**Are we talking about inputs (cause of the change) or outputs (the actual change that happened?)**

- Logarithms reveal the inputs that caused the growth
- Exponents find the final result of growth

**Are we talking about the grower’s perspective, or an observer’s?**... Read article

**Are we talking about inputs (cause of the change) or outputs (the actual change that happened?)**

- Logarithms reveal the inputs that caused the growth
- Exponents find the final result of growth

**Are we talking about the grower’s perspective, or an observer’s?**

- e and the natural log are from the grower’s instant-by-instant perspective
- Base 10, Base 2, etc. are measurements convenient for a human observer

In my head, I put the options in a table:

and I have thoughts like *“I need the cause, from the grower’s perspective… that’s the natural log.”*. (Natural log is abbreviated with lowercase LN, from the high-falutin’ *logarithmus naturalis*.)

I was frustrated with classes that described the inner part of the table, the raw functions, without the captions that explained when to use them!

That won’t fly, let’s get direct practice thinking with logs and exponents.

Here’s a typical example of growth:

- From 2000 to 2010, the US GDP changed from 9.9 trillion to 14.4 trillion

Ok, sure, those numbers show change happened. But we probably want insight into the cause: What average annual growth rate would account for this change?

Immediately, my brain thinks “logarithms” because we’re working backwards from the growth to the rate that caused it. I start with a thought like this:

A good start, but let’s sharpen it up.

First, which logarithm should we use?

By default, I pick the natural logarithm. Most events end up being in terms of the grower (not observer), and I like “riding along” with the growing element to visualize what’s happening. (Radians are similar: they measure angles in terms of the mover.)

Next question: what change do we apply the logarithm to?

We’re really just interested in the ratio between start and finish: 9.9 trillion to 14.4 trillion in 10 years. This is the same growth *rate* as going from $9.90 to $14.40 in the same period.

We can sharpen our thought:

Ok, the cause was a rate of .374 or 37.4%. Are we done?

Not yet. Logarithms don’t know about how long a change took (we didn’t plug in 10 years, right?). They give us a rate as if all the change happened in a single time period.

The change could indeed be a single year of 37.4% continuous growth, or 2 years of 18.7% growth, or some other combination.

From the scenario, we know the change took 10 years, so the rate must have been:

From the viewpoint of instant, continuous growth, the US economy grew by 3.74% per year.

Are we done now? Not quite!

This continuous rate is from the grower’s perspective, as if we’re “riding along” with the economy as it changes. A banker probably cares about the human-friendly, year-over-year difference. We can figure this out by letting the continuous growth run for a year:

The year-over-year gain is 3.8%, slightly higher than the 3.74% instantaneous rate due to compounding. Here’s another way to put it:

- From an instant-by-instant basis, a given part of the economy is growing by 3.74%, modeled by e
^{.0374 · years} - On a year-by-year basis, with compounding effects worked out, the economy grows by 3.81%, modeled by 1.0381
^{years}

In finance, we may want the year-over-year change which can be compared nicely with other trends. In science and engineering, we prefer modeling behavior on an instantaneous basis.

I detest contrived examples like “Assume bacteria doubles every 24 hours, find its growth formula.”. Do bacteria colonies replicate on clean human intervals, and do we wait around for an exact doubling?

A better scenario: “Hey, I found some bacteria, waited an hour, and the lump grew from 2.3 grams to 2.32 grams. I’m going to lunch now. Figure out how much we’ll have when I’m back in 3 hours.”

Let’s model this. We’ll need a logarithm to find the growth rate, and then an exponent to project that growth forward. Like before, let’s keep everything in terms of the natural log to start.

The growth factor is:

That’s the rate for one hour, and the general model to project forward will be

If we start with 2.32 and grow for 3 hours we’ll have:

Just for fun, how long until the bacteria doubles? Imagine waiting for 1 to turn to 2:

We can mechanically take the natural log of both sides to “undo the exponent”, but let’s think intuitively.

If 2 is the final result, then ln(2) is the growth input that got us there (some rate × time). We know the rate was .0086, so the time to get to 2 would be:

The colony will double after ~80 hours. (Glad you didn’t stick around?)

