Does the square root of 2 exist?

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In a recent OP I looked at a discovery by mathematician Norman Wildberger, who found a general method for generating power series solutions of polynomial equations of any degree. Wildberger has an interesting, extremely unconventional and (in my opinion) flawed philosophy of mathematics, which among other things denies the existence of irrational numbers. Here he explains why √2 doesn’t exist, at least not in the way that mainstream mathematicians thinks it does:

There’s lots to criticize about this, but I’ll save it for the comments.

221 thoughts on “Does the square root of 2 exist?”

  1. I quote Keith, “Jock’s number line is, as he describes it, “a series of dots”. It’s what you get when you remove all the irrationals, leaving gaps behind. The “dots” are what remain, each corresponding to a rational number.”

    But even the rationals have no “gaps” and are not a series of “dots”, because between any two rationals there are an infinite number of other rationals.

  2. aleta:

    But even the rationals have no “gaps” and are not a series of “dots”, because between any two rationals there are an infinite number of other rationals.

    It’s true that between any two rationals there are infinitely many other rationals, but that doesn’t mean that they “fill the space” in between. There are still gaps, and those gaps are where the irrationals live. The rationals do not by themselves form a continuum.

    All four of these statements are true:

    1. There are infinitely many rationals between any two rationals.
    2. There are infinitely many irrationals between any two irrationals.
    3. There are infinitely many rationals between any two irrationals.
    4. There are infinitely many irrationals between any two rationals.

  3. I should add another statement:

    5. There are infinitely many numbers between any rational number and any irrational number.

    That feels really weird, because intuitively you would think that there would have to be somewhere where a rational number is next to an irrational, with no gap. Not so.

    It’s analogous to the fact that there is no greatest number less than 3. Pick any number that is less than 3, and there is always a greater number that is also less than 3.

  4. The “operational definition” of ln 7 would be “the power of e which gives you 7”, but then what is the operational definition of e, as it also is irrational?

  5. aleta:

    One serious downside is that you also have sqrt 3 and cube root 17 and sin 31 and ln 7 and (dare I say it) an infinite of cases which would have to be handled this way.

    Yep. And since he’s infiniphobic, he’d have to add the new objects piecemeal as the need arose. What a mess.

    Every irrational would need to have an “algebraic” definition.

    Which would be impossible for the uncomputable numbers. He’d probably be OK with that, though, because although they exist as a class, I don’t think anyone ever uses specific uncomputable numbers for any purpose.

    I’m not sure how he would handle transcendentals. He has a video on π that I haven’t yet watched.

    What would be the operational definition of ln 7?

    Logarithms, like roots, would be a nightmare. Not only would he have to define all these new objects — he’d also have to define rules for how they interact with each other, pairwise.

    For instance, there couldn’t be a general rule that

        \[ \log(a) + \log(b) = \log(ab) \]

    You’d need individual rules saying that

        \[\log(2) + \log(3) = \log(6)\]

        \[\log(4) + \log(2) = \log(8)\]

        \[\log(1/2) + \log(10) = \log(5)\]

    …and so on.

    That’s because “log(2)” would just be a name, and the named object would in no way derive from the number 2. You could just as easily name it “Evelyn”.

    You’d have to create rules saying things like

        \[ \text{Doug + Evelyn = Stanley} \]

        \[ \text{Englebert + Priscilla = Horst} \]

    …and so on, for every possible pair.

    Ugh.

  6. Keith, you write, “It’s true that between any two rationals there are infinitely many other rationals, but that doesn’t mean that they “fill the space” in between. There are still gaps, and those gaps are where the irrationals live.”

    Hmmm. I know that there are lots of irrationals (of a different order of infinity) of irrationals on the number line with the rationals. But I don’t think it is accurate to say there are “gaps”. As you also wrote, “It’s analogous to the fact that there is no greatest number less than 3. Pick any number that is less than 3, and there is always a greater number that is also less than 3.”

    In fact, couldn’t we write, “There is no greatest rational number less than 3. Pick any number that is less than 3, and there is always a greater rational number that is also less than 3.”

    Is that true? Just because there are also all those irrationals, I don’t see how it very accurate to say there are gaps, or, in Jock’s words, “a series of dots with gaps in between.”

