A man is lost at sea in a hot air balloon. He sees a lighthouse approaching in the fog. “Where am I?” he shouts desperately through the wind. “You’re in a balloon!” he hears as he drifts off into the distance.

The response is correct but unhelpful. When people ask about 0.999… they aren’t saying “Hey, could you find the limit of a convergent series under the axioms of the real number system?” (Really? Yes, Really!)

No, there’s a broader, more interesting subtext: What happens when one number gets infinitely close to another?

It’s a rare thing when people wonder about math: let’s use the opportunity! Instead of bluntly offering technical definitions to satisfy some need for rigor, let’s allow ourselves to explore the question.

Here’s my quick summary:

• The meaning of 0.999… depends on our assumptions about how numbers behave.
• A common assumption is that numbers cannot be “infinitely close” together — they’re either the same, or they’re not. With these rules, 0.999… = 1 since we don’t have a way to represent the difference.
• If we allow the idea of “infinitely close numbers”, then yes, 0.999… can be less than 1.

Math can be about questioning assumptions, pushing boundaries, and wondering “What if?”. Let’s dive in.

Do Infinitely Small Numbers Exist?

The meaning of 0.999… is a tricky concept, and depends on what we allow a number to be. Here’s an example: Does “3 – 4″ mean anything to you?

Sure, it’s -1. Duh. But the question is only simple because you’ve embraced the advanced idea of negatives: you’re ok with numbers being less than nothing. In the 1700s, when negatives were brand new, the concept of “3-4″ was eyed with great suspicion, if allowed at all. (Geniuses of the time thought negatives “wrapped around” after you passed infinity).

Infinitely small numbers face a similar predicament today: they’re new, challenge some long-held assumptions, and are considered “non-standard”.

So, Do Infinitesimals Exist?

Well, do negative numbers exist? Negatives exist if you allow them and have consistent rules for their use.

Our current number system assumes the long-standing Archimedean property: if a number is smaller than every other number, it must be zero. More simply, infinitely small numbers don’t exist.

The idea should make sense: numbers should be zero or not-zero, right? Well, it’s “true” in the same way numbers must be there (positive) or not there (zero) — it’s true because we’ve implicitly excluded other possibilities.

But, it’s no matter — let’s see where the Archimedean property takes us.

The Traditional Approach: 0.999… = 1

If we assume infinitely small numbers don’t exist, we can show 0.999… = 1.

First off, we need to figure out what 0.999… means. Most mathematicians see the problem like this:

• 0.999… represents a series of numbers: 0.9, 0.99, 0.999, 0.9999, and so on
• The question: does this series get so close (converge) to a result that we cannot tell it apart?

This is the reasoning behind limits: Does our “thing to examine” get so darn close to another number that we can’t tell them apart, no matter how hard we try?

“Well,” you say, “How do you tell numbers apart?”. Great question. The simplest way to compare is to subtract:

• if a – b = 0, they’re the same
• if a – b is not zero, they’re different

The idea behind limits is to find some point at which “a – b” becomes zero (less than any number); that is, we can’t tell the “number to test” and our “result” as different.

The Error Tolerance

It’s still tough to compare items when they take such different forms (like an infinite series). The next clever idea behind limits: define an error tolerance:

• You give me your tolerance for error / accuracy level (call it “e”)
• I’ll see whether I can get the two things to fall within that tolerance
• If so, they’re equal! If we can’t tell them apart, no matter how hard we try, they must be the same.

Suppose I sell you a raisin granola bar, claiming it’s 100 grams. You take it home, examine the non FDA-approved wrapper, and decide to see if I’m lying. You put the snack on your scale and it shows 100 grams. The scale is accurate to 1 gram. Did I trick you?

You couldn’t know: as far as you can tell, within your accuracy, the granola bar is indeed 100 grams. Our current problem is similar: I’m selling you a “granola bar” weighing 1 gram, but sneaky me, I’m actually giving you one weighing 0.999… grams. Can you tell the difference?

Ok, let’s work this out. Suppose your error tolerance is 0.1 gram. Then if you ask for 1, and I give you 0.99, the difference is 0.01 (one hundredth) and you don’t know you’ve been tricked! 1 and .99 look the same to you.