Figuring out whether you want the input (cause of growth) or output (result of growth) is pretty straightforward. But how do you visualize the grower’s perspective?

Imagine we have little workers who are building the final growth pattern (see the article on exponents):

If our growth rate is 100%, we’re telling our initial worker (Mr. Blue) to work steadily and create a 100% copy of himself by the end of the year. If we follow him day-by-day, we see he does finish a 100% copy of himself (Mr. Green) at the end of the year.

But… that worker he was building (Mr. Green) starts working as well. If Mr. Green first appears at the 6-month mark, he has a half-year to work (same annual rate as Mr. Blue) and he builds Mr. Red. Of course, Mr. Red ends up being half done, since Mr. Green only has 6 months.

What if Mr. Green showed up after 4 months? A month? A day? A second? If workers begin growing immediately, we get the instant-by-instant curve defined by e^{x}:

The natural log gives a growth rate in terms of an individual worker’s perspective. We plug that rate into e^{x} to find the final result, with all compounding included.

Switching to another type of logarithm (base 10, base 2, etc.) means we’re looking for some pattern in the overall growth, not what the individual worker is doing.

Each logarithm asks a question when seeing a change:

- Log base e: What was the instantaneous rate followed by each worker?
- Log base 2: How many doublings were required?
- Log base 10: How many 10x-ings were required?

Here’s a scenario to analyze:

- Over 30 years, the transistor counts on typical chips went from 1000 to 1 billion

How would you analyze this?

- Microchips aren’t a single entity that grow smoothly over time. They’re separate editions, from competing companies, and indicate a general tech trend.
- Since we’re not “riding along” with an expanding microchip, let’s use a scale made for human convenience. Doubling is easier to think about than 10x-ing.

With these assumptions we get:

The “cause of growth” was 20 doublings, which we know occurred over 30 years. This averages 2/3 doublings per year, or 1.5 years per doubling — a nice rule of thumb.

From the grower’s perspective, we’d compute ln(text(1 billion)/1000) / text(30 years) = 46% continuous growth (a bit harder to relate to in this scenario).

We can summarize our analysis in a table:

Learning is about finding the hidden captions behind a concept. When is it used? What point view does it bring to the problem?

My current interpretation is that exponents ask about cause vs. effect and grower vs. observer. But we’re never done; part of the fun is seeing how we can recaption old concepts.

Happy math.

Here’s how to think about switching bases. Assuming a 100% continuous growth rate,

- ln(x) is the time to grow to x
- ln(2) is the time to grow to 2

Since we have the time to double, we can see how many would “fit” in the total time to grow to x:

For example, how many doublings occur from 1 to 64?

Well, ln(64) = 4.158. And ln(2) = .693. The number of doublings that fit is:

In the real world, calculators may lose precision, so use a direct log base 2 function if possible. And of course, we can have a fractional number: Getting from 1 to the square root of 2 is “half” a doubling, or log_{2}(1.414) = 0.5.

Changing to log base 10 means we’re counting the number of 10x-ings that fit:

Neat, right? Read Using Logarithms in the Real World for more examples.

]]>For example, look at two businesses:

- Annie’s Art Gallery sells a single, $1000 piece every day
- Frank’s Fish Emporium sells 250 trout at $4/each every day

By the numbers, they’re identical $1000/day operations, right?... Read article

]]>For example, look at two businesses:

- Annie’s Art Gallery sells a single, $1000 piece every day
- Frank’s Fish Emporium sells 250 trout at $4/each every day

By the numbers, they’re identical $1000/day operations, right? Hah.

Here’s how each business actually behaves:

Transactions are the workhorse that drive income, but they’re lost in the dollars/day description. When studying an idea, separate the results into Oomph and Often:

With Oomph and Often, I visualize two distinct levers to increase. A ratio like dollars/day makes me stumble through thoughts like: *“For better results, I need 1/day to improve… which means the day gets shorter… How’s that possible? Oh, that must be the portion of the day used for each transaction…”*.

Why make it difficult? Rewrite the ratio to include the root case: What’s the Oomph, and how Often does it happen?

