For technical reasons, subindices cannot be used. Please be mindful of this when seeing expressions such as $a1$ or $bn$.

The controversial identity

The identity $0.999\dots=1$ seems to be a controversial one, at least in the math-oriented demographics on the internet. Some people argue that $0.999...$ cannot be equal to $1$, since it sits ever so close but never reaching $1$. One can see this by iterating over its decimal digits: $$0.9, 0.99, 0.999, 0.9999, 0.99999, \dots$$ This argument summons -knowingly or otherwise- the existence of surreal numbers, popularised by D. Knuth on his eponymous 1974 book.

While the topic of surreal numbers is undeniably very interesting from the point of view of real analysis (see this article for a general, albeit somewhat technical overview of what “works” and what “does not” when one considers surreal numbers), it fails to address that assuming the field of real numbers when speaking about numbers with decimals is an unspoken convention which is understood by context. Indeed, just like how one does not specify between degrees Celsius or Fahrenheit when discussing the weather in the school cafeteria, one does not specify the real number $0.7316\dots$ during a Calculus I class.

On the other end of the scale, one finds the mathematics popularisers, which most often justify the identity $0.999\dots=1$ by using a geometric argument: if we set $x=0.999\dots$, then $10x=9.999\dots$. The algorithm of subtraction then implies that $9x=10x-x=9$, and thus $x=1$. However, some have pointed out that this argument makes the use of unjustified implicit assumptions. Indeed, the indiscriminate use of the geometric argument can lead to identities such as $1+2+3+\dots=-1/12$ ([1,2,3]).

While this identity has a place in physics and can be interpreted as a valid expression under the correct assumptions, this identity remains as a cautionary tale of how computations are only valid so long as they are framed in their appropriate context. A more approachable example is perhaps how Pythagoras' Theorem states that $a^2+b^2=c^2$, but when solving for $c$, we need to rule out the negative solution since $c$ is meant to be a length, hence $c > 0$.

Making sense out of infinite decimals

We begin by addressing the topic of infinite -repeating or otherwise- decimals. We begin with a number with a finite amount of decimals: $$0.619=0.6+0.01+0.009=6\cdot\frac{1}{10}+1\cdot\frac{1}{100}+9\cdot \frac{1}{1000}$$ We can extend this logic to define a number with infinite, repeating decimals, $$0.999\dots=0.9+0.09+0.009+\dots=\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\dots$$ This defines an infinite sum, which converges to a real number between $0$ and $1$. The same logic can be used with any real number. Moreover, this infinite sum has a special property: it is absolutely convergent.

To see what this means, we consider the alternating harmonic series, $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots$$ It is well known that if we swap all the $–$ signs for $+$ signs, this sum diverges; it is the well-known harmonic series. Nonetheless, if we leave it as-is, this sum converges to the natural log of $2$. This is called conditional convergence. On the other hand, the sum $$1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\dots$$ converges regardless of the signs. This is called absolute convergence.

In particular, the number $0.a1a2a3\dots$ can be written as the infinite sum $$ \frac{a1}{10} + \frac{a2}{100} + \frac{a3}{1000}+\dots $$ where $a1, a2, a3, \dots $ is an integer between $0$ and $9$. Therefore, any number defines an absolutely convergent infinite sum.

When it is allowed to use the subtraction algorithm

The algorithm of subtraction with carryover does not behave well for numbers with infinite, non-repeating digits (such as $\pi$), since this algorithm require us to start at the rightmost digit (which does not exist in a number with infinitely many decimals). The algorithm of subtraction without carryover does work when both numbers have a finite sequence of repeating decimals, such as $0.545454\dots$ (which we shall denote $0.[54]$ for simplicity) or $0.888\dots$.

We shall denote the number $0.a1a2\dots ana1a2\dots an\dots$ as $0.[a1\dots an]$.

