Note from Emily and Denise: We are noticing that EdTech companies seem to be shifting uncomfortably in their metaphorical chairs these days— increasingly so as the school year approaches and concerns about their products’ presence in schools grows. We believe we are witnessing an “EdTech industry rebrand” and we’re here to call it out. This is Part 1 of a two-part essay.

From summer conference themes to repurposed terminology to doubled down efforts to insert AI into more classrooms this fall, educational technology companies and their ilk are feeling the heat from the growing backlash of parents, teachers, neuroscientists, lawmakers, and advocates opposed to the overreliance on for-profit products in schools that do not improve learning outcomes, threaten the teaching profession, and expose children to serious risks.

In response, instead of pulling their health-harming products from the classrooms or seeking external audits and independently verifiable scientific research to prove their claims, they’re getting a makeover.

But the trouble is, “Children do not need tech; technology companies need children,” as Emily pointed out recently in her testimony to the Kentucky State Senate on education and screentime. At the end (and beginning) of the day, EdTech is an industry– a massive one worth nearly $200 billion currently, with some projections putting it at nearly double that over the next ten years. (Can you imagine if schools had budgets like that?)

“How do we grow when the easy money is gone?”

EdTech products are heavily pushed into school districts by an industry that is well-funded and skilled at marketing. As one example (and there are many), the Tech & Learning EdExec Summit scheduled for September 2026 is pitched to attract:

• CEOs & Founders “who need to steer the ship through economic uncertainty”;

• Sales & Marketing Directors “who need to shorten complex sales cycles and bypass gatekeepers”;

• Product Managers “who need to build for ‘Evidence-Based’ compliance and interoperability”; and

• Strategic Partnership Directors/Business Development “looking to build relationships with key industry players.”

This “learning” summit promises that attendees will have access not just to industry executives, but “school district administrators” too.

The “two-day strategic summit” will seek to answer one question: “How do we grow when the easy money is gone?”

The hosts of the Tech & Learning EdExec Summit say the experience is built to “deconstruct the real friction points in today’s district sales cycle—guided by the leaders who control the budgets.“ Attendees will “get valuable insight into the education market in time for the Back-to-School buying season.” Who are these attendees?

Here’s a list:

It seems to us that the only “learning” occurring here is how to keep milking the cow even when she’s gone dry.

Profits are clearly the motive at such events, not education.

Which is What the Backlash to EdTech Really Threatens

Activists and teachers and concerned parents like us often get painted as “anti-tech”-- but this couldn’t be farther from the truth. Yes, we are deeply concerned about the kinds of content our children can see on their school-issued Chromebooks and yes, we see the potential for serious risks to the mining and selling of their personal data, and yes, we would much rather see kids engaging in classrooms with human teachers and books and pencils. But we do not believe that all technology is inherently evil— we just see a big difference between EdTech and TechEd, for example.

The problem is the business model of the EdTech companies.

These companies rely on “time on device” and “engagement” to compete for a piece of that $200 billion pie, but this business model is fundamentally in opposition to healthy child development.

It doesn’t matter if an EdTech company has a noble mission, or if they have high quality materials, or if they swear they do not collect and sell student data— at the end of the day, if their business model depends on children spending time on a device this is no different than the business models of social media companies.

That’s why we say “EdTech is just Big Tech in a sweater vest”— the business models of Instagram and Snap and Meta are the same as iReady and Seesaw and Canvas. (Which may explain why Emily is also finding she has to fight to protect her intellectual property and trademarks from being co-opted by the very industry she is battling.)

This is how we find ourselves in the middle of Extreme Makeover: EdTech Edition.

The EdTech Industry’s New Look: “Purpose-Built EdTech”

The EdTech industry has taken notice of our movement…and they are concerned. They’re seeing and feeling the impact (and trolling us on social media). They are trotting out the same tired talking points that we’ve come to expect.

But a new report from Instructure reveals a new tactic: “purpose-washing.” You may remember Instructure as the parent company of Canvas, an EdTech company that was the target of a massive data breach in the spring of 2026. Canvas is used by over 40% of higher education in North America and the breach nearly paralyzed institutions around the country.