But that’s child’s-play. Let’s say your scale is accurate to 1e-9 (.000000001, a billionth of a gram). Well then, I’ll sell you a candy bar that is .999999999999 (only one trillionth of a gram off) and you’ll be fooled again! Hah!

In fact, instead of picking a specific tolerance like 0.01, let’s use a general one (e):

• Error tolerance: e
• Difference: Well, suppose e has “n” digits of precision. Let 0.999… expand until we have a difference requiring n+1 digits of precision to detect.
• Therefore, the tolerance can always be less than e! And the difference appears to be zero.

See the trick? Here’s a visual way to represent it:

The straight line is what you’re expecting: 1.0, that perfect granola bar. The curve is the number of digits we expand 0.999… to. The idea is to expand 0.999… until it falls within “e”, your tolerance:

At some point, no matter what you pick for e, 0.999… will get close enough to satisfy us mathematically.

(As an aside, 0.999… isn’t a growing process, it’s a final result on its own. The curve represents the idea that we can approximate 0.999… with better and better accuracy — this is fodder for another post).

With limits, if the difference between two things is smaller than any margin we can dream of, they must be the same.

Assuming Infinitesimals Exist

This first conclusion may not sit well with you — you might feel tricked. And that’s ok! We seem to be ignoring something important when we say that 0.999… equals 1 because we, with our finite precision, cannot tell the difference.

Newer number systems have developed the idea that infinitesimals exist. Specifically:

• Infinitely small numbers can exist: they aren’t zero, but look like zero to us.

This seems to be a confusing idea, but I see it like this: atoms don’t exist to cavemen. Once they’ve cut a rock into grains of sand, they can go no further: that’s the smallest unit they can imagine. Things are either grains, or not there. They can’t imagine the concept of atoms too small for the naked eye.

Compared to other number systems, we’re cavemen. What we call “tiny numbers” are actually gigantic. In fact, there can be another “dimension” of numbers too small for us to detect — numbers that differ only in this tiny dimension look identical to us, but are different under an infinitely powerful microscope.

I interpret 0.999… like this: Can we make a number a bit less than 1 in this new, infinitely small dimension?

Hyperreal Numbers

Hyperreal numbers are one system that uses this “tiny dimension” to examine infinitely small numbers. In this, infinitesimals are usually called “h”, and are considered to be 1/H (where big H is infinity).

So, the idea is this:

• 0.999… < 1 [We're assuming it's allowed to be smaller, and infinitely small numbers exist]
• 0.999… + h = 1 [h is the infinitely small number that makes up the gap]
• 0.999… = 1 – h [Equivalently, we can subtract an infinitely small amount from 1]

So, 0.999… is just a tiny bit less than 1, and the difference is h!

Back to Our Numbers

The problem is, “h” doesn’t exist back in our macroscopic world. Or rather, h looks the same as zero to us — we can’t tell that it’s a tiny atom, not the lack of any matter altogether. Here’s one way to visualize it:

When we switch back to our world, it’s called taking the “standard part” of a number. It essentially means we throw away all the h’s, and convert them to zeroes. So,

• 0.999… = 1 – h [there is an infinitely small difference]
• St(0.999…) = St(1 – h) = St(1) – St(h) = 1 – 0 = 1 [And to us, 0.999... = 1]

The happy compromise is this: in a more accurate dimension, 0.999… and 1 are different. But, when we, with our finite accuracy, try to describe the difference, we cannot: 0.999… and 1 look identical.

Lessons Learned

Let’s hop back to our world. The purpose of “Does 0.999… equal 1?” is not to spit back the answer to a limit question. That’s interpreting the query as “Hey, within our system what does 0.999… represent?”

The question is about exploration. It’s really, “Hey, I’m wondering about numbers infinitely close together (.999… and 1). How do we handle them?”

Here’s my response:

• Our idea of a number has evolved over thousands of years to include new concepts (integers, decimals, rationals, reals, negatives, imaginary numbers…).
• In our current system, we haven’t allowed infinitely small numbers. As a result, 0.999… = 1 because we don’t allow there to be a gap between them (so they must be the same).
• In other number systems (like the hyperreal numbers), 0.999… is less than 1. Here, infinitely small numbers are allowed to exist, and this tiny difference (h) is what separates 0.999… from 1.

