Speaker 1: Okay, let's unpack this today. We are diving deep into
Speaker 1: something well, pretty fundamental if you work with data and
Speaker 1: numbers in Python. We're talking about the numpi library.
Speaker 2: Oh yeah, it's absolutely bedrock. If you've ever felt the
Speaker 2: pain of doing complex math or managing big data sets
Speaker 2: with just standard Python lists, numb pie is basically the answer.
Speaker 1: And we've got a great guide for this deep dive.
Speaker 1: It's a book called Learning numb Pi. Array really takes
Speaker 1: you from the core ideas right through to tackling some
Speaker 1: real world stuff.
Speaker 2: It does, yeah, and it's good because it doesn't just
Speaker 2: throw syntax at you. It explains why numpi is built
Speaker 2: the way it is that's really key to you using
Speaker 2: it effectively exactly.
Speaker 1: So our mission here is to pull out the most
Speaker 1: valuable insights, those aha moments from the material give you
Speaker 1: a shortcut basically to understanding why numb pi is so critical,
Speaker 1: how it gets that amazing speed and efficiency, and how
Speaker 1: you can actually use it from basic array stuff to
Speaker 1: data analysis, maybe even some predictive modeling later on.
Speaker 2: Sounds good, Let's jump in.
Speaker 1: Okay, So, starting right at the beginning, what is numb
Speaker 1: pi really at its core? And why should someone doing
Speaker 1: numerical work in Python absolutely care well?
Speaker 2: Think of it as Python's high performance engine for well,
Speaker 2: anything involving a raise or matrices of numbers. It gives
Speaker 2: you this core object, the end array, right the end
Speaker 2: of it, and that's specifically built for numerical work.
Speaker 1: And the difference compared to a standard Python list is huge,
Speaker 1: isn't it? That efficiency? That speed? That's one of the
Speaker 1: first big takeaways.
Speaker 2: Absolute massive. You know, Python lists are super flexible, they
Speaker 2: can hold anything, but that flexibility it costs you when
Speaker 2: you're doing math. Right, NUMBPI arrays, they're homogeneous. All elements
Speaker 2: are the same data type. That means NUMBPI can store
Speaker 2: them way more efficiently in memory. Okay, Plus a lot
Speaker 2: of numbpi is actually written in C, so the computations
Speaker 2: themselves are just lightning fast.
Speaker 1: And that homogeneity enables one of numpis like superpowers. Right.
Speaker 2: Vectorization exactly vectorized operation instead of writing for loops to
Speaker 2: manually iterate.
Speaker 1: Through numbers, it's tedious and slow.
Speaker 2: Really slow. Yeah, you just apply operations to the entire
Speaker 2: array at once, kind of like how you'd write it
Speaker 2: in math notation.
Speaker 1: Much cleaner code, much cleaner.
Speaker 2: And crucilly much much faster because it's running optimized C
Speaker 2: code underneath.
Speaker 1: That's where the real performance game kicks in, especially with
Speaker 1: bigger data sets. I guess.
Speaker 2: Oh. Definitely, getting started is usually just a simple install.
Speaker 2: The book covers that for Windows, Linux, Mac, you know,
Speaker 2: But the real proof is seeing it in action.
Speaker 1: Okay, yeah, and this is where it gets really interesting.
Speaker 1: I thought that vector edition example, right, adding two lists
Speaker 1: in pure Python with loops. Sure it works, but it's slow.
Speaker 1: Then you see the numb pie version one line and
Speaker 1: dramatically faster, dramatically faster.
Speaker 2: And the key thing there it's not just that it's
Speaker 2: faster for one small example. That performance gap scales. With
Speaker 2: small arrays, maybe you don't notice much, but when you
Speaker 2: have millions billions of elements, numb pi isn't just nice
Speaker 2: to have, it's essential. It makes operations feasible that would
Speaker 2: just be impossibly slow with lists.
Speaker 1: Okay, so the core idea is speed an efficiency built
Speaker 1: around the special array type. Let's talk more about the
Speaker 1: nundarray itself, the fundamental building.
Speaker 2: Block, right, the number. So, like we said, the defining
Speaker 2: thing is homogeneity, all elements, same data type. This predictability
Speaker 2: knowing how big each element is. Let's numb pi lay
Speaker 2: out the data really efficiently in memory contiguously.