  7. aleta:

    The “operational definition” of ln 7 would be “the power of e which gives you 7”…

    It can’t be, for the same reason I gave in my previous comment. “ln(7)” would just be a name that has no relationship with the number 7. The object it represents isn’t even a number, so it cannot be a power of e.

    …but then what is the operational definition of e, as it also is irrational?

    I don’t yet know how Wildberger handles transcendentals such a π and e.

  8. aleta:

    Hmmm. I know that there are lots of irrationals (of a different order of infinity) of irrationals on the number line with the rationals.

    That’s true, but it’s not why there are gaps. Note that these statements are symmetrical:

    1. There are infinitely many rationals between any two rationals.
    2. There are infinitely many irrationals between any two irrationals.
    3. There are infinitely many rationals between any two irrationals.
    4. There are infinitely many irrationals between any two rationals.

    aleta:

    But I don’t think it is accurate to say there are “gaps”.

    If there weren’t gaps, there wouldn’t be room for the irrationals. Which would make Jock very happy. But there are gaps. Name any two rationals, and I can give you an irrational that fits between them. That irrational number lies in a gap.

    For example, if the two rationals you give me are 37.285 and 37.286, the irrational number 37.2854749207… fits between them. It’s always possible to do this, which means the gaps are there.

    In fact, couldn’t we write, “There is no greatest rational number less than 3. Pick any number that is less than 3, and there is always a greater rational number that is also less than 3.”

    Yes. It’s true for rational numbers, for irrational numbers, and for numbers in general.

    Is that true? Just because there are also all those irrationals, I don’t see how it very accurate to say there are gaps, or, in Jock’s words, “a series of dots with gaps in between.”

    Again, the existence of the gaps doesn’t depend on the fact that the set of irrationals is larger than the set of rationals, though that’s true. There’s always a gap between any two rationals where an irrational can live, but there’s also always a gap between any two irrationals where a rational can live.

  9. faded_Glory:

    Moreover, if √2 can’t be placed on the number line because its exact value as a decimal expression is unknown, then neither can 1/3 which has an infinite series of 3’s behind the decimal point.

    √2 and 1/3 are already on the number line, so they don’t need to be placed there. Every real number is on the number line.

  10. keith, I see. I was interpreting the word “gap” differently than you were meaning. You mean a space in which more numbers can fit, which is true. I was taking it to mean “spots” between which there is nothing – a true empty gap between two numbers. Different issues.

  11. aleta:

    keith, I see. I was interpreting the word “gap” differently than you were meaning. You mean a space in which more numbers can fit, which is true. I was taking it to mean “spots” between which there is nothing – a true empty gap between two numbers. Different issues.

    I’m not seeing the distinction you’re trying to draw. In my mind, the only way you can fit more numbers into the line is if there are gaps to stick them in.

    Here’s how I think about it. The number line is an infinite, connected set of points, each of which corresponds to a number. If you remove even one point from the line, you create a gap, albeit an infinitely small one. It’s a true, empty gap. There’s nothing there. The point that used to be there has been removed.The line is no longer connected. There’s room for one point in the gap you’ve created.

    Jock is doing point removal, but on an industrial scale. In effect, he’s taking the number line, scanning it from -∞ to ∞, and removing all of the points that correspond to irrational numbers. That necessarily leaves gaps, and of course we can turn around and stick the irrationals back into those gaps, at which point the number line becomes whole again and fully connected.

    Another way of convincing yourself that there are gaps in his line is to consider the fact that the remaining points constitute literally 0% of the points that were originally on the line. In other words, speaking informally, the set of the rationals is infinitely smaller than the set of the reals, which means that the percentage is zero, despite the fact that there are infinitely many rationals. If you’ve gone from 100% to 0%, you’ve definitely created some gaps.

  12. faded_Glory,

    You wrote:

    Moreover, if √2 can’t be placed on the number line because its exact value as a decimal expression is unknown, then neither can 1/3 which has an infinite series of 3’s behind the decimal point.

    I responded:

    √2 and 1/3 are already on the number line, so they don’t need to be placed there. Every real number is on the number line.