There are life lessons here: can we extend our mental model of the world? Negatives gave us the conception that every number can have an opposite. And you know what? It turns out matter can have an opposite too (Dark matter destroys regular mass when they come in contact, just like 3 + (-3) = 0).

Let’s think about infinitesimals, a tiny dimension beyond our accuracy:

• Some theories of physics reference tiny “curled up” dimensions which are embedded into our own. These dimensions may be infinitely small compared to our own — we never notice them. To me, “infinitely small dimensions” are a way to describe something which is there, but undetectable to us.
• The physical sciences use “significant figures” and error margins to specify the inherent inaccuracy of our calculations. We know that reality is different from what we actually measure: infinitesimals help make this distinction explicit.
• Making models: An infinitely small dimension can help us create simple but accurate models to solve problems in our world. The idea of “simple but accurate enough” is at the heart of calculus.

Math isn’t just about solving equations. Expanding our perspective with strange new ideas helps disparate subjects click. Don’t be afraid wonder “What if?”.

Appendix: Where’s the Rigor?

When writing, I like to envision a super-pedant, concerned more with satisfying (and demonstrating) his rigor than educating the reader. This mythical(?) nemesis inspires me to focus on intuition. I really should give Mr. Rigor a name.

But, rigor has a use: it helps ink the pencil-lines we’ve sketched out. I’m not a mathematician, but others have written about the details of interpreting 0.999… and 1 or less than 1:

“So long as the number system has not been specified, the students’ hunch that .999… can fall infinitesimally short of 1, can be justified in a mathematically rigorous fashion.”

My goal is to educate, entertain, and spread interest in math. Can you think of a more salient way to get non-math majors interested in the ideas behind analysis? Limits aren’t going to market themselves.

Math helps us model the world. We can break a complex idea (a wiggly curve) into simpler parts (rectangles):

But, we want an accurate model. The thinner the rectangles, the more accurate the model. The simpler model, built from rectangles, is easier to analyze than dealing with the complex, amorphous blob directly.

The tricky part is making a decent model. Limits and infinitesimals help us create models that are simple to use, yet share the same properties as the original item (length, area, etc.).

The Paradox of Zero

Breaking a curve into rectangles has a problem: How do we get slices so thin we don’t notice them, but large enough to “exist”?

If the slices are too small to notice (zero width), then the model appears identical to the original shape (we don’t see any rectangles!). Now there’s no benefit — the ’simple’ model is just as complex as the original! Additionally, adding up zero-width slices won’t get us anywhere.

If the slices are tiny but measurable, the illusion vanishes. We see that our model is a jagged approximation, and won’t be accurate. What’s a mathematician to do?

We want the best of both: slices so thin we can’t see them (for an accurate model) and slices thick enough to create a simpler, easier-to-analyze model. A dilemma is at hand!

The Solution: Zero is Relative

The notion of zero is biased by our expectations. Is “0 + i”, a purely imaginary number, the same as zero?

Well, “i” sure looks like zero when we’re on the real number line: the “real part” of i, Re(i), is indeed 0. Where else would a purely imaginary number go? (How far East is due North?)

Here’s a different brain bender: did your weight change by zero pounds while reading this sentence? Yes, by any scale you have nearby. But an atomic measurement would show some mass change due to sweat evaporation, exhalation, etc.

You see, there are two answers (so far!) to the “be zero and not zero” paradox:

• Allow another dimension: Numbers measured to be zero in our dimension might actually be small but nonzero in another dimension (infinitesimal approach — a dimension infinitely smaller than the one we deal with)

• Accept imperfection: Numbers measured to be zero are probably nonzero at a greater level of accuracy; saying something is “zero” really means “it’s 0 +/- our measurement error” (limit approach)

These approaches bridge the gap between “zero to us” and “nonzero at a greater level of accuracy”.

Overview of Limits & Infinitesimals

Let’s see how each approach would break a curve into rectangles:

• Limits: “Give me your error margin (I know you have one, you limited, imperfect human!), and I’ll draw you a curve. What’s the smallest unit on your ruler? Inches? Fine, I’ll draw you a staircasey curve at the millimeter level and you’ll never know. Oh, you have a millimeter ruler, do you? I’ll draw the curve in nanometers. Whatever your accuracy, I’m better. You’ll never see the staircase.”

