In English, we often drop the subject of a phrase:

Who are these signs written for? It's really *You, stop* or *You, yield* or *You, be alert for bears* (I'm not doing it).

After internalizing a language, we can take hints without explicit instructions. But, to put it politely, math isn't usually well-internalized.

Let's get clear about who the "math signs" are referring to.

## Imaginary Numbers

Imaginary numbers are often defined as $i^2 = -1$, and written this way they're utterly baffling.

A better restatement is:

which is still confusing. But what about this:

It's getting clearer. The instructions are: "Start with 1, multiply by i, multiply by i again, and (somehow) end up at -1".

What could do that? Realizing we *start at 1 and end up at -1* helps us visualize something like this:

Aha! $i$ is a change (visualized as a rotation) that moves us from positive to negative in two steps (More about imaginary numbers.)

Writing $i^2 = -1$ without a clear subject is confusing. (Don't get me started with $i = \sqrt{-1}$)

Missing the implied subject of "1" in $1* i * i$ caused me years of confusion. I wish this sign was hanging on the classroom wall:

## Exponents

Why is $x^0 = 1$ for any value of x? How do we ask for 0 of something and get 1 back out?

Again, let's break it down with a simple example. Here's a typical exponent:

But it's missing a subject. It's written better as:

We start at 1 (our default multiplicative scaling factor), scale by x, then scale by x again, ending up at $x^2$. The size of the exponent (2) tells us the number of times to use our "times x" scaling machine.

Stepping back, multiplication is about scaling: 3 is really "1 * 3", or the unit quantity enlarged 3 times.

If we want to scale by x (just once), we write:

What if we don't plan on using our scaling machine at all?

The notation is a bit weird, but I'm using empty parenthesis to indicate a lack of action. See, the zero in $x^0$ is that of indifference -- taking no action -- and not obliteration. "Using" $x^0$ means we haven't scaled our original quantity at all.

Subtle, right?

We can take this "growth machine" idea further with the Expand-o-tron 3000.

## Imaginary Exponents

Let's combine insights. What does a strange exponent like $e^{i \pi}$ represent?

With our new "implicitly start at 1" perspective, it's really:

Start at 1 and then apply the growth engine. Here, growth is aimed sideways ($i$) with enough fuel to last for half a circle ($\pi$).

The essence of Euler's Identity is that we are starting at 1 and transforming it with a spin. We aren't creating a negative number out of seemingly positive exponents directly. (See article and video.)

## Calculus

Calculus has numerous notational shortcuts. When we write:

it really means:

which really means:

Here's the tricky part. There isn't a single "dx", there's a whole chain of them along the number line. The sentence is something like:

*"Hey everyone on the number line! You're all spaced "dx" apart. Take your current position and square it. Then I'll come by and add you all up."*

The integral addresses not a single "you" like 1.0, but "them", the countless positions on the number line.

Find the implied subject in an equation, then work to shorten it (*Bears!*).

Happy math.

JoeJune 26, 2018 at 11:35 amLove it! Going to try to think in terms of 1*(*x)(*y) when looking at formulas and mentally play with these concepts. The simplest ideas are the trickiest to find, so congrats on finding some incredibly simple ideas over the years. And thank you for continuing to post them. (The simplest ideas also seem to require repetition and re-wording for them to really sink in :-)

Dr. Parthasarathy SJune 26, 2018 at 6:53 pmI would rather say “The simplest ideas are the trickiest to explain”

kalidJune 27, 2018 at 5:18 pmThanks Joe, great to hear from you!

Richard RuffJune 26, 2018 at 3:34 pmAs always, a stimulating idea to ponder on first thing in the day! Thanks Kalid, beautifully written, beautifully explained.

kalidJune 27, 2018 at 5:18 pmThanks Richard!

Dr. Parthasarathy SJune 26, 2018 at 6:40 pmA good teacher explains things this way. A great teacher makes sure his students can understand and grasp the concept.

kalidJune 27, 2018 at 5:19 pmExactly, you want to make sure students have their Aha moment.

Sachin DateJune 27, 2018 at 2:22 amMultiplication by i = anti-clockwise rotation by 90 degrees. A simple concept but so non-intuitive! If only my initiation into complex numbers had started with this concept…

Thanks Kalid. This is why I like to read your blog.

kalidJune 27, 2018 at 5:19 pmThanks Sachin! One of my favorite math visualizations.

otcasmdrJune 27, 2018 at 10:07 pmWhat all on earth are you talking about? Just gimme the book. It’s fine. I tried following but it just seems…convoluted?

I’ll keep trying to grasp your reasoning (methodology), something has to take hold sooner or later. For me it’s frustratingly slow and laborious

I’m hoping I might be able to use it on slower to adapt students, But it seems to take me out of the scheme of reasoning. Maybe I’m too entrenched.

Satish GodaJune 28, 2018 at 2:15 amWhat an amazing and packed post!!! Thank you Kalid.

kalidJune 29, 2018 at 11:04 amThanks Satish!

MVJune 28, 2018 at 5:11 pmHello Kalid, can you please explain why a log decade compress at higher end ?

Thanks.

sweetspiceniceJune 29, 2018 at 10:04 amis it possible to say e^[(i)(pi)] is equivalent to (i)^2 just from the examples above rotating 180deg? :O I didn’t do any proofs though so I wouldn’t say it’s always valid, just seemed true IF you started at 1 with both examples.

kalidJune 29, 2018 at 11:04 amYep, that’s correct! e^{i * pi} = i^2 = -1

Sharath Bhargava.RJune 30, 2018 at 9:33 pmThanks Khalid,It is indeed a great write up…you made my day.

And one of my request for you…can you write some details on Laplace transforms and why it is like that so?

ShirleyJuly 18, 2018 at 8:50 amGreat article! I found it hard to understand multiplication without that implicit ‘1’. To explain the continuous compounding (1+1/n) is awkward without the ‘one’ in front” 1x(1+1/n): starting with the ‘one’, followed by the ‘x’, math language is more complete: a noun (1 unit principle) and a verb (grows). Only when the language is complete can we be more comfortable making up a story about what happened. Otherwise all I can do is simply re-iterate the dry math symbols: one plus one over n to the power of n—- which makes almost no sense.

What I also found subtle is the implicit interchanges between numbers as nouns and numbers as actions (operators). My personal experiences in dealings of lots of math formulas in physics is that, lot of times it would be much easier to understand if you insert the implicit ‘one’ to start with, interpreting the numbers applied as operations to make up a story for more insights.

Without the implicit one and the following operations, formulas, numbers seems just like a coincidence to `just happen’ to work. Accordingly, if equations are taken as mere coincidences, they will elude our comprehension. Over time, our only way to handle them seem to be through pure rote memorization.

Jean CortesJuly 27, 2018 at 7:45 amThat is the best explanation of 0 as an exponent that I have seen. In the past I had my students explore the base ten system to understand the exponent of 0 but this is more intuitive. Now I will make sure there is a subject!

Thank you!

kalidJuly 27, 2018 at 8:02 amThanks Jean, glad it was helpful!

atzilut1 AharonAugust 13, 2018 at 12:33 amThank you for your amazing site. I struggled with math and physics at school and haven’t visited them since. Although I specialized in Indian Philosophy, I remained totally incapable of ‘understanding’ anything beyond basic math. Your site has really given me an insight into why I haven’t been able to grasp it at all. Thanks very much.

parastoo bakhshiJanuary 2, 2019 at 10:46 pmIt was a good post I liked very much. good luck

https://dehkadeh-web.ir/