Speaker 1: And they're zero indexed like Python lists.
Speaker 2: Yep, zero indexed. First element is that index zero, and
Speaker 2: the array object itself knows things about its structure metadata exactly.
Speaker 2: You can ask it for its shape, like is it.
Speaker 1: A row, a matrix, a three D.
Speaker 2: Cube precisely, or it's d typed to know what type
Speaker 2: of data it holds right size for the total number
Speaker 2: of elements. Item size sells you how many bytes each
Speaker 2: element uses, and in them for the number of dimensions
Speaker 2: you use these attributes all the time.
Speaker 1: And the data types themselves, the D types, there's a
Speaker 1: decent range.
Speaker 2: Oh yeah, everything you'd expect from numeror work integers, signed
Speaker 2: and unsigned, different sizes like selling two bits, sixty four
Speaker 2: bit floats, single double precision boollions, complex numbers, and these
Speaker 2: are represented by special D type objects like in sixty
Speaker 2: four float thirty two. You can convert between types two,
Speaker 2: but you have to.
Speaker 1: Be careful, like you can't just stuff a complex number
Speaker 1: into an integer array exactly.
Speaker 2: NUMPI will complain rightly so with a type error. It
Speaker 2: knows that doesn't make sense mathematically.
Speaker 1: Well, you stress homogeneity is key, But the book mentions
Speaker 1: something called record data types. That sounds like it breaks
Speaker 1: the rule.
Speaker 2: Oh, it's a clever exception. It's for structured data. Think
Speaker 2: of like a spreadsheet row okay, where each cell in
Speaker 2: the row might be different in name, which is text,
Speaker 2: an age, an integer, a height of float.
Speaker 1: Mixed types.
Speaker 2: A record d type lets you define named fields within
Speaker 2: the array element structure. Each field can have its own
Speaker 2: specific data type.
Speaker 1: Ah. Okay, so the elements are all the same record type.
Speaker 1: But inside that record types can do exactly.
Speaker 2: The inventory example in the book nails it item name, string, count, integer, price, float,
Speaker 2: all stored together efficiently in a NUMPI array structure. It's
Speaker 2: great for tabular data.
Speaker 1: Okay, that makes sense. So we have these efficient arrays,
Speaker 1: how do we work with them? Slicing comes first feels
Speaker 1: familiar from lists.
Speaker 2: Yeah, for one dimensional arrays, slicing is exactly like Python lists. Yeah,
Speaker 2: grab a chunk using start, stop step.
Speaker 1: But it gets more interesting with multiple dimensions.
Speaker 2: Right, you can slice along each axis or dimension, and
Speaker 2: then you get into reshaping and flattening.
Speaker 1: Flattening like ravel or flatten That takes a multidimensional array
Speaker 1: and just makes it one long row.
Speaker 2: Yep, turns it into one D. But there's a really
Speaker 2: critical difference between those two you mentioned. Ravel might give
Speaker 2: you back a view of the original data. They could
Speaker 2: share the same memory, whereas flatten always creates a brand new,
Speaker 2: independent copy.
Speaker 1: And knowing that difference is vital, right, because changing a
Speaker 1: view changes the original absolutely vital.
Speaker 2: We definitely need to circle back to that view versus
Speaker 2: copy thing. It catches so many people out, Okay.
Speaker 1: You can also reshape arrays by setting the shape attribute
Speaker 1: directly or using re size and transpose, or just two
Speaker 1: for swapping rose and columns.
Speaker 2: Yeah, teat is super common, especially if you're doing anything
Speaker 2: with linear algebra.
Speaker 1: Then they're stacking combining a rays together, right.
Speaker 2: You can stab them horizontally side by side with.
Speaker 1: Each stack, or vertically one above the other.
Speaker 2: Withou B stack exactly, or even depth wise with D
Speaker 2: stack for three D arrays. The general function behind these
Speaker 2: is concatenat, where you explicitly say which access to join
Speaker 2: along and.
Speaker 1: The opposite is splitting them apart.
Speaker 2: Yeah, each split V split, D split, or the general
Speaker 2: split function takes a big array, cars it into smaller ones.
Speaker 1: Okay, these seem like the day to day tools for
Speaker 1: juggling array shapes.
Speaker 2: They really are your bread butter for manipulation.