• Infinitesimals: “Forget accuracy: there’s an entire infinitely small dimension where I’ll make the curve. The precision is totally beyond your reach — I’m at the sub-atomic level, and you’re a caveman who can barely walk and chew gum. It’s like getting to the imaginary plane from the real one — you just can’t do it. To you, the rectangular shape I made at the sub-atomic level is the most perfect curve you’ve ever seen.”

Limits stay in our dimension, but with ‘just enough’ accuracy to maintain the illusion of a perfect model. Infinitesimals build the model in another dimension, and it looks perfectly accurate in ours.

The trick to both approaches is that the simpler model was built beyond our level of accuracy. We might know the model is jagged, but we can’t tell the difference — any test we do shows the model and the real item as the same.

That trick doesn’t work, does it?

Oh, but it does. We’re tricked by “imperfect but useful” models all the time:

• Audio files don’t contain all the information of the original signal. But can you tell the difference between a high-quality mp3 and a person talking in the other room?

• Computer printouts are made from individual dots too small to see. Can you tell a handwritten note from a high-quality printout of the same?

• Video shows still images at 24 times per second. This “imperfect” model is fast enough to trick our brain into seeing fluid motion.

On and on it goes. We resist because of our artificial need for precision. But audio and video engineers know they don’t need a perfect reproduction, just quality good enough to trick us into thinking it’s the original.

Calculus lets us make these technically imperfect but “accurate enough” models in math.

Working In Another Dimension

We need to be careful when reasoning with the simplified model. We need to “do our work” at the level of higher accuracy, and bring the final result back to our world. We’ll lose information if we don’t.

Suppose an imaginary number (i) visits the real number line. Everyone thinks he’s zero: after all, Re(i) = 0. But i does a trick! “Square me!” he says, and they do: “i * i = -1″ and the other numbers are astonished.

To the real numbers, it appeared that “0 * 0 = -1″, a giant paradox.

But their confusion arose from their perspective — they only thought it was “0 * 0 = -1″. Yes, Re(i) * Re(i) = 0, but that wasn’t the operation! We want Re(i * i), which is different entirely! We square i in its own dimension, and bring that result back to ours. We need to square i, the imaginary number, and not 0, our idea of what i was.

Beware similar mistakes in calculus: we deal with tiny numbers that look like zero to us, but we can’t do math assuming they are (just like treating i like 0). No, we need to “do the math” in the other dimension and convert the results back.

Limits and infinitesimals have different perspectives on how this conversion is done:

• Limits: “Do the math” at a level of precision just beyond your detection (millimeters), and bring it back to numbers on your scale (inches)

• Infinitesimals: “Do the math” in a different dimension, and bring it back to the “standard” one (just like taking the real part of a complex number; you take the “standard” part of a hyperreal number — more later)

Nobody ever told me: Calculus lets you work at a better level of accuracy, with a simpler model, and bring the results back to our world.

A Real Example: sin(x) / x

Let’s try a conceptual example. Suppose we want to know what happens to sin(x) / x at zero. Now, if we just plug in x = 0 we get a nonsensical result: sin(0) = 0, so we get 0 / 0 which could be anything.

Let’s step back: what does “x = 0″ mean in our world? Well, if we’re allowing the existence of a greater level of accuracy, we know this:

• Things that appear to be zero may be nonzero in a different dimension (just like i might appear to be 0 to us, but isn’t)

We’re going to say that x can be really, really close to zero at this greater level of accuracy, but not “true zero”. Intuitively, you can think of x as 0.0000…00001, where the “…” is enough zeros for you to no longer detect the number.

(In limit terms, we say x = 0 + d (delta, a small change that keeps us within our error margin) and in infinitesimal terms, we say x = 0 + h, where h is a tiny hyperreal number, known as an infinitesimal)

Ok, we have x at “zero to us, but not really”. Now we need a simpler model of sin(x). Why? Well, sine is a crazy repeating curve, and it’s hard to know what’s happening. But it turns out that a straight line is a darn good model of a curve over short distances:

Just like we can break a filled shape into tiny rectangles to make it simpler, we can dissect a curve into a series of line segments. Around 0, sin(x) looks like the line “x”. So, we switch sin(x) with the line “x”. What’s the new ratio?

