Math, Better Explained: Learn to Unlock Your Math Intuition

by Kalid Azad

The log of a times b = log(a) + log(b). This relationship makes sense when you think in terms of time to grow. If we want to grow 30x, we can wait ln(30) all at once, or simply wait ln(3), to triple, then ln(10), to grow 10x again.

How about division? ln(5/3) means: How long does it take to grow 5 times and then take 1/3 of that? Well, growing 5 times is ln(5). Growing 1/3 is –ln(3) units of time. So ln(5/3) = ln(5) – ln(3)

Now the question is easy: How long to double at 100% interest? ln(2) = .693. It takes .693 units of time (years, in this case) to double your money with continuous compounding with a rate of 100%.

Ok, what if our interest isn’t 100% What if it’s 5% or 10%? Simple. As long as rate × time = .693, we’ll double our money: rate · time = .693 time = .693/rate So, if we only had 10% growth, it’d take .693 / 10% or 6.93 years to double. To simplify things, let’s multiply by 100 so we can talk about 10 rather than .10: time to double = 69.3/rate, where rate is assumed to be in percent. Now the time to double at 5% growth is 69.3/5 or 13.86 years. However, 69.3 isn’t the most divisible number. Let’s pick a close neighbor,

The intuitive human: ln(e) is the amount of time it takes to get “e” units of growth (about 2.718). But e is the amount of growth after 1 unit of time, so ln(e) = 1. Think intuitively.

In the beginning, you might have had 100 gold coins and were paid 12% per year (percent = per cent = per hundred — those Roman numerals still show up!).

discuss interest rates: APR (annual percentage rate): The rate someone tells you (“12% per year!”). You’ll see this as “r” in the formula. APY (annual percentage yield): The rate you actually get after a year, after all compounding is taken

Most natural phenomena grow continuously. As mentioned earlier, physical phenomena

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ert is the adjustable, one-size-fits-all exponential. It sounds strange, but e can even model the jumpy, staircase-like growth we’ve seen with compound interest.

Updated Mental Models

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And we get the cube root! For me, this is an intuitive reason why dividing the exponents gives roots: we split the time into equal amounts, so each “partial growth” period must have the same effect. If three identical effects are multiplied together, it means they’re each a cube root.

At last, the dreaded 00. What does it mean? The expand-o-tron to the rescue: 00 means a 0x growth for 0 seconds!

7 = (70.5)2 means “We can jump to 7 all at once. Or, we can plan on growing to 7 but only use half the time (√7). But we can do that process for 2 seconds, which gives us the full amount (√7 squared = 7).” We’re like kids learning that 3 × 7 = 7 × 3.

There’s more details, but remember this: The instantaneous growth rate controlled by the bacteria The overall rate measured at the end of each interval by the observer Underneath it all, every exponential curve is just a scaled version of ex: ax = (eln(a))x = eln(a)

EULER’S FORMULA Euler’s formula looks utterly baffling: eix = cos(x) + isin(x) This means eiπ = cos(π) + isin(π) = –1 + i(0) = –1 which is so surreal

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 ea and the angle is determined by eib.

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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?

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

Also, because eix can be converted to sine and cosine, we can rewrite every trig formula and identity into variations of e (which is extremely handy – no need to memorize sin(a + b)).

Here’s my take: Calculus does to algebra what algebra did to arithmetic. Arithmetic is about manipulating numbers (addition, multiplication, etc.). Algebra finds patterns between numbers: a2 + b2 = c2 is a famous relationship, describing the sides of a right triangle. Algebra finds entire sets of numbers — if you know a and b, you can find c. Calculus finds patterns between equations: you can see how one equation (circumference = 2πr) relates to a similar one (area = πr2). Using calculus, we can ask all sorts of questions: How does an equation grow and shrink? Accumulate over time? When does it reach its highest/lowest point? How do we use variables that are constantly changing? (Heat, motion, populations, ...). And much, much more! Algebra & calculus are a problem-solving duo: calculus finds new equations, and algebra solves them. Like evolution, calculus expands your understanding of how Nature works.

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