In physics, we define everyday concepts like “power” with a formal ratio:

Ok. Power can be explained by a ratio, but we’re already in inverted-thinking mode. Just another hassle when exploring an already-tricky concept.

How about this:

Easier, I think. What could Oomph and Often mean?

Well, Oomph is probably the work we do (such as moving a weight) and Often is how frequently we do it (how many reps did you put in?).

In the same minute, suppose Frank lifted 100lbs ten times, while Annie lifted 1000lbs once. From the equation, they have the same power (though to be honest, I’m more frightened by Annie.)

An engine mechanic might internalize power like this:

What does that mean?

Torque is the Oomph, or how much weight (and how far) can be moved by a turn of the engine (i.e., moving 500lbs by 1 foot)

RPM (revolutions per minute) is how frequently the engine turns

A motorcycle engine is designed for reps, i.e. spinning the wheels quickly. It doesn’t need much torque — just enough to pull itself and a few passengers — but it needs to send that to the wheels again and again.

A bulldozer is designed for “Oomph”, such as knocking over a wall. We don’t need to tap into that work very frequently, as one destroyed wall per minute is great, thanks.

I’m not a physicist or car guy, but I can at least conceptualize the tradeoffs with the Oomph/Often metaphor.

Gears can change the tradeoff between Oomph and Often in a given engine. If you’re going uphill, fighting gravity, what do you want more of? If you’re cruising on a highway? Trying to start from a standstill? Driving over slippery snow? Lost the brakes and need to slow down the car?

Oomph/Often gets me thinking intuitively, Work/Time does not.

Electric power has the same ratio as mechanical power:

Yikes. It’s not clear what this means. How about:

It’s hard to have ideas out of the blue, but we might imagine *something* (a mini-engine?) is moving the Oomph around inside the wire. If we call it a “charge” then we have:

And we can give those subparts formal names:

Voltage (Oomph): How much work each charge contributes

Current (Often): How quickly charges are moving through the wire

Now we get the familiar:

Boomshakala! I don’t have a good intuition for electricity, at least my goal is clear: find analogies where voltage means Oomph, and current means Often.

And still, we can take a crack at intuitive thinking: when you get zapped by a doorknob in winter, was that Oomph or Often? What attribute should batteries maximize? What’s better for moving energy through stubborn power lines? (*Vive la rĂ©sistance!*)

The ratios think every type of power reduces to a generic Work/Time calculation. The Oomph/Often metaphor gets us thinking about Torque/RPM in one scenario and Voltage/Current in another.

The Oomph/Often viewpoint lets us think about the true cause of the ratio. Instead of dollars and days, we wonder how the actual transactions affect the outcome:

Can we increase the size of each transaction?

Can we increase the number each day?

In formal terms, we’ve introduced a new parameter to explain the interaction. To change a ratio from a/b to one parameterized by x, we can do:

We change our viewpoint to see x as the key component. In math, we often switch viewpoints to simplify problems:

Instead of asking what happens to the observer, can we change parameters and ask what the mover sees? (Degrees vs. radians.)

Can we see a giant function as being parameterized by smaller ones? (See the chain rule.)

Can we express probabilities as odds, instead of percentages? (It makes Bayes Theorem easier.)

Adjusting parameters is a way to morph an idea that doesn’t click into one that does. Since I don’t naturally think with inverted units, I’ve made it easier on myself: deal with two multiplications, instead of a division.

Happy math.

]]>TOA explains the tangent about as well as x^{2} + y^{2} = r^{2} describes a circle. Sure, if you’re a math robot, an equation is enough. The rest of us, with organic brains half-dedicated to vision processing, seem to enjoy imagery.... Read article

TOA explains the tangent about as well as x^{2} + y^{2} = r^{2} describes a circle. Sure, if you’re a math robot, an equation is enough. The rest of us, with organic brains half-dedicated to vision processing, seem to enjoy imagery. And “TOA” evokes the stunning beauty of an abstract ratio.

I think you deserve better, and here’s what made trig click for me.