    Something else I want to stress is that the points on the number line correspond to values, not to representations. Any number that has a value (which is all of them) has a place on the line. The fact that some values can’t be represented as finite decimals is irrelevant. If it has a value, it has a place on the number line. One and only one place.

  13. Keith, I don’t think I am disagreeing with you. Jock thinks there are really gaps. I know there are not.

  14. faded_Glory,

    When you speak of placing numbers on the number line, I suspect your intuition goes something like this: You’re working your way through a pile of numbers, placing each one on the number line. The number 5 shows up in your pile. You pick it up, and since it’s the number 5, you know you need to place it 5 units away from the origin on the positive side. You pick up your ruler and measure 5 units from the origin, depositing the 5 there.

    The next number in your pile is √2. You need to place it on the number line, but how many units from the origin should it be? No matter how many digits you calculate, you don’t know precisely where to place it. If you don’t know where it’s supposed to go, you can’t place it on the number line. Is that more or less how you’re thinking about this?

    I think Jock’s intuition is similar. He writes:

    √2 has a value, but we can never get to it. Therefore there is a fundamental problem with “placing it on the number line” with the rationals.

    5 and √2 are already on the line, in exactly the right positions. You don’t need to place them — you just need to refrain from removing them. 5 is exactly where it belongs — to the right of all numbers less than 5 and to the left of all numbers greater than 5. √2 is also exactly where it belongs — to the right of all numbers whose squares are less than 2 and to the left of all numbers greater than zero whose squares are greater than 2. That’s its one and only spot. You don’t have to place it — you can just leave it where it is.

  15. aleta:

    Keith, I don’t think I am disagreeing with you. Jock thinks there are really gaps. I know there are not.

    Jock is actually correct about the gaps, because he’s talking about his version of the number line, not the standard one in which there are no gaps. The irrationals aren’t allowed on his number line, so there are gaps where the irrationals would otherwise appear. Hence his description of his number line as a series of dots, not an actual connected line.

  16. keiths: Is that more or less how you’re thinking about this?

    I’m just going with the conversation. I think that the number line is as fictional as the numbers themselves. It all reminds me of trying to decide how many angels can dance on the head of a pin. A hypothetical pin, no less.

    As you say, if someone defines the number line as a line on which you have to place the numbers, then √2 can never be on there. If you think of it as a line that already exists and that contains all the numbers, then they are already on there.

    The next question then becomes, if that line already exists, *where* does it exist and where does it come from 😉 ?

  17. FG asks, “The next question then becomes, if that line already exists, *where* does it exist and where does it come from 😉 ?.”

    I’m reading a fairly simple and not particularly good book called “Infinite Powers” by Steven Strogatz which tells the story, of which I am quite familiar, of the development of the notion of infinity and its role in calculus. Just last night I read,

    “To them [the Greeks], numbers meant exclusively discrete quantities, like whole numbers and fractions. By contrast, continuous quantities of the sort measured by the length of a line were regarded as magnitudes, a conceptually distinct category from numbers. So for the nearly two thousand years from Archimedes to the beginning of the seventeenth century, numbers were absolutely not seen as equivalent to the continuum of points on a line. ”

    Then Descartes (and also Fermat) had the brilliant idea of merging the two concepts, inventing the idea of not one but two number lines (the x and y axes), and thus was born the extremely powerful tool, analytic geometry, which brought together algebra and geometry, and made the invention of calculus possible.

    So the answer to FG’s question is that the number line exists as an idea, is represented in various ways in words, pictures, and mathematical symbols, and came from Descartes.

  18. faded_Glory:

    I think that the number line is as fictional as the numbers themselves.

    I agree.

    It all reminds me of trying to decide how many angels can dance on the head of a pin. A hypothetical pin, no less.

    It’s similar in the sense that both angels and number lines are fictional creations, but that doesn’t make it unimportant. Whether there’s room for all the real numbers on the number line, and whether they form a continuum, are important questions for mathematicians because the answers tell us something fundamental about the nature of the reals.

    And yes, the reals are fictions too, but they’re useful fictions, and their definition imbues them with certain properties. Mathematical lines are also fictional, with certain properties. The reals are an infinite set, and so are the points that make up a line. Whether they are the same “size” of infinity is an important question, because it tells us whether the set of reals can be fully represented in the form of a number line. They are, and it can.