Theorem. Let $$S1=a1+a2+a3+\dots;\quad S2=b1+b2+b3+\dots$$ be absolutely convergent infinite sums. Then, $$S1-S2=(a1-b1)+(a2-b2)+(a3-b3)+\dots$$ In particular, if $x=0.[a1\dots an]$ and $y=0.[b1\dots bn]$ with $a1 \geq b1$, ..., $an \geq bn$, then $$x-y=0.[(a1-b1)\dots (an-bn)].$$

Note that we have assumed for simplicity that all numbers are between $0$ and $1$, but any real number $x > 1$ can be written as $x=n+0.a1a2a3...$ for some natural number $n$. Thus the theorem above works for, say, $42.[54]$.

There is another result we still need to finally justify the popular method used to prove that $0.999\dots=1$:

Theorem. Let $$S=a1+a2+a3+\dots$$ be an absolutely convergent infinite sum, and $\alpha$ be a real number. Then, $$\alpha S=(\alpha a1)+(\alpha a2)+(\alpha a3)+\dots$$

Observe that neither of these theorems holds without the condition of absolute convergence. If we choose $S=0.[54]$ and $\alpha=100$, then $$\alpha S=100\left(\frac{5}{10}+\frac{4}{100}+\frac{5}{1000}+\frac{4}{10000}+\dots\right)=$$ $$=5\cdot 10+4\cdot 1+\frac{5}{10}+\frac{4}{100}+\frac{5}{1000}+\frac{4}{10000}+\dots=54.[54].$$ More generally, if $S=0.[a1\dots an]$ and $\alpha=10^n$, then $\alpha S=a1\dots an.[a1\dots an]$. This not only justifies the step used in the title (that $10x=9.999\dots$), but it also justifies the algorithm see in middle school to turn an infinitely repeating decimal number to fraction form!

Conclusion

Despite my best intentions, this has turned out to be a rather lengthy and technical explanation. I can only hope that the examples left out along the way (and more importantly, the references) are enough for a high school student with an interest in mathematics to grasp the gist of what is going on.

At its core, the problem with $0.999\dots$ is that it is a subtle way to write a limit, an expression that infinitely approaches a value. Zeno's arrow paradox echoes in the words of those who struggle to grasp the idea that $0.999\cdot$ can be equal to $1$, as indeed how can a number which stands apart from $1$ at each step could somehow reach that value.

There are indeed mathematical models where this can and does happen, but the real numbers have a property known as sequential closure, meaning that a series of numbers which inches ever-so-close to a fixed value $x$ must inevitably be equal to $x$.

Last night, this thread on Mastodon inspired me to pull out my copy of The Structure of Evolutionary Theory and read what Gould had to say about “evolutionary psychology” towards the end.

It's great, so I wanted to quote it in its entirety:

As a primary correlation regulating the distribution and importance of spandrels vs. primary adaptations, increasing complexity of an organ must imply a rising relative frequency of nonadaptive side consequences with potential future utility. With greater complexity in number and form of components, cooptable side consequences must rise to exceed, or even to overwhelm, primary adaptations. The chief example in biology may be a unique feature of only one species, but we obviously (and properly) care for legitimate reasons of parochial concern. The human brain may have reached its current size by ordinary adaptive processes keyed to specific benefits of more complex mentalities for our hunter-gatherer ancestors on African savannahs. But the implicit spandrels in an organ of such complexity must exceed the overt functional reasons for its origin. (Just consider the obvious analogy to much less powerful computers. I may buy my home computer only for word processing and keeping the family spread sheet, but the machine, by virtue of its inherent internal complexity, can also perform computational tasks exceeding by orders of magnitude the items of my original intentions—the primary adaptations, if you will—in purchasing the device.)

A failure to appreciate the central role of spandrels, and the general importance of nonadaptation in the origin of evolutionary novelties, has often operated as the principal impediment in efforts to construct a proper evolutionary theory for the biological basis of universal traits in Homo sapiens—or what our vernacular calls “human nature.”

I welcome the acknowledgment of self-proclaimed “evolutionary psychologists” (compared with the greater stress placed by the “sociobiology” of the 1970's on a search for current adaptive value) that many universal traits of human behavior and cognition need not be viewed as current adaptations, but may rather be judged as misfits, or even maladaptations, to the current complexities of human culture. But most evolutionary psychologists have coupled this acknowledgment with a belief that the origins of such features must be sought in their adaptive value to our hunter- gatherer African ancestors. (Much of the daily practice of current “evolutionary psychology” focuses upon efforts to identify and characterize the EEA (their term), or “environment of evolutionary adaptation,” for the origin of cognitive universals as direct adaptations in the common ancestral population of all modern humans.)