- Visualize a dome, a wall, and a ceiling
- Trig functions are
*percentages*to the three shapes

Imagine Bob The Alien visits Earth to study our species.

Without new words, humans are hard to describe: “There’s a sphere at the top, which gets scratched occasionally” or “Two elongated cylinders appear to provide locomotion”.

After creating specific terms for anatomy, Bob might jot down typical body proportions:

- The armspan (fingertip to fingertip) is approximately the height
- A head is 5 eye-widths wide
- Adults are 8 head-heights tall

How is this helpful?

Well, when Bob finds a jacket, he can pick it up, stretch out the arms, and estimate the owner’s height. And head size. And eye width. One fact is linked to a variety of conclusions.

Even better, human biology explains human thinking. Tables have legs, organizations have heads, crime bosses have muscle. Our biology offers ready-made analogies that appear in man-made creations.

Now the plot twist: *you* are Bob the alien, studying creatures in math-land!

Generic words like “triangle” aren’t overly useful. But labeling sine, cosine, and hypotenuse helps us notice deeper connections. And scholars might study haversine, exsecant and gamsin, like biologists who find a link between your fibia and clavicle.

And because triangles show up in circles…

…and circles appear in cycles, our triangle terminology helps describe repeating patterns!

Trig is the anatomy book for “math-made” objects. If we can find a metaphorical triangle, we’ll get an armada of conclusions for free.

Instead of staring at triangles by themselves, like a caveman frozen in ice, imagine them in a scenario, hunting that mammoth.

Pretend you’re in the middle of your dome, about to hang up a movie screen. You point to some angle “x”, and that’s where the screen will hang.

The angle you point at determines:

- sine(x) = sin(x) = height of the screen, hanging like a sign
- cosine(x) = cos(x) = distance to the screen on the ground
- the hypotenuse, the distance to the top of the screen, is always the same

Want the biggest screen possible? Point straight up. It’s at the center, on top of your head, but it’s *big* dagnabbit.

Want the screen the furthest away? Sure. Point straight across, 0 degrees. The screen has “0 height” at this position, and it’s far away, like you asked.

The height and distance move in opposite directions: bring the screen closer, and it gets taller.

Nobody ever told me in my years of schooling: **sine and cosine are percentages**. They vary from +100% to 0 to -100%, or max positive to nothing to max negative.

Let’s say I paid $14 in tax. You have no idea if that’s expensive. But if I say I paid 95% in tax, you know I’m getting ripped off.

An absolute height isn’t helpful, but if your sine value is .95, I know you’re almost at the top of your dome. Pretty soon you’ll hit the max, then start coming down again.

How do we compute the percentage? Simple: divide the current value by the maximum possible (the radius of the dome, aka the hypotenuse).

*That’s* why we’re told “Sine = Opposite / Hypotenuse”. It’s to get a percentage! A better wording is “Sine is your height, as a percentage of the max”. (Sine becomes negative if your angle points “underground”. Cosine becomes negative when your angle points backwards.)

Let’s simplify the calculation by assuming we’re on the unit circle (radius 1). Now we can skip the division and just say sine = height.

Every circle is really the unit circle, scaled up or down to a different size. So work out the connections on the unit circle and apply the results to your particular scenario.

Try it out: plug in an angle and see what percent of the height and width it reaches:

The growth pattern of sine isn’t an even line. The first 45 degrees cover 70% of the height, and the final 10 degrees (from 80 to 90) only cover 2%.

This should make sense: at 0 degrees, you’re moving nearly vertical, but as you get to the top of the dome, your height changes level off.

One day your neighbor puts up a wall *right next* to your dome. Ack, your view! Your resale value!

But can we make the best of a bad situation?

Sure. What if we hang our movie screen on the wall? You point at an angle (x) and figure out:

- tangent(x) = tan(x) = height of screen on the wall
- distance to screen: 1 (the screen is always the same distance along the ground, right?)
- secant(x) = sec(x) = the “ladder distance” to the screen

We have some fancy new vocab terms. Imagine seeing the Vitruvian “TAN GENTleman” projected on the wall. You climb the ladder, making sure you can “SEE, CAN’T you?”. (Yeah, he’s naked… won’t forget the analogy now, will you?)