Instructure’s newest report claims to have analyzed the evidence, compliance and interoperability, and accessibility and usability of the top 150 EdTech products used in K-12 schools during Fall 2025. Before we dive into their report, we should point out that Instructure’s partnerships with companies like Google and Microsoft render such a “report” as scientifically compromised.

According to Instructure, “families, communities, and policymakers” are starting to ask if “all this screen time is beneficial for students.”

As a result of these questions being asked, Instructure believes, school leaders will need to defend their use of technology in the classroom to “boards, families, educators, and their communities.”

Of course, the most burning question is “Why weren’t they doing this before?”

Instructure, of course, has a “solution”: school leaders should make a distinction between “purpose-built” screentime and “consumer products,” and be sure parents and policymakers see Instructure products as the former not the later.

By attempting to put distance between “purpose-built EdTech” (which Instructure claims includes Kahoot! and iReady) and “Consumer Tech” (YouTube and ChatGPT), Instructure attempts to weave a story that “purpose-built” products like these offer more evidence of effectiveness and are safer.

Except we know from research, journalistic deep-dives, and lawsuits that products like iReady are anything but “effective” or “safe” or “legal.”

Real-Time Rebrand: What It Looks Like

A recent LinkedIn post by Curriculum Associates, creators of the troubled iReady Math program and current defendant in a lawsuit about data privacy concerns, is a great example of an EdTech company attempting to assuage parental and school administrator anxiety by claiming that by partnering with “Digital Promise,” Curriculum Associates will be able to provide a “framework” to justify inserting AI into schools.

The only problem is that Digital Promise, an organization that describes itself as “a global nonprofit working to expand opportunity for every learner,” has numerous funders and partners with the tech industry.1

Emily and Denise were both lead authors on Fairplay’s recent Call for a Pause on AI in PreK-12th Grade and we firmly oppose the use of any AI products in education. We reject the notion that in an era of “rapid AI experimentation” that a “framework” given out by tech-funded “non-profit” and bestowed by a problematic EdTech company facing lawsuits for taking kids’ data without parental consent should be the arbiters of such designations.

At the same time, the language now being used by Curriculum Associates (and other EdTech companies) sounds very similar to the “Tech-Intentional” framework created by Emily. We’re seeing this in multiple places, and as Emily has written about, it appears to be a growing issue.

It’s Not just EdTech Companies. It’s the Organizations Who Defend Them, Too.

We have written previously about the need to follow the money, not just behind EdTech companies, but the organizations that defend them, too.

One well-known example is ISTE, formerly the International Society for Technology in Education, who merged with ASCD (the Association for Supervision and Curriculum Development) in 2022 and recently rebranded themselves as the International Society for Transforming Education. In a not-so-transparent attempt to distance themselves from the growing skepticism of the products they push in schools, note how the word “Technology” has been replaced with “Transforming.”

ISTE also offers an “ISTE Seal”2 for EdTech products that “align” to their ISTE Standards.3 However, the ISTE Seal, which seems to require only a self-audit, appears to be a pay-to-play rubber stamp for well-funded EdTech companies to lend their for-profit products false merit. To “receive” an ISTE Seal, companies must pay $5K to apply and pay a $3K fee every two years.

ISTE also targets educators through tech-funded conferences sponsored by tech industry giants such as Microsoft, Google, META, Pinterest, Intel, Epson, Logitech, Samsung, Dell and the Chan Zuckerberg Initiative.

ISTE is obviously tech-funded, but claims to be a “non-profit” serving educators and administrators. They’re doing quite well for a non-profit. In 2024, ISTE announced that Google had invested $10 million “to support ISTE+ASCD in its efforts to provide AI skills training to educators and students across the country.” According to CauseIQ, in 2024 ISTE had a revenue of approximately $46 million.

What the Makeover Reveals

ISTE is attempting to reframe the conversation around screentime in schools. Language in a recent document titled “From Screentime to Screen Value”4 cites research that is a decade old or comes from compromised, tech-funded organizations; pitches four “themes” rooted in industry myths and propaganda (more on this in Part 2); and implies that parents just don’t understand the value of “quality” screentime.

ISTE has it all wrong, of course.

This is not just a conversation about “quality” screentime or “consumer tech vs EdTech.”