I applaud this use and recognition of the Nietzsche-Darwin principle of discordance between reasons for historical origin and bases of current utility (or disutility). But I also believe that “evolutionary psychology” will remain limited and stymied in its worthy and vital goal—to understand the human mind in evolutionary terms—so long as its practitioners place such unwarranted and effectively exclusive weight upon conventional adaptationist explanations for the origin of universal cognitive traits, and fail to recognize the central role (I would say dominant, but the issue obviously remains open) of constraints and nonadaptations in the initial construction of the cognitive and emotional modules and attributes that we collectively designate as “human nature.”

A central principle about constraint from each of my two chapters (10 and 11) on the subject would broaden the range of hypotheses and lead to a richer and ultimately more accurate “evolutionary” psychology, both in immediate empirical terms of understanding the human mind, and in conformity with the true depth and range of modern evolutionary theory, rather than invoking an almost caricatured version of adaptationism as the only ground of evolutionary explanation for the origin of traits.

1. At a sufficient depth and distance, original adaptations now act primarily as historical constraints, and must be so characterized and analyzed (the central theme of Chapter 10). When we recognize a cognitive universal of human mentality as ill-fit to the complexities of modern social life, we do not then achieve an explanation of its human origin in adaptationist terms simply because we can state a good case for its initial phyletic appearance as an adaptation. We need to specify the evolutionary distance and the environmental context of initial appearance before we can render any judgment. In general, I would accept the statement that if we can locate the feature's adaptational origin in the last common ancestor of Homo sapiens, or even as far back as the common ancestor of the hominid line (after splitting from the lineage of great apes), then we may legitimately argue that this initial adaptive context establishes the “evolutionary meaning” of the feature in our quest to understand its appearance in human phyletic history.

But suppose that the feature had a far more ancient, but still fully adaptational, origin in a distant ancestor of very different form and neurological function, and also living in a very different environment—say, in the basal gnathostome fish of early Paleozoic times. Suppose also that this mental attribute has persisted ever since as a plesiomorphic aspect of the basic operation of the vertebrate brain. When we then try to explain the evolutionary significance of this mental mode in contemporary human life—especially when we try to identify its role in quirky and clearly suboptimal characteristics of human reasoning in the modern world—would we wish to claim that an adaptational analysis (in recognition of the feature's Darwinian origin in such a distant ancestor) will provide our best understanding? Clearly, we do not so proceed in most evolutionary analyses—and for good reasons discussed at length in Chapter 10 on the evolution of development. Rather, we treat such features predominantly as historical constraints because, as invariant and plesiomorphic traits of our entire clade (not only of all hominids and primates, but also of all mammals and tetrapods), they operate as unchanging constraints upon any subsequent evolution of mental modes, despite their adaptational origin in such a distant ancestor of such different form and environment.

I suspect that many puzzling features of human mentality would be better resolved if we conceptualized them as historical constraints derived from distant adaptational origins. To cite a hypothetical example (that would attract my substantial and favorable wager were I a betting man): I agree with a major theme of structuralist philosophy and research, as developed most cogently in our times by Claude Levi- Strauss and his followers, that identifies our tendencies to parse natural variety into pairs of opposed and dichotomous categories as an inherent property of human mental functioning—with male and female, night and day, and culture vs. nature as primary examples. I think that most people would also identify this strong preference as a constraint with highly unfortunate consequences for human life—not only because we so often construct invalid dichotomous taxonomies in our real world of complex continua, but primarily because we so often impose another conceptual module for moral judgment upon our pairings (the Manichean good vs. bad), and then proceed to identify one side of the dichotomy (including ourselves and our preferences) as righteous, and the other side (including “foreigners” and competitors) as worthy of anathematization or even ripe for burning. (I need hardly add that yet another aspect of human mentality, our capacity to devise grisly means of death and torture, and our technological ability to apply such means to large numbers of people in short periods of time, makes our innate preferences for dichotomization particularly dangerous.)