Let’s notice a few things about tangent, the height of the screen.

It starts at 0, and goes infinitely high. You can keep pointing higher and higher on the wall, to get an infinitely large screen! (That’ll cost ya.)

Tangent is just a bigger version of sine! It’s never smaller, and while sine “tops off” as the dome curves in, tangent keeps growing.

How about secant, the ladder distance?

- Secant starts at 1 (ladder on the floor to the wall) and grows from there
- Secant is always longer than tangent. The leaning ladder used to put up the screen must be longer than the screen itself, right? (At enormous sizes, when the ladder is nearly vertical, they’re close. But secant is always a smidge longer.)

Remember, the values are *percentages*. If you’re pointing at a 50-degree angle, tan(50) = 1.19. Your screen is 19% larger than the distance to the wall (the radius of the dome).

(Plug in x=0 and check your intuition that tan(0) = 0, and sec(0) = 1.)

Amazingly enough, your neighbor now decides to build a ceiling on top of your dome, far into the horizon. (*What’s with this guy? Oh, the naked-man-on-my-wall incident…*)

Well, time to build a ramp to the ceiling, and have a little chit chat. You pick an angle to build and work out:

- cotangent(x) = cot(x) = how far the ceiling extends before we connect
- cosecant(x) = csc(x) = how long we walk on the ramp
- the vertical distance traversed is always 1

Tangent/secant describe the wall, and COtangent and COsecant describe the ceiling.

Our intuitive facts are similar:

- If you pick an angle of 0, your ramp is flat (infinite) and never reachers the ceiling. Bummer.
- The shortest “ramp” is when you point 90-degrees straight up. The cotangent is 0 (we didn’t move along the ceiling) and the cosecant is 1 (the “ramp length” is at the minimum).

A short time ago I had *zero* “intuitive conclusions” about the cosecant. But with the dome/wall/ceiling metaphor, here’s what we see:

Whoa, it’s the same triangle, just scaled to reach the wall and ceiling. We have vertical parts (sine, tangent), horizontal parts (cosine, cotangent), and “hypotenuses” (secant, cosecant). (Note: the labels show where each item “goes up to”. Cosecant is the full distance from you to the ceiling.)

Now the magic. The triangles have similar facts:

From the Pythagorean Theorem (a^{2} + b^{2} = c^{2}) we see how the sides of each triangle are linked.

And from similarity, ratios like “height to width” must be the same for these triangles. (Intuition: step away from a big triangle. Now it looks smaller in your field of view, but the internal ratios couldn’t have changed.)

This is how we find out “sine/cosine = tangent/1″.

I’d always tried to memorize these facts, when they just jump out at us when visualized. SOH-CAH-TOA is a nice shortcut, but get a real understanding first!

*Psst*… don’t over-focus on a single diagram, thinking tangent is always smaller than 1. If we increase the angle, we reach the ceiling before the wall:

The Pythagorean/similarity connections are always true, but the relative sizes can vary.

(But, you might notice that sine and cosine are always smallest, or tied, since they’re trapped inside the dome. Nice!)

For most of us, I’d say this is enough:

- Trig explains the anatomy of “math-made” objects, such as circles and repeating cycles
- The dome/wall/ceiling analogy shows the connections between the trig functions
- Trig functions return percentages, that we apply to our specific scenario

You don’t need to memorize 1^{2} + cot^{2} = csc^{2}, except for silly tests that mistake trivia for understanding. In that case, take a minute to draw the dome/wall/ceiling diagram, fill in the labels (a tan gentleman you can see, can’t you?), and create a cheatsheet for yourself.

In a follow-up, we’ll learn about graphing, complements, and using Euler’s Formula to find even more connections.

You may see tangent defined as the length of the tangent line from the circle to the x-axis (geometry buffs can work this out).

As expected, at the top of the circle (x=90) the tangent line can never reach the x-axis and is infinitely long.

I like this intuition because it helps us remember the name “tangent”, and here’s a nice interactive trig guide to explore:

Still, it’s critical to put the tangent vertical and recognize it’s just sine projected on the back wall (along with the other triangle connections).