What the industry doesn’t seem to understand is that the growing backlash from parents, teachers, lawmakers, and a few brave school administrators is not about what type of screentime children are getting at school; it is driven by the fact that children can and should experience quality learning experiences without the use of technology products created by for-profit companies in the first place.

As criticism grows, EdTech companies will continue to dig into their deep pockets and spend their marketing budgets to “purpose-wash” their products in the best light possible.

But we know what this makeover really shows— that the EdTech industry is worried enough that a hasty remodel seems the best path forward to protecting their bottom line.

Just like slapping a coat of paint on a mold-infested wall only hides the dark spots temporarily, however, attempting to rebrand the same old EdTech products as somehow new and improved does not hide the problem— it reveals it.

In Part 2, we will break down ISTE’s “Four findings that cut across the evidence” and challenge the dubious research they cite to defend these “findings.”

Thanks to AI, famous mathematical conjectures are seemingly getting overturned faster than red cards at a World Cup. But there’s something that’s been bugging me about all these new AI proofs.

Take the recent mathematical proof – or rather disproof – of the Jacobian conjecture by Fable, announced on X:

It’s a remarkable finding, which has solved a puzzle that has frustrated mathematicians for decades. And in many ways, it’s the best in mathematical elegance and accessibility: a proof that can fit in a tweet and could be verified by any mathematics undergrad, or even a bright high school student.

Like many others, the first thing I did when I saw it was put the (x, y, z) co-ordinates into the equation to check it held up. But, like others, I then wondered how Fable had managed to spot such an elusive equation. And that’s where I hit a wall. It’s unlikely Fable found it by brute force alone – after all, humans have been trying for decades – but it’s still not clear exactly what it did behind the scenes.

What should one do?

Back in 1985, mathematician Jean-Pierre Serre expressed caution about the rise of proofs that ran to hundreds of pages, which hardly any humans could be expected to understand or check:

‘What should one do with such theorems, if one has to use them? Accept them on faith? Probably. But it is not a very comfortable situation.’

Serre was talking of proofs that were hard to verify. In contrast, AI proofs like the above are fairly easy to check. But the path the AI took to get there is often beyond reach. We have the what, but we frequently lack the why.

To paraphrase Serre: what should we do with such proofs?

In a recent talk, mathematician Terence Tao points out that there is more to mathematical research than just showing things are true:

For a proof to actually contribute to the broader field, it is not enough for it to be correct and easy to read. It also needs to be accepted and valued by the community. Other mathematicians need to digest the result and incorporate it into their own work.

Authors can assist in the digestion process by describing their own insights and stories from when they were working on the problem. However, current AI tools are quite opaque about their problem-solving process. This is particularly true for proprietary models whose inner workings remain a corporate secret.

Tao concluded with a suggestion that the ultimate goal of mathematical research is to have verified solutions that are ‘digested, accepted, and incorporated into the definitive theory of the field’.

Solved conjectures and stamp collecting

Nobel laureate Ernest Rutherford supposedly once said that ‘all science is either physics or stamp collecting’. In his view, the distinction came about because physics aimed to derive fundamental laws and mathematical principles, whereas other fields merely observe and categorise.

Many fields would rightly take issue with this generalisation, but the quote still came to my mind this week. Because I realised that I now knew a famous conjecture was false, but I didn’t have much beyond that.

Are we just collecting lots of little AI-proof-shaped stamps? In the long run, I expect it won’t be that simple. AI has the potential to show us a lot of things we’ve so far missed. But ‘show’ is the operative word here. It won’t be enough to tell us lots of things are true (or false); we’ll need the tales and theories that others can build on.

If you’re interested in reading more about mathematical proof and AI, you might like my latest book Proof: The Uncertain Science of Certainty.

How the words we teach English language learners changed

A dark history that is in danger of returning as support for people with disabilities withers

The post When Stupid Was a Diagnosis appeared first on Nautilus.

In the beginning...

I first heard about the Fourier Transform in a stuffy lecture theater. I was fighting off sleep at the time and thought the whole thing was just too abstract  and so did my fellow students judging by their nodding heads. Still, I was good at math and I learned what I needed to get through the exam. It's only years later that I realized how fundamental and important the Fourier Transform is in the real world.  The professor could have made his lectures come alive if only he'd only given us a taste of why it's so consequential in so many areas. 

