It emerges from a more general formula:

Yowza -- we're relating an imaginary exponent to sine and cosine! And somehow plugging in pi gives -1? Could this ever be intuitive?

Not according to 1800s mathematician Benjamin Peirce:

It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.

Argh, this attitude makes my blood boil! Formulas are not magical spells to be memorized: we must, must, must find an insight. Here's mine:

Euler's formula describes two equivalent ways to move in a circle.

That's it? This stunning equation is about spinning around? Yes -- and we can understand it by building on a few analogies:

• Starting at the number 1, see multiplication as a transformation that changes the number (1 * e^(i*pi))
• Regular exponential growth continuously increases 1 by some rate; imaginary exponential growth continuously rotates a number
• Growing for "pi" units of time means going pi radians around a circle
• Therefore, e^(i*pi) means starting at 1 and rotating pi (halfway around a circle) to get to -1

That's the high-level view -- let's dive into the details. By the way, if someone tries to impress you with "e^(i*pi) = -1", ask them about i to the ith power. If they can't think it through, Euler's formula is still a magic spell to them.

Update: While writing, I thought a companion video might help explain the ideas more clearly:

It follows the post -- watch together, or at your leisure.

Understanding cos(x) + i * sin(x)

The equals sign is overloaded. Sometimes we mean "set one thing to another" (like x = 3) and others we mean "these two things describe the same concept" (like sqrt(-1) = i).

Euler's formula is the latter: it gives two formulas which explain how to move in a circle. If we examine circular motion using trig, and travel x radians:

• cos(x) is the x-coordinate (horizontal distance)
• sin(x) is the y-coordinate (vertical distance)

The statement

is a clever way to smush the x and y coordinates into a single number. The analogy "complex numbers are 2-dimensional" helps us interpret a single complex number as a position on a circle.

When we set x to pi, we're traveling "pi" units along the outside of the unit circle. Because the total circumference is 2*pi, plain old pi is halfway around, putting us at -1.

Neato: The right side of Euler's formula (cos(x) + i*sin(x)) describes circular motion with imaginary numbers. Now let's figure out how the e side of the equation accomplishes it.

What is Imaginary Growth?

Combining x- and y- coordinates into a complex number is tricky, but manageable. But what does an imaginary exponent mean?

Let's step back a bit. When I see "3^4" I think of it like this:

• 3 is the end result of growing instantly (using e) at a rate of ln(3). 3 = e^ln(3)
• 3^4 is the same as growing to 3, but then growing for 4x as long. So 3^4 = e^(ln(3) * 4) = 81

Instead of seeing numbers on their own, you can think of them as something e had to "grow to". Real numbers, like 3, give an interest rate of ln(3) [1.1] and that's what e "collects" as its going along, growing continuosly.

Regular growth is simple -- it keeps "pushing" a number in the same (real) direction it was going. 3 × 3 pushes in the original direction, making it 3 times larger (9).

Imaginary growth is different -- the "interest" we earn is in a different direction! It's like a jet engine that's was strapped on sideways -- instead of going forward, we start pushing at 90 degrees.

The neat thing about a constant orthogonal (perpendicular) push is that it doesn't speed you up or slow you down -- it rotates you! Taking any number and multiplying by i will not change its magnitude, just the direction it points.

Intuitively, here's how I see continuous imaginary growth rate: "When I grow, don't push me forward or back in the direction I'm already going. Rotate me instead."

But Shouldn't We Spin Faster and Faster?

I wondered that too. Regular growth compounds in our original direction, so we go 1, 2, 4, 8, 16, multiplying 2x each time and staying in the real numbers. We can consider this e(ln(2)*x): grow instantly at a rate of ln(2) for "x" seconds.

And hey -- if our growth rate was twice that (2*ln(2)), it would look the same as growing for twice as long (2x vs x). The magic of e lets us swap rate and time; 2 seconds at ln(2) is the same growth as 1 second at 2*ln(2).

Now, imagine we have some imaginary growth rate (R*i) which rotates us: e^R*i becomes imaginary and grows to "i". Well, if we double that, we get i^2 (-1) and as we keep going we just spin around the circle.

Now imagine we double that rate (2R*i). Would that spin us off the circle? Nope! Having a rate of 2R*i means we just spin twice as fast, or alternatively, spin at a rate of R for twice as long.

Once we realize that some exponential growth rate can take us from 1 to i, increasing that rate just spins us faster. We'll never escape the circle.