Speaker 1: So let's double back, like you said to that really
Speaker 1: critical idea, views versus copies. The book really emphasis and for.
Speaker 2: Very good reason, is probably one of the most important,
Speaker 2: maybe subtle things to grasp to avoid weird bugs. Okay,
Speaker 2: when you do certain Numpi operations slicing is a big one,
Speaker 2: or using the view method. Explicitly, Numpie tries to be efficient.
Speaker 2: It doesn't want to waste time and a memory copying
Speaker 2: data if it doesn't have to.
Speaker 1: So it gives you a view. What does that actually mean.
Speaker 2: It means the new array object you get back, yeah,
Speaker 2: shares the same underlying data in memory as the original array.
Speaker 2: It's just looking at potentially at a different part of it,
Speaker 2: or maybe the same part with a different shape or
Speaker 2: data type.
Speaker 1: Which means if I changed the data in the view,
Speaker 1: you are changing.
Speaker 2: The data in the original array too, because it's the
Speaker 2: same data. It's not a separate snapshot.
Speaker 1: WHOA, Okay, that's huge.
Speaker 2: It really is. The Lena image example in the book
Speaker 2: makes it super clear. You take a slice of the
Speaker 2: image array, maybe representing her hat. You set all the
Speaker 2: pixel values in that slice to black, thinking you're just
Speaker 2: modifying the slice. But then you look at the original.
Speaker 1: Image and her hat is blacked out on the original too.
Speaker 2: Exactly because the slice was just a view into the
Speaker 2: original images data buffer.
Speaker 1: Okay, so how do you avoid that? If you don't
Speaker 1: want to change the original?
Speaker 2: You have to explicitly ask for a copy using the
Speaker 2: dot copy method. That tells numbpi, no, I want a
Speaker 2: completely separate version of this data in new memory. Then
Speaker 2: if you modify the copy, the original is totally unaffected.
Speaker 1: Right. So views are efficient but dangerous if you're not careful.
Speaker 1: Copies are safe, but use more memory and take time.
Speaker 2: That's the trade off. You need to know when you're
Speaker 2: getting a view and when you're getting a copy, and
Speaker 2: use dot copy when you need independence. It's fundamental.
Speaker 1: Oh, okay, fundamental. Indeed, Now beyond basic slicing, numb pi
Speaker 1: has more advanced ways to index arrays.
Speaker 2: Fancy indexing, yeah, fancy indexing. It basically lets you select
Speaker 2: elements using things other than simple integer slices. Primarily you
Speaker 2: use lists of indices or booleon arrays.
Speaker 1: So Instead of describing a block like two point five,
Speaker 1: I could give it a list, say one, five, seven,
Speaker 1: to pick out just those specific rows.
Speaker 2: Or columns exactly. You can pinpoint specific scattered elements. The
Speaker 2: books example using nymex with lists to kind of shuffle
Speaker 2: parts of the Lina image around visually shows this power.
Speaker 1: Right, and Boolean indexing that sounds like filtering, it is.
Speaker 2: It's incredibly useful for filtering data based on conditions. You
Speaker 2: create an array of true and false values.
Speaker 1: Usually by applying some comparison to your data array like
Speaker 1: data ten precisely.
Speaker 2: Yeah, and then you use that Booleon array as the
Speaker 2: index for your original data array.
Speaker 1: And numb I just gives you back the elements where
Speaker 1: the Booleon array was true.
Speaker 2: Yep, only those elements that meant the condition. The example
Speaker 2: of putting dots along the diagonal of the Lina image
Speaker 2: is a neat way to see its selecting pixels based
Speaker 2: on whether their row and column index are equal.
Speaker 1: Okay, that seems really powerful for data selection. Now stride
Speaker 1: tricks that sounds a bit more arecane.
Speaker 2: It is definitely a more advanced concept. Yeah, but the
Speaker 2: idea behind it is fascinating and it really shows off
Speaker 2: how numb pi thinks about memory. Okay, so we know
Speaker 2: numbpi stores array data in one contiguous block of memory,
Speaker 2: right because of howmogin eighty huh. Stride tricks let you
Speaker 2: create views of that same block of memory, but you
Speaker 2: tell numb pi to interpret it with a completely different structure.
Speaker 2: You do this by specifying the strides.
Speaker 1: Strides like how many bytes to jump to get to
Speaker 1: the next element.