Well, “x/x” is 1. Remember, we aren’t really dividing by zero because in this super-accurate world: x is tiny but non-zero (0 + d, or 0 + h). When we “take the limit or “take the standard part” it means we do the math (x / x = 1) and then find the closest number in our world (1 goes to 1).

So, 1 is what we get when sin(x) / x approaches zero — that is, we make x as small as possible so it becomes 0 to us. If x became pure, true zero, then the ratio would be undefined (and it is at the infinitesimal level!). But we’re never sure if we’re at perfect zero — something like 0.0000…0001 looks like zero to us.

So, “sin(x)/x” looks like “x/x = 1″ as far as we can tell. Intuitively, the result makes sense once we read about radians).

Visualizing The Process

Today’s goal isn’t to solve limit problems, it’s to understand the process of solving them. To solve this example:

• Realize x=0 is not reachable from our accuracy; a “small but nonzero” x is always available at a greater level of accuracy
• Replace sin(x) by a straight line as a simpler model
• “Do the math” with the simpler model (x / x = 1)
• Bring the result (1) back into our accuracy (stays 1)

Here’s how I see the process:

In later articles, we’ll learn the details of setting up and solving the models.

Caveats: The Trick Doesn’t Always Work

Some functions are really “jumpy” — and they might differ on an infinitesimal-by-infinitesimal level. That means we can’t reliably bring them back to our world. It looks like the function is unstable at microscopic level and doesn’t behave “smoothly”.

The rigorous part of limits is figuring out which functions behave well enough that simple yet accurate models can be made. Fortunately, most of the natural functions in the world (x, x², sin, e^x) behave nicely and can be modeled with calculus.

Limits Or Infinitesimals?

Logically, both approaches solve the problem of “zero and nonzero”. I like infinitesimals because they allow “another dimension” which seems a cleaner separation than “always just outside your reach”. Infinitesimals were the foundation of the intuition of calculus, and appear inside physics and other subjects that use it.

This isn’t an analysis class, but the math robots can be assured that infinitesimals have a rigorous foundation. I use them because they click for me.

Summary

Phew! Some of these ideas are tricky, and I feel like I’m talking from both sides of my mouth: we want to be simpler, yet still perfectly accurate?

This famous dilemma about “being zero sometimes, and non-zero others” is a famous critique of calculus. It was mostly ignored since the results worked out, but in the 1800s limits were introduced to really resolve the dilemma. We learn limits today, but without understanding the nature of the problem they were trying to solve!

Here are the key concepts:

• Zero is relative: something can be zero to us, and non-zero somewhere else
• Infinitesimals (”another dimension”) and limits (”beyond our accuracy”) resolve the dilemma of “zero and nonzero”
• We create simpler models in the more accurate dimension, do the math, and bring the result to our world
• The final result is perfectly accurate for us

My goal isn’t to do math, it’s to understand it. And a huge part of grokking calculus is realizing that simple models created beyond our accuracy can look “just fine” in our dimension. Later on we’ll learn the rules to build and use these models. Happy math.

I wish I had a minute with myself in high school calculus:

“Psst! Integrals let us ‘multiply’ changing numbers. We’re used to “3 × 4 = 12″, but what if one quantity is changing? We can’t multiply changing numbers, so we integrate.

You’ll hear a lot of talk about area — area is just one way to visualize multiplication. The key isn’t the area, it’s the idea of combining quantities into a new result. We can integrate (”multiply”) length and width to get plain old area, sure. But we can integrate speed and time to get distance, or length, width and height to get volume.

When we want to use regular multiplication, but can’t, we bring out the big guns and integrate. Area is just a visualization technique, don’t get too caught up in it. Now go learn calculus!”

That’s my aha moment: integration is a “better multiplication” that works on things that change. Let’s learn to see integrals in this light.

Understanding Multiplication

Our understanding of multiplication changed over time:

• With integers (3 × 4), multiplication is repeated addition
• With real numbers (3.12 x sqrt(2)), multiplication is scaling
• With negative numbers (-2.3 * 4.3), multiplication is flipping and scaling
• With complex numbers (3 * 3i), multiplication is rotating and scaling

We’re evolving towards a general notion of “applying” one number to another, and the properties we apply (repeated counting, scaling, flipping or rotating) can vary. Integration is another step along this path.