    Jock’s number “line” is fictional too, and of course he is free to define it however he likes. But here’s the thing: His fictional number line is motivated by the false belief that the standard number line is incapable of representing the entire set of the reals. He’s wrong about that, and so he’s needlessly rejecting the beautiful, clean number line of mainstream mathematics and replacing it with a tattered, idiosyncratic number line that is literally full of holes.

    It’s analogous to a mistake that he and Flint made during our marathon 8-month discussion. They thought the real numbers were inadequate for expressing measurements, so they invented the “flintjock numbers” to solve that nonexistent problem. Likewise, Jock thinks the number line is inadequate for representing the reals, so he has invented his own number line to solve that nonexistent problem.

    It isn’t just that Jock’s number line and the flintjock numbers are unnecessary; they also create major problems. For instance, the Cartesian coordinate plane is defined by two number lines, the x- and y-axes, as aleta noted. If Jock were correct, it would be impossible to plot points such as (√3, 2) or, when operating in the complex plane, 5 + √7i. It isn’t just that the axes would be full of holes — functions would be, too. The beautiful, smooth curve of y = x² would turn into a choppy set of disconnected points.

    As you say, if someone defines the number line as a line on which you have to place the numbers, then √2 can never be on there. If you think of it as a line that already exists and that contains all the numbers, then they are already on there.

    The next question then becomes, if that line already exists, *where* does it exist and where does it come from 😉 ?

    The number line is an invented concept, and the concept as invented already includes all of the real numbers. They don’t have to be placed on the line one by one, just as the natural numbers don’t have to be placed in their set one by one.

    And even if they did need to be placed on the line one by one, that wouldn’t present a problem. Jock’s number line includes all the rationals, and there are infintely many of those. If Jock is allowing himself to (conceptually) place the rationals on the line, one by one, then he is allowing himself to complete an infinite task. But if infinite tasks are allowed, then it’s possible to perform the infinite task of determining the full decimal expansion of √2, whereupon it can be placed on the number line in precisely the right spot.

  19. Here’s another problem with Wildberger’s algebraic extensions to the rational numbers: ordering becomes wonky. Example: In normal (vs Norman) math, √2 is a number, and we can say that

        \[ (1 + \sqrt{2}) < (1 + 450\sqrt{2}) \]

    It’s obvious. The number on the right is much larger. But in Wildberger World, √2 is just an algebraic object with no numerical value. It contributes nothing to the value of the complete number. Does that mean that in Wildberger world, those two numbers are equal? That’s definitely not what we want.

    Perhaps we can borrow an ordering method from the realm of the complex numbers, since they, like Wildberger’s extended rationals, have separate components. You can order the complex numbers based on magnitude — that is, based on the distance from the origin to the point in the complex plane that represents the number in question. It’s just geometry. For a complex number a + bi, the magnitude is

        \[ |a + bi| = \sqrt{a^2 + b^2} \]

    That only forms a partial ordering, though, because there are infinitely many (a,b) pairs that share a given magnitude. They can’t be ordered relative to each other. Restating that geometrically, for a given distance from the origin — say, 5 — there are infinitely many lines of that length that have one endpoint at the origin, each going in a different direction and ending at a different point — a different complex number. Thus the magnitude approach won’t work with Wildberger numbers, because for any two distinct numbers, we want one to be “greater” than the other, just as in mainstream math. That doesn’t happen with the magnitude approach.

    Here’s another way of ordering the complex numbers: we can order them based on the real components, if those are unequal. Otherwise, we can order them based on the imaginary components. For example,

        \[ (6 + 3i) > (5 + 20i) \ \text{because } 6 > 5 \]

        \[ (7 + 24i) > (7 + 9i) \ \text{because } 7 = 7 \text{ and } 24 > 9 \]

    But if you apply that method in Wildberger World, you get nonsensical results like

        \[ (3 + 450\sqrt{2}) < (4 + \sqrt{2}) \]

    …so that won't work either.