However, if our growth rate is complex (a+bi vs Ri) then the real part (a) will grow us like normal, while the imaginary part (bi) rotates us. But let's not get fancy: Euler's formula, e^(i*x), is about the purely imaginary growth that keeps us on the circle (more later).

A Quick Sanity Check

While writing, I had to clarify a few questions for myself:

Why e^x -- aren't we rotating 1?

e represents the process of starting at 1 and growing continuously at 100% interest for 1 unit of time.

When we write e we're capturing that entire process in a single number -- e represents all the whole rigamarole of continuous growth. So really, e^x is saying "start at 1 and grow continuously at 100% for x seconds", and starts from 1 like we want.

But what does i as an exponent do?

For normal exponents like 3^4 we ask:

• What is the implicit growth rate? It's growing from 1 to 3.
• How do we change that growth rate? We make it 4x larger (^4).

Converting to "e" format, we say our instantaneous growth rate is ln(3) and make it 4x larger: ln(3) * 4. The top exponent (4) just scaled our growth.

When the top exponent is i (like 3^i), we get the same growth (ln(3)) but multiply it by i. So instead of growing at ln(3), we're growing at ln(3) * i.

The top exponent modifies the growth rate inherent in the bottom part.

The Nitty Gritty Details

Let's take a closer look. Remember this definition of e:

That 1/n represents the interest we earned in each microscopic period. We assumed the interest was real -- but what if it were imaginary?

Now, our newly formed interest adds to us in the 90-degree direction. Surprisingly, this does not change our length -- this is a tricky concept, because it appears to make a triangle where the hypotenuse must be larger. We're dealing with a limit, and the extra distance is within the error margin we specify. This is something I want to tackle another day, but take my word: continuous perpendicular growth will rotate you. This is the heart of sine and cosine, where your change is perpendicular to your current position, and you move in a circle.

We apply i units of growth in infinitely small increments, each pushing us at a 90-degree angle. There is no "faster and faster" rotation - instead, we crawl along the perimeter a distance of |i| = 1 (magnitude of i).

And hey -- the distance crawled around a circle is an angle in radians! We've found another way to describe circular motion!

To get circular motion: Change continuously by rotating at 90-degree angle (aka imaginary growth rate).

So, Euler's formula is saying "exponential, imaginary growth traces out a circle". And this path is the same as moving in a circle using sine and cosine in the imaginary plane.

In this case, the word "exponential" is confusing because we travel around the circle at a constant rate. In most discussions, exponential growth is assumed to have a cumulative, compounding effect.

Some Examples

You don't really believe me, do you? Here's a few examples, and how to think about them intuitively.

Example: e^i

Where's the x? Ah, it's just 1. Intuitively, without breaking out a calculator, we know that this means "travel 1 radian along the unit circle". In my head, I see "e" trying to grow 1 at 100% all in the same direction, but i keeps moving the ball and forces "1" to grow along the edge of a circle:

Not the prettiest number, but there it is. Remember to put your calculator in radian mode when punching this in.

Example: 3^i

This is tricky -- it's not in our standard format. But remember, -- the real question is "How do we transform 1"?

We want an initial growth of 3x at the end of the period, or an instantaneous rate of ln(3). But, the i comes along and changes that rate of ln(3) to "i * ln(3)":

We thought we were going to transform at a regular rate of ln(3) (a little faster than 100% continuous growth since e is about 2.718). But oh no, i spun us around: now we're transforming at an imaginary rate which means we're just rotating about. If i was a regular number like 4, it would have made us grow 4x faster. Now we're growing at a speed of ln(3), but sideways.

We should expect a complex number on the unit circle -- there's nothing in the growth rate to increase our size. Solving the equation:

So, rather than ending up "1" unit around the circle (like e^i) we end up ln(3) units around.

Example: i^i

A few months ago, this would have had me tears. Not today! Let's break down the transformations:

We start with 1 and want to change it. Like solving 3^i, what's the instantaneous growth rate represented by i as a base?

Hrm. Normally we'd do ln(x) to get the growth rate needed to reach x it the end of 1 unit of time. But for an imaginary rate? We need to noodle this over.

In order to start with 1 and grow to i we need to start rotating at the outset. How fast? Well, we need to get 90 degrees (pi/2 radians) in 1 unit of time. So our rate is "i * pi/2". Remember our rate must be imaginary since we're rotating, not growing! Plain old "pi/2" is about 1.57 and results in regular growth.