In this blog post, I'm going to make up for my professor's shortcomings and explain why it's so important. I'm also going to share some of the math because that shows where it all comes from and how it can be used. But first, let's have a really high-level explanation of what a Fourier Transform is.

The ELI5 version of Fourier Transforms

The Fourier Transform converts data measured in one domain to data in the inverse domain. For example, it can convert music in the time domain to its constituent frequencies in the frequency domain. 

It works by decomposing a signal into a series of overlapping sine waves. Even a square pulse can be decomposed into sine waves, in this case yielding an infinite number of them. Once you have the sine waves, you can work out their frequencies and so do the conversion from the time to to frequency domain. Understanding that any signal is a sum of sine waves opens the door to understanding natural phenomena like analog circuit behavior or heat flow.

There are two types of Fourier Transform: a version which works on discrete data (like sampled music) and a version which works on continuous data (for example, an equation describing the orbit of a planet).

What use is it?

The easiest example to explain is processing audio data like music. Digitized music is in the time domain and applying the Fourier Transform puts the data in the frequency domain. Once it's in the frequency domain we can do things like filtering out frequencies we don't want. For example, by removing frequencies humans can't hear, we can compress the data (which is partly how MP3 works).

It's not just music signals; the Fourier Transform is used extensively in signal processing, including processing radar, sonar, radio signals, images (including medical images), and seismology data. It's the same idea in each field: by transforming the data into the inverse domain some types of processing become a lot easier. 

It turns out it's fundamental to quantum mechanics too. The Heisenberg Uncertainty Principle states that you can't know a particle's position and momentum both with 100% accuracy; the more you know about one, the less you know about the other. In quantum theory, position and momentum are Fourier Transform pairs, which means one is the Fourier Transform of the other. This relationship directly leads to the Heisenberg Uncertainty Principle as I'll show later.

Back in the 19th century, the transform was originally developed to model the flow of heat. Legend has it, Napoleon wanted more efficient cannons, so he engaged Jean-Baptiste Joseph Fourier to study the limiting factor for cannons, which was heat flow. In solving this problem, one of Fourier's key insights was that any complex signal or wave could be decomposed into a series of overlapping sine waves. At the time, this idea was very controversial and it took some time for the scientific community to accept it.

These uses cases should have been enough to wake up sleepy students, but there's more.

It turns out, Fourier Transforms play a key role in probability theory, which is why they're used to model risk in insurance companies. I'll come back to this in a later blog post.

Given what we already know, it's probably not a surprise that they're used to solve some differential equations; an intractable problem in one domain might be very solvable in the inverse domain. Perhaps more surprisingly, they also creep into number theory.

I haven't covered all of its uses, but let's move on to defining some quantities and the Fourier Transform itself.

WTF - too many fs and notation issues

There's a really dumb problem with the Fourier Transform, there are two many things beginning with f. We have f for frequency, function, and Fourier. Capital letters only take us so far so we're going to have to make some choices about what characters we use for what. I'm following the herd with my choice of notation.

Why do we care about angular frequency? Because \(2 \pi \nu\) crops up a lot in Fourier math and using \(\omega\) just simplifies things. 

Discrete Fourier Transforms

Let's imagine we have a signal we've sampled in the time domain, say we sample some live music at a rate of 96 kHz (or one sample every 10 micro-seconds). We'll represent our signal like this:

\[\{x_n\} = \{x_0, x_1, x_2, ..., x_{N-1}\}\]

The Fourier Transform of \(x_n\) is:

\[ X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-i2\pi \frac{k}{N} n} \]

which gives us the transformed data set:

\[\{X_k\} = \{X_0, X_1, X_2, ..., X_{N-1}\}\]

This is all very abstract, so let me show you what this means in reality. Let's imagine that our music is a simple sine wave that looks like this:

\[x_n = \sin(2\pi \cdot 9000 \cdot t)\]

Of course, we're dealing with discrete data in this section, so we're going to sample the data at 48 kHz. If we plot it out, it looks like this (the light blue line is just to guide the eye, the dots are the measured data):

Now, let's Fourier Transform the data. After the transform, we get data in the frequency domain that looks like this:

See the peak in Fourier Transform plot? That tells us the frequencies present in the data. In this case, we know there's only one frequency and we know what it is, but that's not always the case. We can use this technique to find out the "active frequencies" in a piece of music or speech. The chart below is the Fourier Transform of some sampled music, you can see the frequencies present and you can see this is much, much more complex than a simple sine wave.