    As far as I can tell, the only way to get an ordering that matches the normal ordering of mainstream math is to create a rule that says "For ordering purposes, the value of (a + b√2) = a + (b times the numerical √2)". But using the numerical √2 would defeat Wildberger's entire purpose in creating the algebraic √2, so that's out.

    Can anyone else come up with an ordering scheme for Wildberger's extended numbers that would match what happens in mainstream math?

  20. Given all the problems we’ve discussed, here’s my advice to Wildberger regarding the algebraic √2:

    If you want a mathematical object that behaves like the number √2, just use the frikkin’ number itself. Fear of infinite decimals can be overcome through exposure therapy.

  21. In my math classes I taught kids to use both decimal approximations in some problems and write out exact answers in others, and to understand the difference. For example, simplify root(12)/root(5) into 2/5 root(15). We need to know how to do that and not even think about decimals.

  22. aleta: We need to know how to do that…

    Do we, though? I just entered “simplify sqrt12 divided by sqrt5” into google search window and I got this in a flurry of explanations.

  23. And whether we do exercises like this with pencil and paper or on a screen, I can’t offhand think of a practical use in my daily life other than hopefully delaying dementia.

  25. faded_Glory,

    Indeed! Not knocking it. Just got 1.55 with pencil and paper. 🤓

    ETA

    √12=2√3
    √5=5/√5
    √12)√5=(2√3)×√5÷5
    Sqrt3 is 1.732 roughly and sqrt2 is 2.24 roughly so:
    2×1.732=3.464
    3.464×2.24=7.76
    Multiply by 2 and divide by 10
    1.55

  26. I’m surprised that you folks don’t have more appreciation for the formal math that is done in the derivation of all sorts of applications without which the modern world would not exist. If all we did was cart around decimal approximations all sorts of understandings, including proofs that we were right, would not be very accessible.

  27. aleta,

    I didn’t start learning decimals till I left the Church of England school at 11. Fractions on the other hand, pounds, gallons, £sd…

  29. aleta:
    I’m surprised that you folks don’t have more appreciation for the formal math that is done in the derivation of all sorts of applications without which the modern world would not exist. If all we did was cart around decimal approximations all sorts of understandings, including proofs that we were right, would not be very accessible.

    I suppose seeking the “actual value” of things like the square root of 2, or pi, or e, is barking up the wrong tree. We do not need a decimal approximation of these symbols to use them in proofs and derivations.

  30. aleta:

    I’m surprised that you folks don’t have more appreciation for the formal math that is done in the derivation of all sorts of applications without which the modern world would not exist. If all we did was cart around decimal approximations all sorts of understandings, including proofs that we were right, would not be very accessible.

    Not to mention that it would uglify math and take a lot of the magic out of it. The most beautiful equation in the world…

        \[ e^{i\pi} + 1 = 0 \]

    …would become…

        \[ 2.718^{i \cdot 3.1416} + 1 \approx 0 \]

    …which is pretty boring.

    This unremarkable approximate sum…

        \[ \sum_{n=1}^\infty \frac{1}{n^2} \approx \frac{(3.1416)^2}{6} \approx 1.6449 \]

    …is amazing when expressed symbolically and exactly:

        \[ \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} \]

    I wouldn’t even expect π to show up in that equation, much less π². It’s magic.

    Speaking of π showing up by magic, check out this video.

  31. Is it possible to define a number for which it is impossible to determine whether it is less than or greater than sqrt 2?

  32. Another example:

    This looks pretty boring…

        \[ 0.3183098861 \approx \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)! \, (1103 + 26390k)}{(k!)^4 \, 396^{4k}}. \]

    …until it’s written like this:

        \[ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)! \, (1103 + 26390k)}{(k!)^4 \, 396^{4k}}. \]

    …at which point it becomes astounding.

    Ramanujan figured that out, and I have no idea how. What’s even more amazing is how efficient it is. Each successive term adds 8 correct digits to the approximation as the sum converges to 1/π.

  33. petrushka:

    Is it possible to define a number for which it is impossible to determine whether it is less than or greater than sqrt 2?

    Yes, but it depends on the nature of the definition. If the definition gives you enough information to calculate the decimal digits of your number, you can always* tell whether it’s greater than or less than √2 by calculating and comparing successive digits from both numbers. You’ll eventually find a digit that differs between the two, whereupon you can determine which is larger.

