181–190 of 425 posts
Proof: The derivative of x^2 is x. x^2 = x * x = x + x ... + x (x times) Therefore, d/dx(x^2) = d/dx(x * x) = d/dx (x + x + ... + x (x times)) = d/dx(x) + d/dx(x) + ... + d/dx(x) (x times) = 1 + 1 + ... + 1 (x times) = x
It's actually 2x...
Earlier quoted context omitted.
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.
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.
The problem with "multiplication is repeated addition" is that the concept breaks down once you move on from integers.
2 X 3: "add 2 together, three times" - works well
½ X ½: oof. You can sorta do it, like "add ½ a half time", but the concept is an impediment that isn't helping anymore.
I can see why teachers who aren't that comfortable with math might be getting confused about this though. Repeated addition is a reasonable algorithm when you've got one operand that is a relatively small integer.
If you spend a long time teaching small integer multiplication you may forget that "You can solve these multiplication problems with repeated addition" does not mean "Multiplication is repeated addition."
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> In all fields and rings, multiplication and addition are related by the distributive property. That's a necessary condition for "repeated addition" to work for multiplication, but not a sufficient one. Try plugging in A = pi to the distributive formula and see how well "repeated addition" works.
pi(3 + 5) == 3pi + 5pi == (Pi + Pi + Pi) + (Pi + Pi + Pi + Pi + Pi) == 8-pi. Am I missing something here? -------- This "multiplication is repeated addition through the distributed property" thing works on freaking __matricies__. They don't even have to be numbers or even related. Pi * ([ 1 0 ; + [ 1 0 ; 0 1 ] 2 1 ]) Pi * [ 1 0 ; + Pi * [ 1 0 ; 0 1 ] 2 1 ] [ Pi 0 ; + [ Pi 0 ; 0 Pi ] 2Pi Pi ] [ 2Pi 0 ; 2Pi 2Pi ] You c…
3 * pi = pi + pi + pi
but how do you represent the other distribution, where 3 is added together pi times?
2 apples + 3 apples = 5 apples
This confuses a lot of people when they later learn algebra and believe that it is equivalent to:
2a + 3a = 5a
Both of these equations look correct at face value but the critical difference is that an apple is not a variable. Do we mean the weight of the apple? Or the price of the apple? Or its energy content? Apple is too concrete to be used as a variable and it's not quantifiable. The word 'apples' invites people to think of the variable as a concrete object instead of a placeholder for a quantifiable property of an object.
Ah yes, there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously. You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be def…
> there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously I don't think his argument (or the similar argument being made in the subject article of this thread) for not teaching students that multiplication is repeated addition is a "pedantic falsehood". I think he has a valid point: that there are downsides…
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…
> 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 pro…
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:
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pi(3 + 5) == 3pi + 5pi == (Pi + Pi + Pi) + (Pi + Pi + Pi + Pi + Pi) == 8-pi. Am I missing something here? -------- This "multiplication is repeated addition through the distributed property" thing works on freaking __matricies__. They don't even have to be numbers or even related. Pi * ([ 1 0 ; + [ 1 0 ; 0 1 ] 2 1 ]) Pi * [ 1 0 ; + Pi * [ 1 0 ; 0 1 ] 2 1 ] [ Pi 0 ; + [ Pi 0 ; 0 Pi ] 2Pi Pi ] [ 2Pi 0 ; 2Pi 2Pi ] You c…
I think the issue is more about: 3 * pi = pi + pi + pi but how do you represent the other distribution, where 3 is added together pi times?
3 * (3 + 0.1 + 0.04 + 0.001 + 0.0005...)
Aka: 9.4245...
You know, how we've been multiplying 3 * pi for our whole lives. We split pi up into an infinite sum of component numbers (3, 1, 4, 1, 5, 9, 2, 6...) and then combine them together by individually multiplying the parts (3 * 3 + 3 * 0.1 + 3 * 0.04...)
Earlier quoted context omitted.
> Multiplication has nothing to do with addition and at some point we have to stop teaching students that it's related. I just finished studying Galois Fields, which LITERALLY redefines multiplication and addition operators to study new forms of math (IE fields in particular) In all fields and rings, multiplication and addition are related by the distributive property. A(b + c) is equal to Ab + Ac, in literally every…
> In all fields and rings, multiplication and addition are related by the distributive property. That's a necessary condition for "repeated addition" to work for multiplication, but not a sufficient one. Try plugging in A = pi to the distributive formula and see how well "repeated addition" works.
Earlier quoted context omitted.
> In all fields and rings, multiplication and addition are related by the distributive property. That's a necessary condition for "repeated addition" to work for multiplication, but not a sufficient one. Try plugging in A = pi to the distributive formula and see how well "repeated addition" works.
pi(3 + 5) == 3pi + 5pi == (Pi + Pi + Pi) + (Pi + Pi + Pi + Pi + Pi) == 8-pi. Am I missing something here? -------- This "multiplication is repeated addition through the distributed property" thing works on freaking __matricies__. They don't even have to be numbers or even related. Pi * ([ 1 0 ; + [ 1 0 ; 0 1 ] 2 1 ]) Pi * [ 1 0 ; + Pi * [ 1 0 ; 0 1 ] 2 1 ] [ Pi 0 ; + [ Pi 0 ; 0 Pi ] 2Pi Pi ] [ 2Pi 0 ; 2Pi 2Pi ] You c…
Sorry, I should have said substitute b = pi or c = pi. Or see the sibling post by iio8999.