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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.

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> 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.

"That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square."

That is one way to derive it, but there are others, including my personal favorite, the resolvents method (which also works for deriving cubic and quartic formulas).

We did that derivation in class, actually! I was so lucky to have a great math teacher in high school.

That's not the norm?

No, 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).

The 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.

The 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.

The best math class I ever had was a drafting class called "Descriptive Geometry". In that college class we used a drafting table to solve math problems.

An easy example would be the length of line of the corner seam of a hip roof. Sadly I have forgotten what the more complex problems were. Fantastic class that isn't offered anymore.

Btw, the game Euclidea on iOS is fun for that. You "discover" and prove concepts and progress through higher-orders of geometry proofs.

The point of the Geometry course in that curriculum is to teach the student how to do proofs. The Geometry itself is secondary.

Unfortunately this is typically done in a counterproductive way.

See http://jwilson.coe.uga.edu/TiMER/Schoenfeld%20(1988)%20Good%...

Beliefs 1–4 on page 151 are exactly what I thought about geometry until later in college. Thanks for the article, it's fascinating.

> Instead of learning that you can use shapes to do useful calculations like square roots

Huh?

You can easily draw shapes that conceptually represent square roots, but how do you get from that to calculating the square root? You'd need an infinitely-graded ruler.

All calculation is inherently imperfect. But if you want to have something of length sqrt(2) you can.

I'd say the opposite; the drawing is guaranteed to be imperfect, but the calculation won't be.

I don't think so. I was on the Calculus track in high school so we derived it in...Pre-Calculus.

Prior to that the quadratic formula was something that seemed to be handed down from on high. We used it in Algebra II and maybe even before that, but I had no idea where it came from.

It was a mind-opening experience when we derived it in class one day. Our teacher didn't ruin the surprise. She just said, let's complete the square on a general quadratic equation. And there it was. The quadratic formula!

> Our teacher didn't ruin the surprise. She just said, let's complete the square on a general quadratic equation.

How is this not ruining the surprise? The only possible outcomes of doing that are that (1) you make a mistake; or (2) you get a formula for solving quadratic equations. Quadratic equations have the same solutions regardless of your methodology, so there's only one formula you can get.

She did not tell us we were deriving the quadratic formula. I had no idea where she was going when she started the exercise.

Oh god, no. Not at all, especially in the US.

I can tell you how great my math education was. I don't even recall the quadratic formula or completing the square is. I'm almost certain they were part of the curriculum at some point.

I recall in later math course, notably calculus, the teacher assumed we we're familiar with some concept because we we're supposedly taught it the prior year, yet not a single student in the class could recall it having been taught before.

Are you sure? My teacher did that.

It doesn't take very long, and I can definitely see someone forgetting all about it.

I distinctly remember "accidentally" deriving it when I forgot the formula on a test; perhaps my proudest math moment (though really I was just scrambling apply any rule I could think of that got me closer to the vague form I remembered)

And then I got really annoyed that no one ever told me to do that before and started discounting teachers for years onwards.. probably to my own detriment.

Yeah, we got it as bonus homework, to derive it ourselves.

> 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.

Multiplication 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.

> 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.

You can do it like this:

First define a*b for integers by addition.

Define the the rational numbers as ordered pairs of integers where the last number may not be 0. Consider the rational number (a,b) to be equivalent to (c, d) if a*d = b*c

(Exercise for the reader: Prove that this is an equivalence relation)

Now define (a, b) + (c, d) = (ad + bc, bd), (a, b) * (c, d) = (ac, bd)

(Exercise for the reader: Prove that the subset of rationals (n, 1) behaves just like the integers)

Hmm, 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) 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.

> The first bit is repeated addition

So now your definition of "repeated addition" is "repeated addition, plus a version of multiplication". Division is the inverse of multiplication, so your definition is circular: you're "defining" multiplication in terms of repeated addition and multiplication.

Similar objections apply to another poster's contention upthread that the "repeated addition" definition is justified because of the distributive law. The distributive law defines A times (b plus c) in terms of A times b plus A times c. So it's useless as a definition of multiplication in terms of addition.