Now I am perfectly willing to believe that our brain's preference for dichotomization arose as a highly adaptive attribute in a very distant and ancient small-brained ancestor that, to enhance its prospects for survival, needed to make limited, quick, and twofold decisions that exhausted the maximal capacity of its judgment in any case: mate or wait, eat or sleep, fight or flee. But, whatever the adaptational basis of origin, dichotomization then persisted throughout the subsequent phylogeny of vertebrates as a historical constraint that became more and more quirky, and more and more limiting, as the brain enlarged into the much more sophisticated instrument of a lineage that eventually generated our exalted, but curiously freighted, selves.

1. At the level of immediate reasons for persistence and flourishing of the hunter-gatherer common ancestor of Homo sapiens in Africa, many distinctive mental attributes of our species, including major features of “human nature” that define our evolutionary success, must have arisen as nonadaptive spandrels (later exapted, in several cases, as vital bases of our current domination), and not as primary adaptations (the central theme of Chapter 11). This conclusion necessarily follows from the previous argument that, at the level of maximal natural complexity represented by the human brain, consequential spandrels must, at least in number, overwhelm the primary adaptations that generate them. Therefore, in terms of exaptive potential for evolutionary futures, the brain includes more cooptable spandrels than primary adaptations. Any “evolutionary psychology” that neglects the nonadaptational origin of many features now useful (or at least used, however dubiously), and that limits the domain of evolutionary inquiry to arguments (often speculative) about initial adaptive causes and benefits, will become more misleading than enlightening in restricting investigation to such a narrow scope of inquiry. We must abandon the largely unconscious bias of an overly strict Darwinian approach that equates all “evolutionary” explanation with adaptationist analysis.

[Edited to fix some misformatting]

I was wanting to compare the real cost of some different teas and coffees recently, and thought I'd write up the process I used. First, the amount used in one session

1 session tea = 5g

This could be a 100mL gaiwan, or a half-litre pot made Western style.

1 session coffee = 20g

This is 3 cup moka, or a 3dL (or 12 oz) mug of filtre coffee. This could also be a strong double espresso. If you make espresso, and you use the classic proportions, that would instead give:

1 session coffee = 15g

Depending on how you make your coffee and your tea, you can adjust these to match what feels to you like equivalent preparations. Using these ratios, we can now compare prices!

5g tea * 4 = 20g coffee 1000/4=250 price/g tea * 250 = price/kg coffee

Examples

I have some Dong Ding that we like a lot, which I got for 600NTD for a 150g packet. That's 17.53€, or 0.1167€/g. Converting that into an equivalent kg of coffee, 0.1167*250=29,25€/kg, which is about 1€ less than the bag of coffee we're currently drinking. Nice!

What about pu-erh prices? There are still some perfectly nice factory bings available for around 18USD : 18USD/357g = 0.0504 USD/g. That gives an equivalent coffee price of $12.60 or about 11.80€ per kg. That's some very cheap coffee! Personally, I'd much rather drink the factory pu-erh.

Finally, let's look at converting coffee prices to tea. I'm currently drinking a 31€/kg coffee, and I generally buy in the 20-40€/kg range. 20/250=0.08, so that gives a range of 0.08-0.16€/g, or 0.085-0.17 USD/g. The equivalent 150g packet of oolong would be 12-24€ or 411-822 NTD. But we already know that the middle of that range is exactly what I paid for my Dong Ding. A 357g bing would be 28.50-57€ or 30.50-61 USD; a 400g bing would be 34-68 USD.

Nespresso

Nespresso charges eye-watering prices per kg, their ordinary offerings starting at 89.60€/kg. But to be fair, you're not using 20g of coffee per session! One capsule contains a mere 4.8g of coffee, so an equivalent double Nespresso gives a tea-conversion factor of 520. The entry-level Nespresso capsules are then equivalent to 89.6/520=0.172€/g tea; that's the same as a 0.172€/g * 357g = 61.45€ or $65.50 bing.

Edited to change the title from “FUD” to “misinformation” as the former might be interpreted as implying a commercial motivation, which is not my intention

This started as a reply to a thread on fedi, but it got long so I moved it to the blog. The thread describes a blog article in Chinese

I’m judging by this article alone (and in translation), so take this with a grain of salt, but this guy looks like a total crank.

