Math, Better Explained: Learn to Unlock Your Math Intuition

by Kalid Azad

Defining 00 as 1 makes many theorems work smoothly. In reality, 00 depends on the scenario (continuous or discrete) and is under debate. The microwave analogy isn’t about rigor: it helps us see why 00 = 1 can be reasonable, in a way that “repeated counting” does not.)

0^0 = 1

Growth curves written like 2x are from the observer’s viewpoint, not the grower.

To get into the grower’s viewpoint, we use the magical number e. There’s much more to say, but we can convert any “observer-focused” formula like 2x into a “grower-focused” one: 2x = (eln(2))x = eln(2)x

every exponential curve is just a scaled version of ex: ax = (eln(a))x = eln(a)x Every exponent is a variation of e, just like every number is a scaled version of 1.

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 I need to write it again: eiπ = –1

Euler's formula

Euler’s formula describes two ways to move around a circle.

Euler’s formula says that eix is the same process as moving around a circle using (cos(x) + isin(x)).

imaginary growth rate should give you “interest” in the imaginary direction and it would start pulling you 90-degrees!

On the contrary, real growth pulls you straight

Imaginary growth: When I grow, don’t push me forward or back in the direction I’m already going. Rotate me instead.

Imaginary growth

After 1 second you’ll be at 90 degrees (i), at two seconds 180 degrees (i2 = –1), and so on.

If your growth rate is a larger imaginary number (2i), you can consider the growth happening for twice as long (remember how e merges rate and time?). But it’s still pushing you in a perpendicular direction, which doesn’t change your speed.

Spinning faster doesn't make sense

if your growth rate is complex (a+bi) then the real part (a) grows or shrinks your magnitude like a normal exponent does, while the imaginary part (bi) rotates you. But Euler’s formula (as written) is about purely imaginary growth (eix).

Circular motion: Change continuously by rotating at 90-degree angle (imaginary growth rate).

Circular motion

we should expect ii to make things smaller. And it does: ii = e ~ .2.

i^i