Speaker 2: Exactly, how many bytes to step to get to the
Speaker 2: next element in the same row, and how many bytes
Speaker 2: to step to get to the next element in the
Speaker 2: same column, which is usually just the side of one row. Okay,
Speaker 2: with stride tricks, using functions like a strided, you can
Speaker 2: manipulate those step sizes. You can tell numb pi to
Speaker 2: get to the next element in this dimension step forward
Speaker 2: this many bytes, even if that overlaps with previous data
Speaker 2: or creates a totally different logical layout.
Speaker 1: The Sudoku example in the book was wild taking a
Speaker 1: nine by nine grid.
Speaker 2: Yeah, and using a stride in to make numb PI
Speaker 2: see that same nine by nine block of memory not
Speaker 2: just as nine rows of nine numbers, but as an
Speaker 2: array of three by three.
Speaker 1: Squares without copying any data without.
Speaker 2: Copying anything, You're just giving NUMPI a new recipe, new
Speaker 2: strides for how to walk through the existing memory to
Speaker 2: precede these three by three blocks.
Speaker 1: Wow, So that efficient memory layout isn't just for raw speed.
Speaker 1: It enables these incredibly clever ways to access structured data
Speaker 1: within the array exactly.
Speaker 2: It really highlights how numb pile leverages that contiguous memory.
Speaker 2: You can instantly get all the three by three blocks
Speaker 2: or overlapping windows for signal processing just by defining the
Speaker 2: right strides. It raises that question, doesn't it How this
Speaker 2: simple contiguous block enables such complex views?
Speaker 1: Mind blown slightly? Okay? One more fundamental concept broadcasting. This
Speaker 1: one seems simpler but pops up everywhere it does.
Speaker 2: Broadcasting is how numb pie handles operations like addition or
Speaker 2: multiplication between arrays that don't have the exact same.
Speaker 1: Shape, like adding a single number to every element in
Speaker 1: an array.
Speaker 2: That's a classic example, or multiplying an entire array by
Speaker 2: a scaler. The rule basically is that numbpi tries to
Speaker 2: stretch or duplicate the smaller arrays dimensions so that its
Speaker 2: shape becomes compatible with the larger array. For the element
Speaker 2: wise operation.
Speaker 1: The audio volume example is perfect for this. You have
Speaker 1: an array of audio.
Speaker 2: Sample right, maybe thousands of numbers, then you.
Speaker 1: Just multiply it by point two to make it quieter.
Speaker 2: Yep, numb Pi doesn't actually create a massive array filled
Speaker 2: with point two's to match the audio data size. That
Speaker 2: would be really inefficient. It just understands the broadcasting rule.
Speaker 2: It sees you're multiplying an n dimensional array by a scaler,
Speaker 2: which is like a zero dimensional array. It knows to
Speaker 2: apply that scaler multiplication to every single element of the
Speaker 2: n dimensional array.
Speaker 1: In one go, using that fast C code again exactly.
Speaker 2: So what does this all mean for you? It means
Speaker 2: you can write really intuitive code like audio dada zero
Speaker 2: point two or ray plus five without writing loops. Numb
Speaker 2: Pile figures out how to make the shapes compatible efficiently.
Speaker 2: It makes a ray math much cleaner and faster.
Speaker 1: Okay, so we've got the foundation efficient arrays, data types, manipulation,
Speaker 1: the crucial views versus copies, fancy indexings, broadcasting. That's a
Speaker 1: powerful toolkit just on its own.
Speaker 2: Absolutely.
Speaker 1: Now let's talk about putting it to work. The book
Speaker 1: moves into actual data analysis prediction linking up with other
Speaker 1: librarries right.
Speaker 2: Applying these tools. The basic data analysis example using weather
Speaker 2: data from a station in the Netherlands to built I
Speaker 2: think it's very practical.
Speaker 1: Shows how you load data from a file maybe a
Speaker 1: CSV or a text file using load TX.
Speaker 2: And it immediately hits a real world issue, messy data
Speaker 2: missing values.
Speaker 1: Yeah, which happens all the time. In that data set,
Speaker 1: missing values were marked with like meta I or something yeah.
Speaker 2: Some special code, and the book shows how you typically
Speaker 2: handle that, maybe filter them out or more often convert
Speaker 2: them into NUMBPI special NAN value not a number, right nan,
Speaker 2: because numbpie's math functions often know how to handle NaN's correctly,
Speaker 2: like ignoring them when calculating a mean.
