Math, Better Explained: Learn to Unlock Your Math Intuition

by Kalid Azad

Learn How to Learn: The essays highlight my favorite learning method: get the context of an idea, formulate analogies, and cover examples using those analogies. This learning technique works with many subjects, not just math. Updates & Mailing List

Missing the big picture drives me crazy: math is about ideas — formulas are just a way to express them.

Math and ideas

Step 1: Find the central theme of a math concept. This can be difficult, but try starting with its history. Where was the idea first used? What was the discoverer doing? This use may be different from our modern interpretation and application. Step 2: Explain a property/fact using the theme. Use the theme to make an analogy to the formal definition. If you’re lucky, you can translate the math equation (x2 + y2 = r2) into a plain-English statement (“All points the same distance from the center”). Step 3: Explore related properties using the same theme. Once you have an analogy or interpretation that works, see if it applies to other properties. Sometimes it will, sometimes it won’t (and you’ll need a new insight), but you’d be surprised what you can discover.

Steps towards full understanding

Here’s a few common definitions of e:

Looking at e’s history, it seems it has something to do with growth or interest rates. e was discovered when performing business calculations (not abstract mathematical conjectures) so “interest” (growth) is a possible theme.

Whablamo! These are four different ways to describe the mysterious e. Once we have the core idea (“e is about 100% continuous growth”), the crazy equations snap into place — it’s possible to translate calculus into English. Math is about ideas!

It’s ok to be visual. We think of math as rigid and analytic — but visual interpretations are ok! Do what develops your understanding. Imaginary numbers were puzzling until their geometric interpretation came to light, decades after their initial discovery.

Math becomes difficult and discouraging when we focus on definitions over understanding.

Remember that the modern definition is the most advanced step of thought, not necessarily the starting point. Don’t be afraid to approach a concept from a funny angle — figure out the plain-English sentence behind the equation. Happy math.

Yes, we can rattle off equations, but do we really understand the nature of area? This fact may surprise you: The area of any shape can be computed from any line segment squared

Did you know this about area

can pick any line segment and figure out area from it: every line segment has an “area factor” in this universal equation: Area = Factor ·(line segment)2

2.3 Can We Pick Any Shape? Sort of. A given area formula works for all similar shapes, where “similar” means “zoomed versions of each other”. For example: All squares are similar (area always s2) All circles are similar, too (area always π r2) All triangles are not similar:

Let’s call the long side c (5), the middle side b (4), and the small side a (3). Our area equation for these triangles is: Area = F · hypotenuse2 where F is some area factor (6/25 or .24 in this case; the exact number doesn’t matter). Now let’s play with the equation: Area(Big) = Area(Medium) + Area(Small) F · c2 = F · b2 + F · a2 Divide by F on both sides and you get: c2 = b2 + a2

Look at that!

Metcalfe’s Law (if you believe it) says the value of a network is about n2 (the number of relationships). In

If a=3 and b=4, then c=5. Easy, right? Well, a key observation is that a and b are at right angles (notice the little red box). Movement in one direction has no impact on the other. It’s a bit like North/South vs. East/West. Moving North does not change your East/West direction, and vice-versa — the directions are independent (the geek term is orthogonal).

The Pythagorean Theorem lets you find the shortest path distance between orthogonal directions. So it’s not really about right “triangles” — it’s about comparing “things” moving at right angles.

In our example, C is 5 blocks of “distance”. But it’s more than that: it contains a combination of 3 blocks East and 4 blocks North. Moving along C means you go East and North at the same time. Neat way to think about it, eh?

As you can guess, the Pythagorean Theorem generalizes to any number of dimensions.

distance between points (x1, y1, z1) and (x2, y2, z2) using the same approach: distance2 = (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)

preferences as a vector, and used the theorem to find the distance between them (and group similar items, perhaps).

3.10 The Point: You Can Measure Anything If you can represent a set of characteristics with numbers, you can compare them with the theorem: Temperatures during the week: (Mon, Tues, Wed, Thurs, Fri). Compare successive weeks to see how “different” they are (find the difference between 5-dimensional vectors). Number of customers coming into a store hour-by-hour, day-by-day, or week-by-week Spacetime distance: (latitude, longitude, altitude, date). Useful if you’re making a time machine (or a video game that uses one)! Differences between people: (Height, Weight, Age) Differences between companies: (Revenue, Profit, Market Cap)

Representations with numbers

is so important I’ll repeat it again: If you can quantify it, you can compare it using the the Pythagorean Theorem.

Math is beautiful, but the elegance is usually buried under mechanical proofs and a wall of equations.

circle has 360 degrees and a year has 365 days?.

Unlike a pirate, I bet you landlubbers can’t determine the seasons by the night sky. Here’s the Big Dipper (Great Bear) as seen from New York City in 2008:

360 is close enough for government work. It fits nicely into the Babylonian base-60 number system,

That’s how we do math! We write equations in terms of “Hey, how far did I turn my head see that planet/pendulum/wheel move?”. I bet you’ve never bothered to think about the pendulum’s feelings, hopes and dreams.