I don't think so. Look more closely. (x * p) is repeated addition with p integer - we can do this no multiplication involved. Then dividing by q involves finding r such that r * q = x * p. Here, again, we can do this by finding r such that r, added to itself q times, equals x * p. Since q is integer, again, this again involves repeated addition. (It does need an algorithm to find the right value of r. I suppose that trying an arbitrary value, then increasing or decreasing it, would work.)

How would you see multiplication by pi as repeated addition?

By taking 3 times as usual and then another .14 approximately. E.g. for 2, it is 3+3+28/100, roughly 6.28. (edit: if you’re confused by /100, read it as “two places next to a point”)

If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:

> By taking 3 times as usual and then another .14 approximately

First, "taking 3 .14 times" doesn't make sense, at least not in the elementary school student's understanding of "repeated addition".

Second, pi is irrational, as you evidently realize since you say "approximately". There is no way to add 3 pi times.

> If you’re asking how to do that exactly, first tell how do you even add pi to some number exactly? Let’s start with one, 1 + pi:

Add 1 to the 3 and keep the part to the right of the decimal point the same. Addition is commutative.

Basically, your argument is that multiplication is repeated addition as long as we're willing to adopt increasingly perverse interpretations of "repeated addition" as we expand the scope of the numbers we use. Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?

But what is that number? You only described an algorithm that relies on a vague definition of “part to the right”. See, you can’t even write it down (unlike 1 + 3.14 = 4.14), because pi is not really a number, in a sense. It is an infinite calculation that happens to converge between 3.14 and 3.15. You can never get rid of “pi” in your calculations unless it cancels out naturally, so for practical reasons it’s 3.14 and for theoretical reasons it’s “pi”.

Let’s make it clear: you cannot add pi or take pi times at all. You only can add to factors to the left of it (same as a+bi). That’s why you cannot multiply by pi by repeated addition, not because the numeric method is wrong.

Wouldn't it be better to just admit up front that multiplication is not repeated addition, although the two are similar for some types of numbers?

For generalized reasoning, yes. For teaching, not sure. Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that? When they learn, they ask themselves why and it’s a rabbit hole. You should stop somewhere before turning into maths professor, if that’s not your goal.

> what is that number?

Pi minus 3. If you are saying you can't actually perform that operation, great! That means you are agreeing with me. See below.

> you cannot multiply by pi by repeated addition

Thank you for agreeing with my main point. See below.

> not because the numeric method is wrong.

If someone is going to claim that multiplication is repeated addition, then their definition of multiplication in terms of repeated addition must cover all cases. You are saying it doesn't cover pi, which means it doesn't cover all cases. So the definition is wrong.

> For generalized reasoning, yes. For teaching, not sure.

Why not? What's wrong with telling kids, multiplication in general is a distinct primitive operation, we are teaching you how to multiply whole numbers using repeated addition, but be aware that this method will not generalize to all cases?

> Which is better, a student who knows that mul is rep-add, or a student who gave up and doesn’t know even that?

A student who has been told what I just described above is not in either of these positions, so you are arguing against a straw man.

> When they learn, they ask themselves why and it’s a rabbit hole.

No, it's natural human curiosity which should be encouraged, not stomped on. At some point they'll either realize that they're up against material they're not yet ready for, and put it aside for later, or they'll end up being a math prodigy. Both outcomes are better ones than them just being told "this is it, don't ask any questions".

I think what they are saying is that if we can't say that multiplication is repeated addition due to the reason you propose, then addition itself is ill-defined as a concept, which is a bigger problem.

That by using your argument we move the whole debate one level down the fundamental operations and now you no longer have a way to simply explain addition or subtraction.

> not in the elementary school student's understanding of "repeated addition".

Pi isn't particularly well understood by elementary school students either.

Funny, my teachers in middle and high school frowned on using the quadratic formula until we could demonstrate we knew how to complete the square.

Exactly. 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.

