8 ms·
These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. You can de
by rustybolt 6y ago
These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.
You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication?
This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you gained enough intuition you are ready to use these more complicated operations as primitives as well. Then, you can generalize and apply your intuition to more abstract objects.
- yakubin 6y ago> You can definitely see multiplication as repeated addition. Only for rational numbers. Doesn't work for real and complex numbers. > It's even a useful thing to do, how else would you define multiplication? Axiomatically, not algorithmically.
- WallWextra 6y agoIf you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.
- yakubin 6y agoNot algorithms. There will be infinite addition involved, and algorithms are finite. Thinking of multiplication as repeated addition also won't explain anything about it. It's a separate operation. Deal with it. For similar reasons, you can't calculate x-th power of a number, when x is irrational, by decomposing it into exponentiation and roots. This metaphor is just training wheels. At some point you should lose it. Together with the notion that "multiplication is repeated addition" comes the notion that numbers are quantities. Only some of them are, and this isn't really what makes them numbers. Now what exactly gets repeated, when you don't have quantities?
- prionassembly 6y agoMultiplication on the naturals is repeated addition. You might see the natural numbers as training wheels for higher mathematics, but number theorists might disagree...
- yakubin 6y agoNumber theorists work with integers, not naturals. And they too will agree with me, as they mostly work with polynomials, which show exactly that. I also didn't say natural numbers are training wheels, just this metaphor, which to use HN lingo: doesn't scale.
- alisonkisk 6y agoNumber theorists work with complex numbers and irrational numbers too https://www.britannica.com/science/Riemann-zeta-function https://www.britannica.com/science/Riemann-zeta-function
- gugagore 6y agoAlgorithms are not finite. E.g. Newton's method.
- yakubin 6y agoFirst sentence on Wikipedia[1]: > In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ (About this soundlisten)) is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation. Newton's method is finite too. You perform finitely many iterations. It doesn't calculate roots. It calculates their approximations. [1]: <https://en.wikipedia.org/wiki/Algorithm https://en.wikipedia.org/wiki/Algorithm>
- gugagore 6y agoIf you use a termination condition that has to do with convergence of iterates instead of a number of iterations (often the case), then you generally don't know beforehand the length of the finite sequences. Maybe you know a bound, but in general you might not even have that. In an important sense, it only becomes a finite algorithm. It isn't one. You cannot write the finite sequence of instructions down. It's got loops. To your point about approximations vs not, if you have an algorithm that, for any desired approximation accuracy can compute the square root to that accuracy in a finite number of steps, then that process is as much "the square root" as anything involving the real numbers.
- qsort 6y agoWell, no, not really. The standard definition of the reals is as the unique nontrivial totally-ordered, Dedekind-complete, Archimedean field up to isomorphism. So what you would really need is a uniqueness proof, with addition and multiplications "provided" by the hypothesis.
- howling 6y agoAnd how do you prove that a totally-ordered, Dedekind-complete, Archimedean field does not lead to contradiction besides constructing it explicitly by bootstrapping from natural numbers?
- qsort 6y agoNothing in the construction requires you to show an algorithm that given x, y \in R allows you to compute x+y and xy. You would make the usual Dedekind construction and show it satisfies the axioms of such a field. (as a matter of fact, no such algorithm exists in full generality!) It's probably a tomato/tomato kind of thing, but I'm only objecting to the 'algorithm' part of parent's comment.
- SAI_Peregrinus 6y agoMultiplication of the Surreals is a recursive operation using sums (addition and subtraction on the left and right sets). Since the Reals are a strict subfield of the Surreals one can define multiplication of the reals using only the same recursive formula and restricting both operands to be Reals.
- eigenket 6y agoIts probably worth emphasising that the recursion you need to "construct" the Surreals is infinite, in other words this does not give a reasonable algorithm to (for example) add two real numbers, you need S_omega in order to have even all rational numbers. The construction is rather involved but if we're only interested in the reals for now you can think of it as defining a real number by a set of rational numbers, in particular define a particular "real number" to be the set of all rational numbers less than it, for example sqrt(2) is defined to be the set of all rationals p/q such that p^2/q^2 < 2. We can "recursively" define addition of these real numbers in terms of addition of rational numbers because to add two reals you "just" have to add all the rationals in their respective sets. In general there is no sensible algorithm to do anything in the real numbers, since most real numbers aren't even computable (there is no way to represent an arbitrary real number on a Turing machine).