Understanding Area

Area is a nuanced topic. For today, let’s see area as a visual representation of of multiplication:

With each count on a different axis, we can “apply them” (3 applied to 4) and get a result (12 square units). The properties of each input (length and length) were transferred to the result (square units).

Simple, right? Well, it gets tricky. Multiplication can result in “negative area” (3 x -4 = -12), even though that can’t exist in the real world.

We understand the graph is a representation of multiplication, and use the analogy as it serves us. If everyone was blind and we had no diagrams, we could still multiply just fine. Area is just an interpretation.

Multiplication Piece By Piece

Now let’s multiply 3 × 4.5:

What’s happening? Well, 4.5 isn’t a count, but we can use a “piece by piece” operation. If 3×4 = 3 + 3 + 3 + 3, then

3 × 4.5 = 3 + 3 + 3 + 3 + 3×0.5 = 3 + 3 + 3 + 3 + 1.5 = 13.5

We’re taking 3 (the value) 4.5 times. That is, we combined 3 with 4 whole segments (3 × 4 = 12) and one partial segment (3 × 0.5 = 1.5).

We’re so used to multiplication that we forget how well it works. We can break a number into units (whole and partial), multiply each piece, and add up the results. Notice how we dealt with a fractional part? This is the beginning of integration.

The Problem With Numbers

Numbers don’t always stay still for us to tally up. Scenarios like “You drive 30mph for 3 hours” are for convenience, not realism.

Formulas like “distance = speed * time” just mask the problem; we still need to plug in numbers and multiply. So how do we find the distance we went when our speed is changing over time?

Describing Change

Our first challenge is describing a changing number. We can’t just say “My speed change from 0 to 30mph”. It’s not specific enough: how fast is it changing? Is it smooth?

Now let’s get specific: every second, I’m going twice that in mph. At 1 second, I’m going 2mph. At 2 seconds, 4mph. 3 seconds is 6mph, and so on:

Now this is a good description, detailed enough to know my speed at any moment. The formal description is “speed is a function of time”, and means we can plug in any time (t) and find our speed at that moment (”2t” mph).

(This doesn’t say why speed and time are related. I could be speeding up because of gravity, or a llama pulling me. We’re just saying that as time changes, our speed does too.)

So, our multiplication of “distance = speed * time” is perhaps better written:

where speed(t) is the speed at any instant. In our case, speed(t) = 2t, so we write:

But this equation still looks weird! “t” still looks like a single instant we need to pick (such as t=3 seconds), which means speed(t) will take on a single value (6mph). That’s no good.

With regular multiplication, we can take one speed and assume it holds for the entire rectangle. But a changing speed requires us to combine speed and time piece-by-piece (second-by-second). After all, each instant could be different.

This is a big perspective shift:

• Regular multiplication (rectangular): Take the amount of time moved in one second, assume it’s the same for all seconds, and “scale it up”.
• Integration (piece-by-piece): See time as a series of instants, each with its own speed. Add up the distance moved on a second-by-second basis.

We see that regular multiplication is a special case of integration, when the quantities aren’t changing.

How large is a “piece”?

How large is a “piece” when going piece by piece? A second? A millisecond? A nanosecond?

Quick answer: Small enough where the value looks the same for the entire duration. We don’t need perfect precision.

The longer answer: Concepts like limits were invented to help us do piecewise multiplication. While useful, they are a solution to a problem and can distract from the insight of “combining things”. It bothers me that limits are introduced in the very start of calculus, before we understand the problem they were created to address (like showing someone a seatbelt before they’ve even seen a car). They’re a useful idea, sure, but Newton seemed to understand calculus pretty well without them.

What about the start and end?

Let’s say we’re looking at an interval from 3 seconds to 4 seconds.

The speed at the start (3×2 = 6mph) is different from the speed at the end (4×2 = 8mph). So what value do we use when doing “speed * time”?

The answer is that we break our pieces into small enough chunks (3.00000 to 3.00001 seconds) until the difference in speed from the start and end of the interval doesn’t matter to us. Again, this is a longer discussion, but “trust me” that there’s a time period which makes the difference meaningless.

On a graph, imagine each interval as a single point on the line. You can draw a straight line up to each speed, and your “area” is a collection of lines which measure the multiplication.