This should make sense: to turn 1.0 to i at the end of 1 unit, we should rotate pi/2 radians (90 degrees) in that amount of time. So, to get "i" we can use e^(i * pi/2).

Phew. That describes i as the base. How about the exponent?

Well, the other i tells us to change our rate -- yes, that rate we spent so long figuring out! So rather than rotating at a speed of i * pi/2, which is what a base of i means, we transform the rate to:

The i's cancel and make the growth rate real again! We rotated our rate and pushed ourselves into the negative numbers. And a negative growth rate means we're shrinking -- we should expect i^i to make things smaller. And it does:

Tada! (Search "i^i" on Google to use its calculator)

Take a breather: You can intuitively figure out how imaginary bases and imaginary exponents should behave. Whoa.

And as a bonus, you figured out ln(i) -- to make e^x become i, make e rotate pi/2 radians.

Example: (i^i)^i

A double imaginary exponent? If you insist. First off, we know what our growth rate will be inside the parenthesis:

We get a negative (shrinking) growth rate of -pi/2. And now we modify that rate again by i:

And now we have a negative rotation! We're going around the circle a rate of -pi/2 per unit time. How long do we go for? Well, there's an implicit "1" unit of time at the very top of this exponent chain; the implied default is to go for 1 time unit (just like e = e^1). 1 time unit gives us a rotation of -pi/2 radians (-90 degrees) or -i!

And, just for kicks, if we squared that crazy result:

It's "just" twice the rotation: 2 is a regular number so doubles our rotation rate to a full -180 degrees in a unit of time. Or, you can look at it as applying -90 degree rotation twice in a row.

At first blush, these are really strange exponents. But with our analogies we can take them in stride.

Complex Growth

We can have real and imaginary growth at the same time: the real portion scales us up, and the imaginary part rotates us around:

A complex growth rate like (a + bi) is a mix of real and imaginary growth. The real part a, means "grow at 100% for a seconds" and the imaginary part b means "rotate for b seconds". Remember, rotations don't get the benefit of compounding since you keep 'pushing' in a different direction -- rotation adds up linearly.

With this in mind, we can represent any point on any sized circle using (a+bi)! The radius is e^a and the angle is determined by e^(b*i). It's like putting the number in the expand-o-tron for two cycles: once to grow it to the right size (a seconds), another time to rotate it to the right angle (b seconds). Or, you could rotate it first and the grow!

Let's say we want to know the growth amount to get to 6 + 8i. This is really asking for the natural log of an imaginary number: how do we grow e to get (6 + 8i)?

• Radius: How big of a circle do we need? Well, the magnitude is sqrt(6^6 + 8^8) = sqrt(100) = 10. Which means we need to grow for ln(10) = 2.3 seconds to reach that amount.
• Amount to rotate: What's the angle of that point? We can use arctan to figure it out: atan(8/6) = 53 degrees = .93 radian.
• Combine the result: ln(6+8i) = 2.3 + .93i

That is, we can reach the random point (6 + 8i) if we use e^(2.3 + .93i).

Why Is This Useful?

Euler's formula gives us another way to describe motion in a circle. But we could already do that with sine and cosine -- what's so special?

It's all about perspective. Sine and cosine describe motion in terms of a grid, plotting out horizontal and vertical coordinates.

Euler's formula uses polar coordinates -- what's your angle and distance? Again, it's two ways to describe motion:

• Grid system: Go 3 units east and 4 units north
• Polar coordinates: Go 5 units at an angle of 71.56 degrees

Depending on the problem, polar or rectangular coordinates are more useful. Euler's formula lets us convert between the two to use the best tool for the job. Also, because e^(ix) can be converted to sine and cosine, we can rewrite formulas in trig as variations on e, which comes in very handy (no need to memorize sin(a+b), you can derive it -- more another day). And it's beautiful that every number, real or complex, is a variation of e.

But utility, schmutility: the most important result is the realization that baffling equations can become intuitive with the right analogies. Don't let beautiful equations like Euler's formula remain a magic spell -- build on the analogies you know to see the insights inside the equation.

Happy math.

Appendix

The screencast was fun, and feedback is definitely welcome. I think it helps the ideas pop, and walking through the article helped me find gaps in my intuition.

References:

• Brian Slesinsky has a neat presentation on Euler's formula
• Visual Complex Analysis has a great discussion on Euler's formula -- see p. 10 in the Google Book Preview

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.