Speaker 1: Okay, so data loaded cleaned up a bit, then doing
Speaker 1: the actual analysis is easy, super easy.
Speaker 2: With NUMPI wenty average temperature maxwindspeed dot max standard deviation
Speaker 2: dot STV. You apply these functions directly to your arrays
Speaker 2: or columns of data.
Speaker 1: The example showed calculating things like the daily temperature range
Speaker 1: max minus men or looking at yearly averages.
Speaker 2: Yeah. And while you know one station's data isn't proof
Speaker 2: of global warming or anything of course, it gives you
Speaker 2: a taste of using these tools for exploring trends. You
Speaker 2: could do the same for wind, pressure, humidity, whatever's in
Speaker 2: the data sets.
Speaker 1: Okay, so that's understanding past data. What about predicting the future?
Speaker 1: The book touches on simple predictive analytics.
Speaker 2: Yeah, moving from description to forecasting. The core idea is
Speaker 2: using historical patterns to guess what might happen.
Speaker 1: Next, like with the temperature data exactly.
Speaker 2: The book mentions basic concepts like autoregressive models or ar models.
Speaker 2: Simple idea, predict tomorrow's temperature based on today's yesterday's the
Speaker 2: day before.
Speaker 1: Using past values to predict the future.
Speaker 2: Right, and it hints at how you'd actually fit a
Speaker 2: model like that to your data. This often involves bringing
Speaker 2: in tools from the wider scientific Python world like SIP. Yeah,
Speaker 2: maybe using something like pip dot optimize at least sq
Speaker 2: to find the model parameters that best match the historical data.
Speaker 2: It also mentions that tools like pandas, which builds on NUMPI,
Speaker 2: are great for summarizing data and looking for correlations before
Speaker 2: you even start modeling.
Speaker 1: Makes sense. Another area is signal processing, analyzing data that
Speaker 1: changes over time, often with cycles right.
Speaker 2: Like the sunspot data example. Sunspots have these known cycles
Speaker 2: roughly eleven.
Speaker 1: Years, and signal processing techniques help analyze those patterns.
Speaker 2: Yep. The book mentions smoothing like using a moving average
Speaker 2: to filter out short term noise and see the underlying
Speaker 2: trend more clearly.
Speaker 1: Though it notes simple moving averages aren't always the best
Speaker 1: for cyclical data like sunspots.
Speaker 2: Correct they can distort the peaks and troughs, so it
Speaker 2: hints it more advanced stuff like decomposing a signal into
Speaker 2: its core components. It mentions techniques like EMD empirical mode
Speaker 2: decomposition to break down the sunspot signal into intrinsic mode
Speaker 2: functions or IMFs to better analyze those cycles.
Speaker 1: And NUMPI is doing the heavy lifting numerically for these
Speaker 1: algorithms exactly.
Speaker 2: It provides the array operations needed to implement these complex
Speaker 2: signal processing methods.
Speaker 1: It really feels like NUMPI isn't just standalone, It's like
Speaker 1: the central hub for this whole ecosystem of scientific Python tools.
Speaker 2: That's a great way to put it. It's the common language,
Speaker 2: the common data structure SCIPI, as we mentioned, builds directly
Speaker 2: on NUMPI. That's tons more advanced scientific tools.
Speaker 1: Like what kind of things, oh.
Speaker 2: Numerical integration, solving, differential equations, interpolation, optimization algorithms, more linear algebras,
Speaker 2: statistical functions, stuff that goes beyond numbpi's core array focus.
Speaker 1: And psychic Learn, the big machine learning.
Speaker 2: Library hugely reliant on NUMBPI. Almost everything in psychic learn
Speaker 2: expects input data as NUMBPI arrays your features, your targets,
Speaker 2: and it often outputs for results as numbi erays two
Speaker 2: predictions model coefficients. The book points to examples like clustering
Speaker 2: stock data or using Psychic's image which is related for
Speaker 2: image processing, like finding corners in a picture, all powered
Speaker 2: by numb pi arrays underneath.
Speaker 1: And what if Python itself, even with numbpis C backend,
Speaker 1: isn't fast enough for some really critical part of your code.
