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Perhaps a refresher in mathematics would help here. A linear equation can be represented on a graph using a straight line. The equation y = x + 1 would show a diagonal line where each point on the y axis has a value that is one unit higher than that point’s location on the x axis. Increasing the value on x by any number would have the same effect on y. Suppose the initial value of x is 1. Here are some examples of the proportional increase:

- y = x + 1
- [x = 1] 2 = 1 + 1
- [add 4 to x] 6 = 5 + 1
- [add another 10] 16 = 15 + 1

The output y is proportional to the input x in linear equations. Nonlinear equations do not behave that way. Trying the same thing with a nonlinear equation, using a square number, the following results are obtained:

- y = x
^{2} - [x = 1] 1 = 1
^{2} - [add 1 to x] 4 = 2
^{2} - [add another 10] 144 = 12
^{2}

Increasing the value of x does not produce a proportional increase of y. While linear equations are homogeneous and additive, nonlinear equations are not.

Controlling output in nonlinear systems can be a problem. Nonlinearity in information processing requires more intricate calculations. Analog signals produce curved rather than straight lines because of the varying wave forms. Amplifying signals may require complicated algorithms. Nonlinear systems may seem chaotic or unpredictable.

Pablo Parrilo of MIT says, “I think that it’s a reasonable statement that we mostly understand linear phenomena.” But the fact that most of the universe is nonlinear makes work more interesting for physicists, mathematicians and computer scientists.

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