# Linearity – I

The word ‘linear’ is used in so many different contexts that it kind of loses its importance. After a lot of contemplation, it turns out that this word as simple as it may sound, has far reaching consequences. The point of this post is to exactly make this clear and precise (maybe not precise). There are lots of ways to describe linearity. Someone who has taken a calculus course in high school or college will say that things are nice when everything is linear, for example linear equations or straight lines have a constant slope, oops we need to define what ‘slope’ is. Let’s say we would like to buy sugar. The cost of a kilogram of sugar is \$10 and to simplify things, we pay a tax of \$5 how much ever the quantity is. If we want to buy 10 kilograms of sugar, we need \$105 i.e., 10 times 10 plus 5(meh).  A linear equation is a thing that can be written in this form i.e., c = ax + b where a and b can be any arbitrary numbers, x is the amount of sugar and c is the total cost to buy x kilograms of sugar. The number a is often called as the slope since it decides how steep the function, that is, if a is a big positive number then we can say that the buying sugar will be expensive and so on, well to be honest it also depends on b but let’s say that we always have \$b. We note at this point that this also allows a to be zero (now what does that mean in our example? I’ll leave that for you). But the situation is bad when a is zero, why? this requires some amount of reasoning and some machinery or terminology to address this issue.