    If your definition doesn’t allow you to calculate the digits of your number, but does allow you to establish a bound, then you may be able to tell whether your number is greater than or less than √2. If your number is bounded above by, say, 1.413 (which is less than √2), then your number — even though you don’t know the digits — must also be less than √2. Similar reasoning on the other side of √2: If your number is bounded below by, say, 1.415 (which is greater than √2), then your number — even though you don’t know the digits — must also be greater than √2.

    For a definition that truly doesn’t allow you to determine whether your number is greater than or less than √2, you could do something like this: construct your number using the first n digits of √2, but then append a bunch of digits you cannot possibly know. For example, this drop of water (pick a specific drop) contains some integer number of water molecules**. Specify that you are taking the last 10 digits of that number, which are completely unknown to you, and appending them to the number you are constructing. You’ll have no idea whether the resulting number is greater than or less than √2. However, you will know that the number has a definite value, even though you don’t know what that value is.

    * If the definition just so happens to define a number that is exactly equal to √2, then of course you will never find a digit that differs. If you’re allowing yourself to instantaneously compare infinitely many digits (since this is just a thought experiment), you’ll be able to figure out that they’re equal. Otherwise, if comparing each digit takes time, then you’ll never know for sure that they’re equal. Even if you’ve gone through 8 gazillion digits so far, and they’ve all matched, it’s possible that the next digit will mismatch. You’ll never be able to confirm that they’re equal by comparing digits, though it’s possible that by analyzing the definition mathematically you’ll be able to prove that the specified number is equal to √2.

    ** In the real world, the number of molecules isn’t static because the drop exchanges molecules with the atmosphere. But since this is a thought experiment, let’s assume that the number stays constant.

  34. Alan:

    I didn’t start learning decimals till I left the Church of England school at 11. Fractions on the other hand, pounds, gallons, £sd…

    That’s because decimals are satanic. The Church was protecting you from their pernicious influence. Fractions are wholesome.

  35. I asked ChatGPT to generate the first four terms of the Ramanujan series. It took five tries to get it right — ChatGPT is so smart, and so stupid, all at the same time.

    Here are the terms, and as indicated by the leading zeros, you can see that it really is generating 8 new decimal digits with each term:

    0.3183098784404701232176844531789199021860
    0.0000000077433204835215119806419083229320
    0.0000000000000000647985705171743502428697
    0.0000000000000000000000005757498449479696

  36. I forget whether I posted this before: Someone mentioned earlier that they knew of no use for Euler’s identity e^(i•pi) = -1. However, we also talked about ln x recently, and I discovered that the definition of ln(negative number) involves Euler’s identity, because if you take the natural log of both sides, you find that ln(-1) = i•pi, so ln(-x) = ln(x) + i•pi.

    So there is a use. 🙂

  37. aleta,

    And not only a use, but a practical use. Logarithms of negative numbers are used in electrical engineering, and Euler’s identity makes that possible.

  38. Watched a video in which Wildberger explains his views on \pi. He doesn’t define an algebraic proxy for \pi, but instead employs something similar to his “applied √2” approach. That is, he defines and uses different approximations depending on the application:

        \begin{equation*} \begin{aligned} \pi_0 &= 3 \\ \pi_1 &= 3.1 \\ \pi_2 &= 3.14\\ \pi_3 &= 3.141\\ &\vdots \\ \pi_{10{,}000{,}000{,}000{,}050} &= 3.1416926\ldots611421 \end{aligned} \end{equation*}

    (That last one accounts for work by Kondo and Yee in 2011, who generated 10 trillion and 50 digits of \pi. The number is higher now, so I suppose that a bunch of new \pis have sprung into existence since then.)

    EDIT: The record as of April 2nd is 300 trillion.

    At least here he is using subscripts for his different \pis. For √2, he pretended that each approximation actually was √2:

    √2 = r refers to a rational number which has the property that r^2 is approximately 2.