Trig functions take an angle and return a percentage. sin(30) = .5 means a 30-degree angle is 50% of the max height.

The inverse trig functions let us work backwards, and are written sin^{-1} or arcsin (“arcsine”), and often written `asin`

in various programming languages.

If our height is 25% of the dome, what’s our angle?

Now what about something exotic, like inverse secant? Often times it’s not available as a calculator function (even the one I built, sigh).

Looking at our trig cheatsheet, we find an easy ratio where we can compare secant to 1. For example, secant to 1 (hypotenuse to horizontal) is the same as 1 to cosine:

Suppose our secant is 3.5, i.e. 350% of the radius of the unit circle. What’s the angle to the wall?

**Example: Find the sine of angle x.**

Ack, what a boring question. Instead of “find the sine” think, “What’s the height as a percentage of the max (the hypotenuse)?”.

First, notice the triangle is “backwards”. That’s ok. It still has a height, in green.

What’s the max height? By the Pythagorean theorem, we know

Ok! The sine is the height as a percentage of the max, which is 3/5 or .60.

**Follow-up: Find the angle.**

Of course. We have a few ways. Now that we know sine = .60, we can just do:

Here’s another approach. Instead of using sine, notice the triangle is “up against the wall”, so tangent is an option. The height is 3, the distance to the wall is 4, so the tangent height is 3/4 or 75%. We can use arctangent to turn the percentage back into an angle:

**Example: Can you make it to shore?**

You’re on a boat with enough fuel to sail 2 miles. You’re currently .25 miles from shore. What’s the largest angle you could use and still reach land? Also, the only reference available is *Hubert’s Compendium of Arccosines, 3rd Ed*. (Truly, a hellish voyage.)

Ok. Here, we can visualize the beach as the “wall” and the “ladder distance” to the wall is the secant.

First, we need to normalize everything in terms of percentages. We have 2 / .25 = 8 “hypotenuse units” worth of fuel. So, the largest secant we could allow is 8 times the distance to the wall.

We’d *like* to ask “What angle has a secant of 8?”. But we can’t, since we only have a book of arccosines.

We use our cheatsheet diagram to relate secant to cosine: Ah, I see that “sec/1 = 1/cos”, so

A secant of 8 implies a cosine of 1/8. The angle with a cosine of 1/8 is arccos(1/8) = 82.8 degrees, the largest we can afford.

Not too bad, right? Before the dome/wall/ceiling analogy, I’d be drowning in a mess of computations. Visualizing the scenario makes it simple, even fun, to see which trig buddy can help us out.

In your problem, think: am I interested in the dome (sin/cos), the wall (tan/sec), or the ceiling (cot/csc)?

Happy math.

**Update:** The owner of Grey Matters put together interactive diagrams for the analogies (drag the slider on the left to change the angle):

- Sine/Cosine: The Dome
- Tangent/Secant: The Wall
- Cotangent/Cosecant: The Ceiling
- Combined visualization

Thanks!

]]>My goals were to be friendly, readable, and easy-to-navigate. Here’s a quick before-and-after:

Neil did a fantastic job here — I’d been looking for a way to convey a welcoming, conversational tone.... Read article

]]>My goals were to be friendly, readable, and easy-to-navigate. Here’s a quick before-and-after:

Neil did a fantastic job here — I’d been looking for a way to convey a welcoming, conversational tone.

A site about explanations should describe what it does simply, right?

The fonts are bumped up, there’s more breathing room, and pages are optimized for iPads/iPhones. Instead of a text-dense cram session, I want an unhurried walkthrough of insights.

My favorite feature is a site summary that reduces insights to a few words. Previously, I had trouble navigating the various articles, and I bet you did too :). Readers of the newsletter got a sneak peek, and I have a PDF version I’ll be sending out to subscribers as well.

Overall, BetterExplained is an excited friend who shares what *really* helps ideas click, not an authority trying to be the grand poombah of math. Let’s have a good time on this journey of learning.

Hope you enjoy the new site, feedback is welcome!

-Kalid

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