This simple question “is a straight line linear?” came to my mind and couldn’t get out of it. It seems to be obvious, so it should have an obvious answer. However, when I dig into the research and try to find proof for this simple answer, I realized it’s more complex than I thought and more interesting!

*Linearity* and *non-linearity* are terms commonly used in Data Science and Math. We try to figure out if a problem can be solved using linear or non-linear models, or if our data is *linearly separable*. If we widely use these terms, it’s important to understand them well, isn’t it? So what does it mean that something is linear by the book?

## It’s simple… draw it!

Let’s think and try to visualize *linearity*. What comes to your mind? For me, it’s a simple straight line. Such a line is described by the following formula:

Linear (?) function formula (image by Author).

And indeed it is called a *linear function* [1]. In this formula, we have a variable *x*, and two parameters *a* and *b*. Let’s assume that *a = 1*,_ b = -2_, and plot it:

Linear function f(x) = x -2 (image by Author).

In the picture we can see that line goes through point *A (0, 2)* as *f(0) = 2* and point *B (-2, 0)*, because *f(-2) = 0*. That’s how we usually imagine linearity.

We also tend to think about linearity in terms of *proportionality* [2]. It means that the input is proportional to the output. Such intuitive thinking is natural for us, however, we need to remember that formally proportionality is a separate, wider term.

By definition proportionality of two variables or quantities is a relation that occurs when ratio or product of these quantities is a *constant*. Having variables *x* and *y*, they are *proportional* if:

Proportionality coefficient (image by Author).

Where *k* is a constant. In this situation, if *x* is increasing, *y* also is increasing. The same applies when we decrease our variables, it’s proportional.

Variables are *inversely proportional* if their product is a constant:

Inverse proportionality coefficient (image by Author).

To check if it fits our intuition, we can transform this formula like this:

Transformed inverse proportionality formula (image by Author).

Assuming that *k* is greater than *0*, and *y* is not equal to *0*, while increasing *y*, *x* is decreasing. So everything is ok, it’s inversely proportional.

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