Where to even start. I'll set aside the half of the article that's setting up his bona fides as the aggrieved party or attacking his critics, because I'm neither familiar with the history nor with sinosphere “discourse” to judge what's going on there, except to say: so-and-so's wife even chimed in, and she's just a singer? Assuming there wasn't a massive failure of machine translation going on, this reeks of both credentialism and anti-woman sentiment.

Fermented food production

This is the biggest red flag for me. He asserts that pu'erh production is completely out of control, and that normal fermented foods must be inoculated to control the fungal composition. This is flat out wrong.

Traditional methods of fermentation have been widely studied, from lacto-fermentation, to cheese production, wines, beers, etc. Sometimes the steps performed alone can produce a reliable fermentation with a predictable mix of bacteria and fungi (as in sauerkraut, kimchi, sour pickles, etc). Sometimes, as with spontaneous beers, and most traditional European cheese production, the environment in which the production takes place plays a key role. In these cases, there is inoculation, but it is informal, and happens from the ambient environment. It is not however, less effective for it.

All of this is well studied.

Shupu

Shu is produced in specific facilities. They are able to turn out a consistently similar product, batch after batch, year after year. This isn't proof of controlled fermentation, but it is highly suggestive. That is what traditional controlled fermentation looks like.

I started seeing studies about the fungal composition of pu'erh tea maybe 20 years ago, and it always seems to contain mostly the same mix: Aspergillus and Penicillium in particular. Here's one random example from about a decade ago.

This is what one would expect from controlled fermentation.

Shengpu

I'm a bit more concerned about sheng, and here he may have a point, to the extent that the post-fermentation is performed at home by non-experts. The warehouses and cellars that produce reliable aging are probably more consistent. Given its age, I'd expect Yee On's cellar for example to inoculate teas with a pretty consistent profile of spores. I believe the studies on shengpu correctly draw their samples from these larger facilities, which is how most sheng is aged. But that probably leaves a larger gap in the data about home-aged teas.

Whither contamination

He just asserts that pu'erh is widely contaminated with carcinogens. From the studies I've seen, this is false. He presumes that this comes from uncontrolled fermentation, and thus doesn't apply to other tea types. The fermentation of pu'erh is certainly well controlled for shu (the results would be disgusting otherwise), and for sheng it really depends on the history of a particular sample.

But what about other teas? This is an interesting study on mycotoxins in various tea styles. The most contaminated tea was a black tea. In particular, the mycotoxin contamination of tea appears to occur in the fields, in the actual growing of the plants. If that is really the primary vector, then no tea style would be spared, and the focus on pu'erh is (almost) completely unwarranted. Almost, because poorly aged shengpu could of course continue to grow the hazardous fungi from the fields. That is a hypothetical, however, and so far the worst offenders seem to be badly produced black tea.

When I first started learning analytic geometry (the complex analogue to algebraic geometry), I recall asking my advisor why the local intersection number is defined with no regards for orientation. At the time, he simply said that complex manifolds are assumed to be positively oriented, which I didn't pay much attention to since I was too busy trying to wrap my head around more elementary concepts, such as the Zariski topology of the spectre of a Noetherian ring.

Fast forward to a few days ago. A friend was telling us about Chern classes, and he mentioned once again orientation and holomorphic manifolds. However, he was more specific on the question:

...and we can assume that [a complex manifold] always has a well-defined orientation since “the complex numbers have an implicit orientation”, or something like that [sic].

I recently happened to have attended a mini-course about Floer homology, which involves endowing real manifolds of even dimension with a complex structure on the tangent spaces, so I realised that you could use the latter to explain the former. For our intents and purposes, a complex structure is a $2\times 2$ matrix $J$ such that $J^2$ is the identity matrix with a minus sign.

Any complex number $x+iy$ can be represented in the real plane in two ways: as a vector $(x,y)$ and as a matrix

$$J(x+iy)=\begin{pmatrix}x&-y \\ y&x\end{pmatrix}.$$

In particular, $J(i)$ is a complex structure (as per our definition)! Not only that, but we also have $J(a+ib)=a J(1)+ b J(i)$, so complex multiplication can be recovered using this $J$ matrix. Note that $J$ (seen as a mapping $\mathbb{R}^2 \to \mathbb{R}^2$) describes an counter-clockwise rotation, which corresponds to the positive orientation on the plane.