    What about \pi_{\infty} , you ask? \pi_{\infty} , which the rest of us call \pi, is not a number according to Wildberger. (Not surprising, because it has an infinite decimal expansion). He likens it to Santa Claus or the Easter Bunny, and says

    However, it might be that there’s something out there called \pi which we have not quite understood yet. So actually my posiiton is that probably in the future someone will figure out what this \piobject really is, and when they do, they will find out that it’s not properly a number in the same sense that 22/7 or -1/3 is a number. This \pi is actually something much more interesting and much more elaborate. So in the meantime, I’d like to call this a “meta-number”.

    What is a meta-number? Wildberger admits that the idea is “vague” and “speculative” and that he doesn’t know what a meta-number is.

    It’s pure hand-wavery.

    I suppose his approach to e is similar, but I can’t be arsed to go looking for it.

  39. keiths: I suppose his approach to e is similar…

    But e is easy to calculate by hand.

    1 + 1/1! + 1/2! +1/3! +…

  40. And logarithms were certainly useful in calculation. Log tables (also containing sines, cosines, tangents, natural logs, and, if you were posh, stuff like cosecants and arctans) were a schoolboy’s daily reference. Then slide rules became status symbols.

  41. Alan:

    But e is easy to calculate by hand.

    1 + 1/1! + 1/2! +1/3! +…

    \pi is easy too. It’s

    4(1–1/3+1/5–1/7+1/9…)

    The issue for Wildberger isn’t complexity — it’s the fact that the decimal expansions are infinite.

  43. I have a question about the ubiquitous summation notation (I have no idea how to make it show up in a post here), the one that sums an expression in steps from k = 0 to, say, infinity.

    Are these steps always integers? Or can they be fractions too, in other words can you go up in steps of 1/2, or 1/3?

    Can √2 be one of the steps?

  44. The intensive 6 weeks maths course I once took many decades ago pretty much ended in triumph with Euler’s equation. Most of us were stupified. We didn’t see that one coming.

  45. faded_Glory: Are these steps always integers?

    k is generally used to indicate an integer. As to the rest, others will have to answer. Though choosing e as a log base transformed navigation.

  46. faded_Glory: I have no idea how to make it show up in a post here

    \LaTeX notation is neat but there’s no preview. So you post and then see your mistakes and then you post edit and then you see your mistakes and then you post edit… 🤓

  47. Alan:

    Is that

        \[\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}=\frac \pi 4\]

    ?

    Yes, except that I multiplied by 4 to get \pi instead of \frac \pi 4:

    4(1–1/3+1/5–1/7+1/9…)

  48. faded_Glory:

    I have a question about the ubiquitous summation notation (I have no idea how to make it show up in a post here), the one that sums an expression in steps from k = 0 to, say, infinity.

    Are these steps always integers? Or can they be fractions too, in other words can you go up in steps of 1/2, or 1/3?

    I’ve never seen a step size other than 1, and there’s a good reason for that. If it were something other than 1, then you’d have to specify the step size somewhere in the notation, and there’s really no good place for it.

    Plus, it isn’t needed. You can effectively convert a step size of 1 into your desired step size if you write the sum correctly. For instance, suppose you want the sum of 1/x from x = √2 to x = 30√2 in increments of √2.

    You can achieve that with a step size of 1 by writing it like this:

        \[ \sum_{n=1}^{30} \frac{1}{n\sqrt{2}} \]

    The index increments by 1, but the step size is effectively √2 because of the multiplication in the denominator. The same thing works in general. If you want your step size to be s instead of 1, just multiply the index by s wherever x appears in the expression you want to sum:

    For example, say you want to sum \frac{x}{x^3 + 5 } from x = 1/2 to infinity in steps of 1/2. Just substitute n/2 for x wherever x appears in the expression, as follows:

        \[ \sum_{n=1}^{\infty} \frac{n/2}{(n/2)^3 + 5}} \]

    fG:

    Can √2 be one of the steps?

    Yes, as demonstrated in my first example.

  49. Alan:

    \LaTeX notation is neat but there’s no preview. So you post and then see your mistakes and then you post edit and then you see your mistakes and then you post edit… 🤓

    If you go to quicklatex.com, you can debug your \LaTeX code there before posting it here. Much more convenient.

  50. That’s Pavel’s work. Tom English kindly persuaded him to improve his plugin, which had developed a few bugs.

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