> 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.

Multiplication 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.

Multiplication 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.

Its 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).

Of 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.

Multiplying 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.

I definitely wasn't trying to indicate that there was a problem with this, just pointing out that the path of using the Surreals (or anything else) to give an "algorithmic" description of real addition or multiplication is probably a bad idea.

I wasn't even attempting to give an "algorithmic" description. Just a formulaic one, that depends on the axiom of infinity (and accepting transfinite induction as a valid process). Since even at least one of the usual process for constructing the Reals (Dedekind cuts) needs this I don't feel it's much of a stretch in reasoning. And the Surreals have a nicer recursive formula for multiplication that eventually turns into repeated sums, and they're a strict superset of the Reals, so it does apply there.

If you want an algorithmic (no possible need an infinite number of steps) explicit construction of anything on the Reals you're going to be disappointed. They're an infinite set.

You can get algorithmic explicit constructions of (for example) integer addition, rational addition and multiplication and even addition and multiplication of algebraic numbers on a Turing machine (all algebraic numbers are computable). All of these sets are infinite. The reals are particularly "badly behaved" even as far as infinite sets go.

If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.

Not 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?

Multiplication on the naturals is repeated addition.

You might see the natural numbers as training wheels for higher mathematics, but number theorists might disagree...

Number 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.

Number theorists work with complex numbers and irrational numbers too

https://www.britannica.com/science/Riemann-zeta-function

> There will be infinite addition involved, and algorithms are finite.

People are just bit sloppy, and say algorithm when they mean something slightly different.

See https://stackoverflow.com/questions/28841260/what-is-the-dif... and https://en.wikipedia.org/wiki/Corecursion

Basically, you don't want an 'algorithm' here to produce the whole number.

All you need is some scheme that will produce the next digit in finite time (and the next one and the next one etc).

Which is a technical way of saying "in real life people use finite rational or algebraic approximations for reals, so uncountability of reals and infinite precision aren't a problem".

Not quite.

Consider the following real number made of binary digits:

Enumerate all Turing machines and all possible inputs, iff the i-th machine/input combination holds, the i-th binary digit in our number is 0, otherwise 1.

This number is well-defined (once you fix your enumeration scheme).

But there's no finite algorithm to produce approximations in your sense.

What I was after were what's also called Computable numbers (https://en.wikipedia.org/wiki/Computable_number). But I used the more general term of co-recursion, that also applies to arbitrary other data-structures like infinite lists, or with some generalization, infinite event-loops where the important condition is that each run through the body of the loop only takes finite time.

Algorithms can work on symbolic formulas, and symbols can represent anything; infinite objects, operations on infinite objects, infinite sets of operations on infinite objects, and so on.

Algorithms aren’t necessarily finite, eg “while True, print 1”.

Also, an irrational exponent is the product of component factors.

b = Prod(0,inf) a^[x_i * 10^(-i)] = a^x

So even with irrational numbers, operations can be decomposed - such as exponentiation into multiplication of integer exponents and roots.

Which makes sense, because in the sequence definition of reals you need a way to generate the resulting sequence from the two original sequences.

I think you’re trying to claim more than is true.

Algorithms are not finite. E.g. Newton's method.

First 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>

If 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.

> 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.

Not really, since approximations, no matter how accurate, don't preserve algebraic properties. You only get to know what it's bigger/smaller than.

I think I understand what you mean, so let me dial back "anything involving the real numbers".

If you are representing or thinking of "sqrt(2)" as "the positive solution to x^2 = 2", then you preserve algebraic properties. But you generally (correct me if I'm wrong) don't get to know whether it's bigger or smaller than something else of the form "the _choose_uniquely_ solution to _some_equation_" unless you rely on an argument where you invoke approximations.

Well, 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.

And 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?

Nothing 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.

>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.

>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.

To 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.

> 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`.

Interesting, what about tetration then?

Good 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

> 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.

> 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.

Anyone 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.

I never learned to complete the square, and I managed to do well enough later to get a degree in math.

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