(BTW - do you think you can identify the piece of music? It's a 10 second burst from the start of a very famous song.)

Once we know all the frequencies present, we can remove some of them and transform the signal back to the time domain, which takes us to the inverse Fourier Transform.

Let's say we've done what we want to do with the frequency domain data \({X_k}\), how do we bring the data back to the time domain? With an inverse Fourier Transform defined like this:

\[ x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{i2\pi \frac{k}{N} n} \]

I'll do one more example to give you a flavor of what you can do with this method. Let's say we have a more complicated time domain signal with two frequencies, like this:

\[x(t) = 0.5\sin(2\pi \cdot 9000 \cdot t) + 1.0\sin(2\pi \cdot 12000 \cdot t)\]

Here's what this looks like on a chart:

How might we get rid of one of the two frequencies? Let's start by doing a Fourier Transform and looking at the data. We can see the two peaks corresponding to the two frequencies. 

Let's cut the higher frequency out so the data looks like this.

Now, let's transform the data back to the time domain. Here's what we get (chart below). 

We've filtered out the higher frequency.

In reality, digital filtering is much more sophisticated than this, but this simple example gives you a taste of what's possible.

Before I move on to talk about continuous data, it's worth noting that the discrete Fourier Transform only works on data that's been sampled at regular intervals. This sounds like it's trivial, but it has important consequences for real-world analysis.

Continuous Fourier Transform

To put it simply, continuous data is data that has values at all points, it has no gaps, meaning it isn't sampled. In the continuous case, we have the equations that govern the data. Good examples of continuous data include the equations governing the motion of planets, probability distributions, and electrical voltage.

We'll start with a continuous function, \(f(x)\), its Fourier Transform is:

\[F(\omega) = \int_{-\infty}^{\infty} f(t)\, e^{-i\omega t} \, dt\]

To go back, there's the inverse Fourier Transform, defined like this:

\[ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega)\, e^{i\omega t} \, d\omega \]

For a continuous signal defined like this:

\[ x(t) = 0.5\sin(2\pi \cdot 9000 \cdot t) + 1.0\sin(2\pi \cdot 12000 \cdot t) \]

It's looks like this in the time domain:

Its Fourier Transform looks like this:

Just as in the discrete case, the transform of the signal lets you clearly see the frequencies present in the data and their relative importance. The frequency at 12 kHz is "louder" and we can see that from the equation and the Fourier Transform.

It's often useful to know how much power is present at different frequencies. We can do this by something called the power spectral density. The Fourier Transform often gives complex data (in the sense of complex numbers, meaning having a real and imaginary component). The power spectral density is the magnitude squared of the real component.

Proof by assertion

I've just quoted the Fourier Transform equations without having derived them. That's deliberate because I want to focus on their use and properties rather than spending too much time on the math (it's also a pain to write so much LaTex). Years ago, my professor focused on their derivation and we spent lecture after lecture on derivations, which for me missed the point. I want to focus on their use.

The dirac delta function and quantum theory

This is a little abstract, but it does help explain how the Heisenberg Uncertainty Principle comes from the Fourier Transform. It's also a nice example of it's use.

The dirac delta function is a weird mathematical function. It's not really a function, it's something else, but it is useful. The function is zero everywhere except at zero where it has the value infinity. Mathematically, this is:

\[ \delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \neq 0 \end{cases}, \quad \int_{-\infty}^{\infty}\delta(x)\,dx = 1 \]

Here's a chart showing the same idea.

Let's say we know the position of a subatomic particle with 100% certainty. That means the uncertainty in /(x/) is 0, so we can model the position using a dirac delta function.

Momentum is the Fourier Transform of position, and here's the Fourier Transform of the dirac delta function.

This chart shows the Fourier Transform has the same value everywhere. In other words, all possible momentum values are equally likely. Which means if we know the position with 100% certainty we know the momentum with 0% certainty.