The transpose of $a+ib$ can be described in terms of the $J$ matrix as $aJ(1)+bJ(-i)$. However, $J(-i)$ corresponds to a clockwise rotation, which is the negative orientation on the plane.

Since the mapping $z \mapsto \overline z$ is not holomorphic, this means that only one of the two orientations on the plane is compatible with the complex derivative!

I put together a quick SRPM for building opus-tools from the github master. The source RPM is here: opus-tools-master-git.src.rpm. To use:

    # Unpack the sources
    rpm -ihv opus-tools-master-git.src.rpm
    
    # Optionally update to the latest from github
    cd ~/rpmbuild/SOURCES
    wget https://github.com/xiph/opus-tools/archive/refs/heads/master.zip
    
    # Build the rpms
    cd ~/rpmbuild/SPECS
    rpmbuild -ba opus-tools.spec
    
    # Install the generated rpm
    sudo dnf install ~/rpmbuild/RPMS/$(arch)/opus-tools-master-git.$(arch).rpm

Additionally, you’ll probably want to add exclude=opus-tools to /etc/dnf/dnf.conf to prevent dnf from trying to “upgrade” you back to the system-provided opus-tools.

There’s a small change to the .spec file to use the latest source download from github, and there’s a small patch to configure.ac to correctly find libm. That’s it!

[Edited in 2023 to add more temperatures for green teas and such]

Here’s some simple recipes for water at different temperatures. If you can measure in grams or mL, you can make about one litre of water at common tea temperatures pretty easily. The power of stoichiometry and basic maths!

Preparation

Have some filtered water at room temperature (20 C ideally, but 16~30 C is fine), and bring at least 1 L to a boil. Pick a container that can hold a bit more than 1L, and wash/preheat it by pouring some of the boiled water into it, then pour it out. If your room is somewhere between 18 and 30 C, you’re fine.

70 C

Mix 700 mL of boiling water with 400 mL room temperature water. This will make 1,1L of 70 C water.

75 C

Mix 700 mL of boiling water with 300 mL room temperature water.

80 C

Mix 750 mL of boiling water with 250 mL room temperature water.

85 C

Mix 900 mL of boiling water with 200 mL room temperature water. This will make 1,1 L of 85 C water.

90 C

Mix 1L of boiling water with 150 mL room temperature water.

95 C

Mix 950 mL of boiling water with 50 mL room temperature water.

The easiest tea to get started brewing is probably rolled oolongs. The second easiest is probably twisted leaf oolongs. So I’ll start with giving brew recipes for these two tea types.

Rolled

The Taiwan oolongs I sent are rolled into small balls, which is typical. A wash is especially important for this style of tea, because it gets the leaf to very literally open up. The amount of tea you put in will look very small at first because it’s compacted. By the second brew it will nearly fill the whole gaiwan or pot.

Rinse for 5-10 seconds, the let the leaf steam for a minute. The first brew should be 30 seconds, the second 40, the third 50, and the fourth 60. These are the brews that have the most typical, nutty-sweet taste associated with high-mountain Taiwan oolongs.

Starting with the fifth brew, things go a little longer: 75 seconds, 90 seconds. If the tea still has a nice taste to you (especially likely for the aged tea), you can keep going: 2 min, 3 min, 4 min.

If you stopped sooner, you can take the used leaves, put them in a litre of room temperature water, and have cold-brew tea tomorrow.

Whole leaf

The Wu Yi oolongs brew a bit quicker, but are still fairly easy.

Rinse for 2-5 seconds, and save the rinse water. This has the flavor of the roast, which in the case of the carefully roasted teas I sent, is interesting. Especially for the aged one. I’d recommend setting it aside, and drinking it at the end of the session.

The first few brews go quicker: 20, 25, 30, 35, 40 seconds. For the next suite, start adding 10 seconds per brew: 50, 60, 70, 80, 90. For the young tea, you might stop here. For the older one, certainly keep going: 105 seconds, 2 min, 2:30, 3:00, 4:00. Keep going or stop when you think the tea is finished.