Where is the “piece” and what is its value?

Separating a piece from its value was a struggle for me.

A “piece” is the interval we’re considering (1 second, 1 millisecond, 1 nanosecond). The “position” is where that second, millisecond, or nanosecond interval begins. The value is our speed at that position.

For example, consider the interval 3.0 to 4.0 seconds:

• “Width” of the piece of time is 1.0 seconds
• The position (starting time) is 3.0
• The value (speed(t)) is speed(3.0) = 6.0mph

Again, calculus lets us shrink down the interval until we can’t tell the difference in speed from the beginning and end of the interval. Keep your eye on the bigger picture: we are multiplying a collection of pieces.

Understanding Integral Notation

We have a decent idea of “piecewise multiplication” but can’t really express it. “Distance = speed(t) * t” still looks like a regular equation, where t and speed(t) take on a single value.

In calculus, we write the relationship like this:

• The integral sign (s-shaped curve) means we’re multiplying things piece-by-piece and adding them together.

• dt represents the particular “piece” of time we’re considering. This is called “delta t”, and is not “d times t”.

• t represents the position of dt (if dt is the span from 3.0-4.0, t is 3.0).

• speed(t) represents the value we’re multiplying by (speed(3.0) = 6.0))

I have a few gripes with this notation:

• The way the letters are used is confusing. “dt” looks like “d times t” in contrast with every equation you’ve seen previously.
• We write speed(t) * dt, instead of speed(t_dt) * dt. The latter makes it clear we are examining “t” at our particular piece “dt”, and not some global “t”
• You’ll often see , with an implicit dt. This makes it easy to forget we’re doing a piece-by-piece multiplication of two elements.

It’s too late to change how integrals are written. Just remember the higher-level concept of ‘multiplying’ something that changes.

Reading In Your Head

When I see

I think “Distance equals speed times time” (reading the left-hand side first) or “combine speed and time to get distance” (reading the right-hand side first).

I mentally translate “speed(t)” into speed and “dt” into time and it becomes a multiplication, remembering that speed is allowed to change. Abstracting integration like this helps me focus on what’s happening (”We’re combining speed and time to get distance!”) instead of the details of the operation.

Bonus: Follow-up Ideas

Integrals are a deep idea, just like multiplication. You might have some follow-up questions based on this analogy:

• If integrals multiply changing quantities, is there something to divide them? (Yes — derivatives)
• And do integrals (multiplication) and derivatives (division) cancel? (Yes, with some caveats).
• Can we re-arrange equations from “distance = speed * time” to “speed = distance/time”? (Yes.)
• Can we combine several things that change? (Yes — it’s called multiple integration)
• Does the order we combine several things matter? (No)

Once you see integrals as “better multiplication”, you’re on the lookout for concepts like “better division”, “repeated integration” and so on. Sticking with “area under the curve” makes these topics seem disconnected. (To the math nerds, seeing “area under the curve” and “slope” as inverses asks a lot of a student).

Reading integrals

Integrals have many uses. One is to explain that two things are “multiplied” together to produce a result.

Here’s how to express the area of a circle:

We’d love to take the area of a circle with multiplication. But we can’t — the height changes as we go along. If we “unroll” the circle, we can see the area contributed by each portion of radius is “radius * circumference”. We can write this relationship using the integral above. (See the introduction to calculus for more details).

And here’s the integral expressing the idea “mass = density * volume”:

What’s it saying? Rho () is the density function — telling us how dense a material is at a certain position, r. dv is the bit of volume we’re looking at. So we multiply a little piece of volume (dv) by the density at that position [], and add them all up to get mass.

We’d love to multiply density and volume, but if density changes, we need to integrate. The subscript V means is a shortcut for “volume integral”, which is really a triple integral for length, width, and height! The integral involves four “multiplications”: 3 to find volume, and another to multiply by density.

We might not solve these equations, but we can understand what they’re expressing.

Onward an upward

Today’s goal isn’t to rigorously understand calculus. It’s to expand our mental model, and realize there’s another way to combine things: we can add, subtract, multiply, divide… and integrate.

See integrals as a better way to multiply: calculus will become easier, and you’ll anticipate concepts like multiple integrals and the derivative. Happy math.