The properties of Fourier Transforms

In professional use of the Fourier Transform, we're often combining multiple functions and transforming them, so it's useful to know how the transform behaves. Let's go through some of its properties.

Scaling

If we have a function \(f(at)\), where \(a\) is a constant, then:

\[ \mathcal{F}\{f(at)\}(\nu) = \frac{1}{|a|}\mathcal{F}\{f\}\left(\frac{\nu}{a}\right) \] \[ \mathcal{F}^{-1}\left\{\frac{1}{|a|}\mathcal{F}\{f\}\left(\frac{\nu}{a}\right)\right\}(t) = f(at) \]

Linearity

If \(f(t)\) and \(g(t)\) are functions with Fourier Transforms \(F(\nu)\) and \(G(\nu)\), and \(a\) and \(b\) are constants, then:

\[ \mathcal{F}\{af(t)+bg(t)\}(\nu) = a\mathcal{F}\{f\}(\nu) + b\mathcal{F}\{g\}(\nu) \]

Time reversal

\[ \mathcal{F}\{f(-t)\}(\nu) = \mathcal{F}\{f\}(-\nu) \] \[ \mathcal{F}^{-1}\{\mathcal{F}\{f\}(-\nu)\}(t) = f(-t) \]

Time shifting

This property says that shifting a signal in time (e.g., delaying it) adds a phase shift to the Fourier Transform of the signal. That's a little too complex (pardon the pun) for me to explain here.

\[ \mathcal{F}\{f(t-t_0)\}(\nu) = e^{-i2\pi\nu t_0}\mathcal{F}\{f\}(\nu) \] \[ \mathcal{F}^{-1}\{e^{-i2\pi\nu t_0}\mathcal{F}\{f\}(\nu)\}(t) = f(t-t_0) \]

Frequency shifting

This is another property I'm not going to explain too much, other than saying it's similar in concept to the time shifting property.

\[ \mathcal{F}\{f(t)e^{i2\pi\nu_0 t}\}(\nu) = \mathcal{F}\{f\}(\nu-\nu_0) \] \[ \mathcal{F}^{-1}\{\mathcal{F}\{f\}(\nu-\nu_0)\}(t) = f(t)e^{i2\pi\nu_0 t} \]

Conjugation

I've shied away talking about the fact that the Fourier Transform relies on complex numbers underneath. My Professor zoomed in on that area and we spent a long time slogging through complex algebra to get to results. Yes, the fact that they're complex underneath is important, but for me it misses the wood for the trees. Anyway, here's the conjugation property.

\[ \mathcal{F}\{f^*(t)\}(\nu) = \mathcal{F}\{f\}^*(-\nu) \] \[ \mathcal{F}^{-1}\{\mathcal{F}\{f\}^*(-\nu)\}(t) = f^*(t) \]

Differentiation

Differentiation in the time domain corresponds to multiplying by frequency in the frequency domain. This is a very powerful property and can massively simplify some calculations.

\[ \mathcal{F}\{f'(t)\}(\nu) = i2\pi\nu\,\mathcal{F}\{f\}(\nu) \] \[ \mathcal{F}\{f'(t)\}(\omega) = i\omega\,\mathcal{F}\{f\}(\omega) \] \[ \mathcal{F}\{f^{(n)}(t)\}(\nu) = (i2\pi\nu)^n\mathcal{F}\{f\}(\nu) \] \[ \mathcal{F}^{-1}\{i2\pi\nu\,\mathcal{F}\{f\}(\nu)\}(t) = f'(t) \]

Integration

This is another very powerful relationship. It relates integration to division and so may very well simplify calculations.

\[ \mathcal{F}\left\{\int_{-\infty}^{t}f(\tau)\,d\tau\right\}(\nu) = \frac{1}{i2\pi\nu}\mathcal{F}\{f\}(\nu) + \frac{1}{2}\mathcal{F}\{f\}(0)\,\delta(\nu) \] \[ \mathcal{F}\left\{\int_{-\infty}^{t}f(\tau)\,d\tau\right\}(\omega) = \frac{1}{i\omega}\mathcal{F}\{f\}(\omega) + \pi\,\mathcal{F}\{f\}(0)\,\delta(\omega) \] \[ \mathcal{F}^{-1}\left\{\frac{1}{i2\pi\nu}\mathcal{F}\{f\}(\nu)\right\}(t) = \int_{-\infty}^{t}f(\tau)\,d\tau \quad \text{(when } \mathcal{F}\{f\}(0)=0\text{)} \]