Adjustments

If you find the taste overwhelming or too bitter at any point, repeat the time for the next brew. For example, if you find the third brew of the Wu Yi too much, instead of doing 20, 25, 30, 35, 40, 50, … you should instead try 20, 25, 30, 30, 35, 40, 50, etc.

If you find the brew too weak, first make sure your water is hot enough. Then skip a number: 20, 25, 30, 50, 60, 70, etc. If you find yourself skipping numbers a lot, you’ve got stronger taste in tea than I do, and we might want to adjust your recipes!

I could write too much just about brewing tea, but I’ll try to keep things simple, and with a focus on having a good experience with the tea. This post is a detailed overview of the technique I’ll have you use, then in later posts I’ll give specific brewing recipes. There are many other ways to do things, but this will work and give you good results.

Water

Use filtered water, or failing that, low-mineral bottled water. Just ordinary good drinking water that doesn’t taste like chlorine.

The teas I sent can all be brewed at boiling: that is, you bring the water up to a boil, then turn your kettle down to keep it just under a boil. If you have a whistling kettle, it should just barely make any noise. Another way to keep your water up to temperature is to use a vacuum thermos: fill it with boiled water, wait 20 seconds, dump it out, the refill and seal it. You can also just bring the water briefly back up to a boil when you need to; that’s slightly worse than the other techniques, but don’t worry about the difference for now.

Equipment

You’ll need a brewing vessel of about 100mL (80~120 or so). And a cup to drink from. A full set-up with a pitcher to pour the tea into, and a filter, is better.

The process

These teas are all brewed similarly: pour the hot water into the teapot or gaiwan. Once it’s heated up, pour the water into your pitcher, and from there into the cup(s). Put one session’s worth of dry leaf in the pot, cover it, and let it heat for 30 seconds to a minute. Lift the lid and smell the dry leaf.

Wash

Before any real brews, you’re going to wash the leaf. Pour hot water in, cover, and after a very short time, pour it out. This “wakes up” the leaf, and gets it ready for reliable brewing (don’t skip this step, I’ve A/B tested and it makes a big difference). Most of the time there’s just a bit of aroma in this brew, and little taste. Pour it into the cup(s) then pour it away. If you have a tea pet like my fatso kitty, you can give it to them.

After rinsing the leaf and letting it steam another half minute or so, look at the wet leaf, and take a smell. It usually smells quite different from the dry leaf.

The brews

Pour the hot water over the leaf, and cover the pot. After a certain amount of time, pour the tea into the pitcher. If you have a gaiwan, you’ll need to push the lid slightly off-center to make a small opening.

For larger leaf teas it’s easier to keep the leaf in and pour the tea out. The smaller and more broken the leaf, the trickier this is, but that’s why it’s good to pour through a filter. If leaves come through, just dump them back into the gaiwan or pot.

Pouring into a pitcher is important if you’re sharing the tea. It lets everyone get a taste or two of each brew. It’s also important if you’re drinking alone out of small cups.

First brews

The first 3-4 or so brews of a tea give you the most aromatics. For most teas (certainly oolongs, whites, and usually shu), the second brew is the most loved.

Middle brews

After the initial few brews, there’s a first change in the flavor of the tea. Some of the initial aromas start to fade, and new flavors come up from deeper in the leaf.

Late brews

Once your brew times get distinctly over 1 minute, you’re into the deep leaf flavors. For some teas this isn’t very interesting, and it’s done. For some teas these brews are also very good and you get yet another set of flavors. Thick tea pots that hold the heat are better here. Make sure your water is as hot as you can get it.

Enjoy

Once you’ve gone through the process once or twice, it’s pretty easy, and shouldn’t be intimidating. It can be fun to get into all the details, and to learn all the different contradictory things you can do and learn why you’d do them … and you’ll eventually end out a tea master if you go far enough along that path. But it’s not necessary, especially when getting started. Enjoy the process, enjoy the tea, and share what you think of it, good or bad!

Welcome to the test blog of the chef-d’œuvre

This is a multi-line post