Limits, Calculus and How Things Change (Explained Without Numbers)


Put your name in the hat to tell a story at the Ale House on Front Parlor nights. Math is sometimes best discussed with beer.

Put your name in the hat to tell a story at the Ale House on Front Parlor nights. Math is sometimes best discussed with beer.

A few years ago for a story night on the theme of “limits” the first definition that came to mind for me as a math nerd was not “a restriction on the size or amount of something permissible or possible,” nor “a point or level beyond which something does not or may not extend or pass,” or any other definition that carried a negative association that most people have with the word. Instead, my mind wandered back to the definition I was taught in September of 1998 during my first days of Calculus at Brooklyn Technical High School.

While the idea of Mathematics is often confused with numbers and equations, at its heart it is a language like any other – we use it to formulate, transcribe, communicate and prove our ideas. It can be used to describe the world around us, either artistically or in explicit detail. Like other languages it contains forms of verbs, nouns, proper nouns, syntax, and other grammatical rules and conventions to be followed. Though it is particularly efficient for expressing and calculating numerical data, it can be equally as powerful in the hands a poet. Further discussion of this topic of this will remain for another day...

The Language of Change

In the language of Mathematics, Calculus contains the words we use to talk about change. Typically when we think about something changing, whether explicitly or not, it happens in relation, or with respect to, one or more other things.

Here are some gifs of things changing:

Like all languages, new words and concepts in mathematics evolve as people need to share new ideas with other people, whether in spoken or written form. In the case of limits, a big part of the backstory is actually already very well known by most elementary school students – when Isaac Newton famously needed to explain his theory of how gravity works. In particular it isn't just that gravity pulls on objects causing them to move towards the ground, it is the subtle detail of what happens when a force (visible or invisible) is applied continuously to an object. When something is falling, it doesn't just fall at a constant speed, it begins to fall at a faster and faster rate the longer it falls.

Common math notation and grammar

Newton wasn't the only mathematician of the time period who needed these kinds of words in their dictionary to describe the world in detail. It took over 200 years, from Descartes (noted with bridging algebraic words with geometric pictures) to Leibniz (who came up with much of the notation we still use today) to Cauchy and Riemann (who refined the definition and grammar of limits) for the new branch to develop and become a formalized addition to the language of Mathematics. Before this you can imagine a world full of mathematicians trying to talk about how things change while speaking slightly different dialects of the same tongue.

When change happens, it's either very small, very big, or somewhere in the middle. Calculus is the mathematical grammar we use when talking about change and it's broken into two main types: Differential Calculus introduces the idea of taking these very small, infinitesimal steps to accurately approximate instantaneous rates of change (like how fast a file is downloading from the Internet right now), whereas Integral Calculus introduces the idea of taking infinitesimally small steps to approximate large sums. Whether you are consciously aware of it or not, when making predications about the future or imagining the past, the thought process is similar to this branch of math and we can use it to express and understand the results of our thinking.

Regardless of the amount of the change, we can also talk about the direction. Is it increasing, decreasing or staying the same? Did the price go up or down? When did it happen? Like Newton and his peers, when we start to ask these questions, we quickly run into one the most important words in all of modern Mathematics: limits.

Approaching The Beginning and Ends of Time and Space

There's actually a bit more imagination involved with understanding limits in the mathematical definition than typically encountered in other branches of math. Unlike the natural language versions, in Mathematics we use limits to talk about what happens as we approach a given point on a path from all possible directions at the same time. There's a bit of poetry to the idea. As we get closer and closer, traveling from any possible direction, if we always arrive on the at the same finite point in space, a limit is said to exist at that point.

Everything is nothing,
with a twist.
— Kurt Vonnegut, Slaughter-house Five

Why is it important that these limits exist on our paths going from one place to another? It's actually more about of what happens when they don't – it's difficult to have a discussion about change without the consistences and continuity that limits bring to the table. Without them such conversations become full of holes and conflicting truths. Or try to imagine walking down a street where you appear in a different place depending on which direction you arrive from. The Universe as we know it is a very tricky place to navigate without limits.

In particular there are two points on our paths that limits are helpful for describing because we often can't actually go there, only approach them; zero (also known as nothing, nil or null) and infinity (the number bigger than the biggest number). In Mathematics these concepts are represented by the symbols “0” and “∞” respectively and they come into play when dealing with very small and very large scale ideas alike. We consistently consider what happens as we get near these places when we are talking about how things are changing at an exact moment in time, or when accumulating the results of continued forces over hundreds of thousands of years.

An xkcd comic uses limits in a punchline in a joke about an infinite number of little pigs in a children's story.

An xkcd comic uses limits in a punchline in a joke about an infinite number of little pigs in a children's story.

The basic concept of limits is just the first step into this toolkit for analysis and expression. When we're talking about information, ideas or even feelings that have this special kind of consistency to them we can use the language and logic that comes with Calculus to reveal deeper meaning and understanding of how things change. In this sense, limits aren't points that can't be crossed, they the interesting places that we can visit, observe from every angle and then move on from. The more limits the better!

However, like any translation, this description is only part of the picture. To get the most insight, humor and beauty out of limits you may need to learn some of the native tongue they are written in. If you remember a little bit of algebra, one place to start is the Khan Academy's introduction to differential calculus which as you would expect, beings with the concept of limits. History and linguistic buffs may prefer to get more of a background of how the of language Calculus has roots that date back thousands of years to the ancient Egyptians and Greeks. Or for a laugh, Look Around You takes a look at Maths.

ps. Thanks to Lisa Rogers Neal for suggested edits and notes.