Speaker 2: That's where Cithon comes in Python. It's a language that's
Speaker 2: kind of a mix of Python and C. You can
Speaker 2: write Python like code, add some static type declarations, and
Speaker 2: Cython compiles it down to.
Speaker 1: Efficient C code, and it works well with NUMPI.
Speaker 2: Very well because NUMPI arrays already have that underlying C structure.
Speaker 2: Cython code can operate on the array data directly at
Speaker 2: C speeds without the Python interpreter overhead. It's great for
Speaker 2: optimizing bottlenecks or for wrapping existing C or C plus
Speaker 2: libraries to use them from Python.
Speaker 1: So the ecosystem is NUMBPI at the core, SIPI for
Speaker 1: more science, math tools, psychic learn for mL, Cython for
Speaker 1: speed optimization. It's quite layered, it is.
Speaker 2: And it's still evolving. The book even looks ahead, mentioning
Speaker 2: projects like Blaze Blaze.
Speaker 1: What's the idea there?
Speaker 2: The goal is to take Numpie's array based way of
Speaker 2: thinking and extend it to data sets that are too
Speaker 2: big for memory.
Speaker 1: Ah big data territory or streaming.
Speaker 2: Data exactly, applying similar principles of efficient array oriented computation
Speaker 2: to distributed systems or data streams. It shows that this
Speaker 2: core idea started by Numpi is still expanding.
Speaker 1: That's really cool. Okay, Before we wrap up one less
Speaker 1: practical point, the book covers good development practices Profiling, debugging, testing, Yeah.
Speaker 2: Really important, especially with numerical code, where small errors can
Speaker 2: sometimes lead to big problems or just slow things down unnecessarily.
Speaker 1: Profiling helps you find where your code is spending its
Speaker 1: time the bottlenecks right.
Speaker 2: Debugging is for tracking down errors when things go wrong,
Speaker 2: and testing. Writing automated test is crucial for making sure
Speaker 2: your code actually works as expected and stays working when
Speaker 2: you make changes later.
Speaker 1: And Python and its ecosystem have tools for these.
Speaker 2: Definitely, Python has built in profilers I Python has magic
Speaker 2: commands like percent debug for jumping into the debugger right
Speaker 2: after an error. There are standalone debuggers like a PDB,
Speaker 2: and for testing, Python's unitist module is standard, but libraries
Speaker 2: like nose or pie test are very popular, especially in
Speaker 2: the scientific community. They have helpful tools like special functions
Speaker 2: for comparing floating point arrays where exact equality is often tricky.
Speaker 2: These are all essential habits for writing robust code.
Speaker 1: Wow, okay, that was definitely a deep dive. We started
Speaker 1: with that fundamental idea, the heniray giving this huge efficiency
Speaker 1: boost over lists.
Speaker 2: Yeah, thanks to homogeneity those vectorized operations the se core.
Speaker 1: We looked at the arrays, structure, its attributes, how to
Speaker 1: manipulate it with reshaping, stacking.
Speaker 2: Grappled with that critical views versus copies distinction definitely critical.
Speaker 1: Then dove into advanced indexing, fancy indexing, boolean indexing, those
Speaker 1: mind bending stride tricks, and the super useful.
Speaker 2: Broadcasting, and then saw how it all comes together for
Speaker 2: real work, basic data analysis, hints of prediction and signal.
Speaker 1: Processing, and its absolutely central role in that wider scientific
Speaker 1: Python world, connecting with SCIPI, psychic learn, cithon.
Speaker 2: Even looking forward with things like Blaze.
Speaker 1: You know, pulling these insights from the book, it really
Speaker 1: feels like understanding why numbpi works this way. The views,
Speaker 1: the broadcasting, the see back end is the key. It's
Speaker 1: the shortcut to writing code that's not just correct, but
Speaker 1: also fast and well pythonic in this numerical context.
Speaker 2: Absolutely so. Thinking about all that power using Numbpi and
Speaker 2: friends for complex analysis, prediction, image stuff, integrating with all
Speaker 2: these libraries, and the future potential for even bigger data,
Speaker 2: it makes you wonder, right, what kinds of problems that
Speaker 2: maybe seem overwhelming today might actually become solvable with these
Speaker 2: tools tomorrow? What for you stands out as the most
Speaker 2: surprising or maybe powerful capability We touched on