Lol

Instead of wondering how far we tilted our heads, consider how far the other person moved. Degrees vs. Radians Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled.

Degrees vs radians

Radians are the empathetic way to do math — a shift from away from head tilting and towards the mover’s perspective.

When a satellite orbits the Earth, we understand its speed in “miles per hour”, not “degrees per hour”. Now divide by the distance to the satellite and you get the orbital speed in radians per hour. Sine, that wonderful function, is defined in terms of radians as sin(x) = x - x3/3! + x5/5! - x7/7!

a bus with wheels of radius 2 meters (it’s a monster truck bus). I’ll say how fast the wheels are turning and you say how fast the bus is moving. Ready? “The wheels are turning 2000 degrees per second”. You’d think: Ok, the wheels are going 2000 degrees per second. That means it’s turning 2000/360 or 5 and 5/9ths rotations per second. Circumference = 2πr, so it’s moving, um, 2 times 3.14 times 5 and 5/9ths... where’s my calculator... “The wheels are turning 6 radians per second”. You’d think: Radians are distance along a unit circle — we just scale by the real radius to see how far we’ve gone. 6 · 2 = 12 meters per second. Next question.

Time for a beefier example. Calculus is about many things, and one concern is what happens when numbers get really big or really small.

When you make x small, like .01, sin(x) gets small as well. And the ratio of sin(x)/x seems to be about .017 — what does that mean? Even stranger, what does it mean to multiply or divide by a degree? Can you have square or cubic degrees? Radians to the rescue! Knowing they refer to distance traveled (they’re not just a ratio!), we can interpret the equation this way: x is how far you traveled along a circle sin(x) is how high on the circle you are So sin(x)/x is the ratio of how high you are to how far you’ve gone: the amount of energy that went in an “upward” direction. If you move vertically, that ratio is 100%. If you move horizontally, that ratio is 0%.

(sin(x)). As x shrinks, the ratio gets closer to 100% — more motion is straight up. Radians help us see, intuitively, why sin(x)/x approaches 1 as x gets tiny. We’re just nudging along a tiny amount in a vertical direction. By the way, this also explains why sin(x) ~ x for small numbers. Sure, you can rigorously prove this using calculus, but the radian intuition helps you understand it.

Remember, these relationships only work when measuring angles with radians. With degrees, you’re comparing your height on a circle (sin(x)) with how far some observer tilted their head (x degrees), and it gets ugly fast.

With natural laws, we’re an observer describing the motion of others. Radians are about them, not us. It took me many years to realize that:

the mover, equations “click into place”. Converting rotational to linear speed is easy, and ideas like sin(x)/x make sense.

Focusing on relationships, not mechanical formulas. Seeing complex numbers as an upgrade to our number system,

Using visual diagrams, not just text, to understand the idea. And our secret weapon: learning by analogy. We’ll approach imaginary numbers by observing its ancestor, the negatives. Here’s your guidebook:

Understanding complex numbers

There’s so much more to these beautiful, zany numbers, but my brain is tired. My goals were simple: Convince you that complex numbers were considered “crazy” but can be useful (just like negative numbers were) Show how complex numbers can make certain problems easier, like rotations

Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).

Importante

e is the base amount of growth shared by all continually growing processes.

e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.

Importante

So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.

Mathematically, if we have x splits then we get 2x times more “stuff” than when we started. With 1 split we have 21 or 2 times more. With 4 splits we have 24 = 16 times more. As a general formula: growth = 2x Said another way, doubling is 100% growth. We can rewrite our formula like this: growth = (1 + 100%)x

the end of 1 year we’d have Our original dollar (Mr. Blue) The dollar Mr. Blue made (Mr. Green) The 25 cents Mr. Green made (Mr. Red) Giving us a total of $2.25. We gained $1.25 from our initial dollar, even better than doubling! Let’s turn our return into a formula. The growth of two half-periods of 50% is: growth = (1 + 100%/2)2 = 2.25

The numbers get bigger and converge around 2.718. Hey... wait a minute... that looks like e! Yowza. In geeky math terms, e is defined to be that rate of growth if we continually compound 100% return on smaller and smaller time periods: This limit appears to converge, and there are proofs to that effect. But as you can see, as we take finer time periods the total return stays around 2.718. 7.7 But What Does It All Mean? The number e (2.718...) represents the maximum compound rate of growth from a process that grows at 100% for one time period.

Importante

e is like a speed limit (like c, the speed of light) saying how fast you can possibly grow using a continuous process. 7.8 What About

Although we picked 1%, we could have chosen any small unit of growth (.1%, .0001%, or even an infinitely small amount!). The key is that for any rate we pick, it’s just a new exponent on e: growth = erate

Given how the natural log is described in math books, there’s little “natural” about it: it’s defined as the inverse of ex, a strange enough exponent already. But there’s a fresh, intuitive explanation: The natural log gives you the time needed to reach a certain level of growth.