- marcosdumay 6y agoOf course, once you decide to iterate over uncountable sets, infinity starts to appear all the time. This isn't the only case of infinite calculations that can't be computed in practice but that mathematics use all the time anyway; and it reflects quite well the fact that multiplying irrational numbers isn't something that one can do practice. There is no problem with it.
- alisonkisk 6y agoMultiplying algebraics is trivial. A rectangle with sides sqrt(3) and sqrt(2) has area sqrt(6), which can be approximated by a decimal if needed. There are countable/computable/constructable subsets of the reals where multiplication has a finite algorithm and is it repeated addition. One example is the algebraics, as well as extensions the including a few special constants like pi. These are the subsets of the reals most commonly used for math and science. So in a wide range of problems areas, multiplication is not just repeated addition.
- xirbeosbwo1234 6y ago>Only for rational numbers. Doesn't work for real and complex numbers. Complex numbers aren't really relevant, in my opinion, because they are usually introduced as an extension of the rules for reals and polynomials. To multiply two complex numbers, you can totally forget that i is imaginary, do the multiplication as if it's just an ordinary variable, then substitute "i" back in. But that relies on being able to multiply polynomials, which would be difficult to define in terms of repeated multiplication. To some extent all of mathematics is a lie. We can do multiplication on the reals because we have decided that it's allowed. It is reasonable to define multiplication at first as repeated addition and then define a way to extend that to the reals that is consistent with the first definition.
- DanielMcLaury 6y agoMultiplication of real and complex numbers is typically defined by starting from repeated addition, extending this notion to rationals, and then extending that notion to reals by taking limits. How exactly are you going to present multiplication of real numbers axiomatically without essentially including an axiom that bootstraps everything from repeated addition? I suppose you can try defining the reals as "the unique complete ordered field" or the complex numbers as "the unique algebraically closed field of characteristic zero with cardinality c," but I don't think either of those are pedagogically useful to someone who is still learning what multiplication is.
- rafaelturk 6y agoYep, it is a clickbait
- parenthesis 6y ago> These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square. In fact, I have a high school textbook that makes that derivation.
- sudosysgen 6y agoWe did that derivation in class, actually! I was so lucky to have a great math teacher in high school.
- eru 6y agoThat's not the norm?
- Jtsummers 6y agoNo, it's not. It depends a bit on your teachers but also on which "level" of math you're in (in the US). Lower level but still algebra/geometry classes tend to teach facts, not derivations from foundational concepts. Those are the classes aimed at non-Honors and maybe non-College Prep students (2 of the 3 typical "tracks" students end up in the US, names may vary by state and decade).
- shadowofneptune 6y agoThe way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.
- throwaway894345 6y agoThe US education system varies tremendously. I don't think you can credibly claim that this teaching method is "standard" in the US. At least my experience was much better than you described. Indeed it wasn't until university-level mathematics that I started to get into the "stop trying to understand how they work and just memorize these formulae". Fortunately my engineering classes provided a through-line for understanding how those formulae worked, and I was much stronger for it than my math-major contemporaries.
- baryphonic 6y agoFunny, my teachers in middle and high school frowned on using the quadratic formula until we could demonstrate we knew how to complete the square.
- Frost1x 6y ago>These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. Education is probably a special case, at least from my educational experience. There's often multiple ways to abstractly represent a problem in mathematics and find the desired solution. The example you pointed out is such a case. When teaching mathematics, I think the goal is to introduce a lot of forms of mathematical thinking and approaches to solving a problem, to make you realize there are often multiple approaches and to take a peek at the insight of some of these approaches and how they often connect or think about different 'branches' of mathematics. Teaching math needs to be explicit with this though: solve this using method X. They should also explain this to kids as to why they're doing it. One of the biggest mistakes I see in mathematics teaching is the perception there's "only one right answer." Well, yes and no. Under certain condition a specific answer exists, sometimes it's a set of answers and sometimes you they're not really assessing so much that you can get the correct answer, but instead that you understand a specific method. The 'answer' is a means to an end to force you to step through, internalize a process, and hopefully at some point understand the deeper insight of that clever process you internalized and apply it to other problems you may encounter in the future. You may never use the exact process or insight behind it as-is, then again you may use facets of the reasoning you internalized later. If you don't work in professions that require this sort of abstract thinking I can see how the entire dance is quite silly but if you work in research, you often appreciate all these nuggets of deep insight and wisdom you've gained you can cobble together, remold, or lead to new insights. As a kid you probably have no idea where you'll work as an adult so maybe a lot of that effort is wasted.
