematical models as simple as possible... but no simpler. A linear function is about the simplest kind of model that it's possible to make. Does this help? Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 02/07/2002 at 23:21:53 From: Abreeka Moore Subject: Importance of Linear Equations Thank you! That was a huge help to me. I look forward to sending questions in the future. |
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