Parseval's theorem

Mathematically, this looks difficult, but the meaning is simple: the energy of a signal is the same in the time domain and the frequency domain. This can be very helpful when you're dealing with some complicated systems.

\[ \int_{-\infty}^{\infty}|f(t)|^2\,dt = \int_{-\infty}^{\infty}|\mathcal{F}\{f\}(\nu)|^2\,d\nu \] \[ \int_{-\infty}^{\infty}|f(t)|^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|\mathcal{F}\{f\}(\omega)|^2\,d\omega \] \[ \int_{-\infty}^{\infty}f(t)g^*(t)\,dt = \int_{-\infty}^{\infty}\mathcal{F}\{f\}(\nu)\mathcal{F}\{g\}^*(\nu)\,d\nu \]

What do the properties mean?

Let's say you're doing some signal processing, maybe trying to process a weak signal in the presence of noise and other signals. You need to do math on the data you receive to filter it, amplify it, and so on. The key signal processing operations all rely on the properties I've just outlined.

Of course, it's not just signal processing. These properties also come into play when you're combining probability distributions to calculate risk. Even something like Parseval's theorem turns out to be important in insurance.

Convolution

One of the most astonishing uses of the Fourier Transform is in convolution. That's such a big topic, I'm going to write my next blog post on it.

The Fast Fourier Transform

The Fourier Transform is important, but it would have remained a theoretical nicety if it weren't for the Fast Fourier Transform.

As I said in the beginning, the key insight Fourier had was that an arbitrary signal can be decomposed into a series of overlapping sine waves. That's great, but to do this kind of decomposition in the real world is computationally very expensive, especially in the digital domain. If it had remained this expensive, frankly we wouldn't be using it much.

The game changer was the discovery of the Fast Fourier Transform (or FFT) algorithm. This slashed the number of computations for digital transformations bringing it well within the reach of computers, even computers in the early days of computing.

The FFT story is an interesting one. It seems to have been discovered several times and the ground work was laid years before it was developed. The math is interesting, but tedious to write out and draw.

Because it's a whole big topic of itself, I'm going to gloss over it for now. I might return to it later in another post. For now, you should know it a very big deal.

Closing thoughts (for now)

I'm going to be bold here. The Fourier Transform (particularly in it's FFT form), has changed the world. It's one of the hidden engines of the modern world, driving areas as diverse as music processing and insurance risk calculation. 

It does neatly illustrate one of the problems we face as a society. It's a crucial core technology to our lives, but it takes serious math to understand it. Which means that the majority of the population don't understand it and aren't in any position to make judgements or decisions about it. If the population doesn't understand key areas, how can a government allocate research funding to those areas and retain public support? Trust is the answer, but that's a whole different blog post.

I would advise that anyone working in math-heavy areas understand the what and why of the Fourier Transform. It's fundamental in many areas. Don't get lost in the derivations, but focus instead on how to use it for your benefit.

Oh, and pay attention to your physics professor when they're lecturing about Fourier Transforms.

AI statement

I (a human being) wrote all of this. I did not use AI for any of the writing. I used AI to generate code to show charts and I used AI to correctly format equations.

Happy Wednesday! I hope you're having a chill week. Do you find this newsletter delightful or thought-provoking? Share it with like-minded friends!

💙 Amanda

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Here's a line from 1,000 to 1 billion. It's a linear scale. Where on this line would you place 1 million?

Have a guess? A 2013 study asked this of nearly 500 people. Roughly half got it wrong. Honestly, I'm surprised it wasn't higher. Here's where 1 million falls on that line:

Most of us are terrible at understanding large values. Yet, we encounter these magnitudes frequently in the news: ChatGPT now has 1 billion users; America's national debt is over $39 trillion; Taylor Swift's net worth recently topped $2 billion; and briefly, Elon Musk became the world's first trillionaire.