- jordan_curve 6y agoTo make this more explicit, mathematics (even at the primary school level) is arguably more about “proving” things than it is about computation. Obviously you’re not writing proofs in 3rd grade, but the emphasis should be on students being able to say why their answer is correct rather than just being able to produce the correct answer.
- scatters 6y ago> how else would you define multiplication? In Church arithmetic, multiplication is defined as function composition; that is, the dot operator `.` in Haskell-like languages. Addition is considerably more complex; it is defined as `lift (.)`. Exponentiation is even simpler; it is function application, so the `id` function - except that we write exponentiation backwards, so actually `flip id`.
- shawnz 6y agoInteresting, what about tetration then?
- scatters 6y agoGood question: you can define m↑↑n as: \m n -> n (exp m) 1' where `exp` is Church exponentiation, `flip id`, and `1'` is the Church numeral for 1, that is, `id`. According to pointfree.io this expands to: flip flip id . flip id . flip id Which isn't particularly illuminating. Pentation isn't much worse, though: flip flip id . flip id . flip flip id . flip id . flip id
- swiley 6y agoAnyone that teaches this way should find something else to do. Memorizing a formula is easy but if you want to do well later you must be able to complete the square.
- daveFNbuck 6y agoI never learned to complete the square, and I managed to do well enough later to get a degree in math.
- pdonis 6y ago> You can definitely see multiplication as repeated addition Only if you restrict to whole numbers. But multiplication does not just apply to whole numbers. How would you see multiplication by pi as repeated addition? > how else would you define multiplication? As a separate operation with its own properties. The subject article gives several examples of properties of multiplication that are simply different from properties of addition.
- dash2 6y agoMultiplication by rational fractions p/q is just multiplication by whole p, followed by multiplication by 1/q. 1/q is just the number that when multiplied by q gives 1. So you can cash that all out as repeated addition. Multiplication by irrationals is just an infinite sum of multiplication by rationals, no? x times pi = x times 3, plus x times 1 / 10, plus.... I don't see there's any conceptual issue here. Mathematicians feel free to correct me.
- pdonis 6y ago> multiplication by 1/q Is not repeated addition. You can't add a number to itself 1/q times. At least, not unless you're willing to adopt increasingly perverse interpretations of "repeated addition" as you try to cover more and more numbers. See my response to wruza upthread.
- dash2 6y agoHmm, so just go about it differently: x * p/q with p,q integers is (x*p) / q. The first bit is repeated addition. The second bit means, find the number r such that r * q = x * p. Even if r is not integer, q is integer, so we can try different numbers, add them to themselves q times, and close in on the answer. So I still think conceptually it's fine to think of it as repeated addition? It might be algorithmically a bad way to do it ("increasingly perverse", taking limits when the number is irrational etc.). I don't know how computers actually implement multiplication - though wikipedia (https://en.wikipedia.org/wiki/Binary_multiplier https://en.wikipedia.org/wiki/Binary_multiplier) says it works via shifts and adds. Or maybe, it's bad to teach it to kids this way? But then I think we need evidence from educationalists.
- whatgoodisaroad 6y agoExactly. Probably every discovery in math starts out with something that's only close to the truth (like multiplication-is-repeated-addition) but generalizing it to open up whole new areas. This part of math is so exciting and compelling, I feel like it should be more of a focus in schools.
- thrwn_frthr_awy 6y ago> how else would you define multiplication? With my six year old I told them to say "groups of" instead of "times". So 3 X 2 isn't read as "three times two" it is "three groups of two" which I think has been helpful.
- alentist 6y ago> You can definitely see multiplication as repeated addition. Doesn't work in every context where multiplication is defined. > It's even a useful thing to do, how else would you define multiplication? Perhaps something along the lines of: A multiplication is an endomorphism on an additive semigroup.