These numbers are so big they're effectively meaningless. Very few of us have real experience with values in the millions, billions or trillions. (If you are an exception, please become a paid subscriber!) We lack an innate understanding of them.

Day-to-day, we deal with small numbers. We're good at small. Picture three rocks in your head. Easy! Now picture 100 rocks... Not so easy. Now a million? Impossible.

But we can't continue to suck at this. Our world is increasingly moving towards extreme values: Damages from climate change, deaths due to genocide, wealth that knows no bounds. And we can't tackle them if we can't wrap our heads around the numbers.

Luckily, a bit of perspective can help.

CONVERT IT

What time is that number? Let's convert these big values into something we understand better: Time. I think this is one of the clearest ways to understand the extreme differences between million, billion and trillion. Here's what each looks like in seconds.

Seconds	Day/year	Roughly
1 million	11.6 days	A vacation
1 billion	31.7 years	A career
1 trillion	31,700 years	Longer than human civilization

What time are dinosaurs? This 24-hour clock compresses the entire history of Earth into a single day. Dinosaurs appear about an hour before humans. And humans don't show up until seconds before midnight.

How far is that number? Another way to grasp large values is to convert them to distances. I don't find this as effective as time, but maybe it works for you! (I'm probably too time stressed). Here's what they look like in millimetres.

Millimetres	Kilometres	Roughly
1 million	1	Down the street
1 billion	1,000	Across France
1 trillion	1,000,000	Around the world 25 times

VISUALIZE IT

Incomprehensible wealth. Mona Chalabi's illustrated piece for the New York Times translates Bezos' bucks into creative comparisons, from cake slices to temperature.

Zoom out to infinity. This 1977 video is a classic. A single, slow zoom takes you from a picnic to the edge of the universe.

PERSONALIZE IT

What's it to you? This piece from the Washington Post personalizes purchases by the ultra-wealthy by comparing them to your own net worth. (It's a few years old, but still useful).

To them, it's trivial. A few days ago, comedian Katherine Ryan popped up on my socials. She tried to put Taylor Swift's $26 million wedding donation into terms an average person might understand — a more normal net worth. It was a great idea, but her math was off. The correct numbers are here:

Net worth	Donation
$2.2 billion	$26 million
$50,000	$590

EXPERIENCE IT

Real-time trillionaire. Watch as Elon Musk's net worth tracks up and down. I had the page open for less than 10 seconds and he'd already made $80,000.

Just a yacht or two. Modelled on Bill Gates' wealth, this game asks you to spend a virtual fortune. Extravagant purchases — yachts, an NBA team — don't even make a dent.

Keep scrolling... Finally, experience large numbers as a horizontal scroll. With this one, you'll feel the difference.

Do those incomprehensible sums feel a little less incomprehensible now? I hope so. Because our numbness to large numbers can affect how we address large-scale problems.

In a famous study from the '90s, researchers asked people how much they would pay to save 2,000 birds from drowning in an oil spill. Then they asked how much they'd pay to prevent 20,000 birds from drowning. And 200,000? Logically, the amounts should have scaled up. But they didn't. On average, people offered to pay roughly the same amount — $80, $78 and $88 — no matter how many birds were at stake.

This is a cognitive bias called scope insensitivity: Failing to adjust our valuation of a problem in proportion to its magnitude. And it happens with human lives, too.

Once a quantity gets big enough, it just becomes "a lot". "A lot" feels the same whether it's "a lot" times a hundred or "a lot" times a million. Our heads do this naturally, but it's worth working on. You and I might never run into a billion or a trillion anything in our day-to-day, but for us to understand modern problems, we need to cultivate an ability to sense the real difference between them.

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HOW THIS WORKS

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FROM ELSEWHERE

Here's what I found interesting, important or delightful this week:

&, #, @ and ¶ are letters. Promise me you'll watch this video about the origin of common typographic symbols. I had to pause it so many times, just to exclaim: What!? Seriously. Watch it.

We need nitrogen. This gorgeous piece from Reuters explains how fertilizer shortages caused by war in Iran will impact global food prices for some time. (Want more? I explored the data behind national food self-sufficiency. Fertilizer plays a big role!)

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